src/HOL/Decision_Procs/Reflected_Multivariate_Polynomial.thy
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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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   166
  "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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   169
  (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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   189
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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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   194
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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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   205
  "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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   209
  "polydivide_aux (a,n,p,k,s) = 
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  (if s = 0\<^sub>p then (k,s)
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   211
  else (let b = head s; m = degree s in
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   212
  (if m < n then (k,s) else 
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   213
  (let p'= funpow (m - n) shift1 p in 
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   214
  (if a = b then polydivide_aux (a,n,p,k,s -\<^sub>p p') 
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   215
  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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   221
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   222
fun poly_deriv_aux :: "nat \<Rightarrow> poly \<Rightarrow> poly" where
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   223
  "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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   224
| "poly_deriv_aux n p = poly_cmul ((int n)\<^sub>N) p"
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   225
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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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   229
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   230
  (* Verification *)
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   231
lemma nth_pos2[simp]: "0 < n \<Longrightarrow> (x#xs) ! n = xs ! (n - 1)"
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   232
using Nat.gr0_conv_Suc
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by clarsimp
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   234
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subsection{* Semantics of the polynomial representation *}
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   236
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   237
consts Ipoly :: "'a list \<Rightarrow> poly \<Rightarrow> 'a::{ring_char_0,power,division_by_zero,field}"
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   238
primrec
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   239
  "Ipoly bs (C c) = INum c"
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   240
  "Ipoly bs (Bound n) = bs!n"
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   241
  "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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   243
  "Ipoly bs (Sub a b) = Ipoly bs a - Ipoly bs b"
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   244
  "Ipoly bs (Mul a b) = Ipoly bs a * Ipoly bs b"
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   245
  "Ipoly bs (Pw t n) = (Ipoly bs t) ^ n"
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   246
  "Ipoly bs (CN c n p) = (Ipoly bs c) + (bs!n)*(Ipoly bs p)"
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   247
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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   248
translations "\<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup>" \<rightleftharpoons> "Ipoly bs p"  
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   249
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   250
lemma Ipoly_CInt: "Ipoly bs (C (i,1)) = of_int i" 
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   251
  by (simp add: INum_def)
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   252
lemma Ipoly_CRat: "Ipoly bs (C (i, j)) = of_int i / of_int j" 
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   253
  by (simp  add: INum_def)
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   254
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   255
lemmas RIpoly_eqs = Ipoly.simps(2-7) Ipoly_CInt Ipoly_CRat
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   256
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   257
subsection {* Normal form and normalization *}
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   258
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   259
consts isnpolyh:: "poly \<Rightarrow> nat \<Rightarrow> bool"
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   260
recdef isnpolyh "measure size"
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   261
  "isnpolyh (C c) = (\<lambda>k. isnormNum c)"
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   262
  "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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   263
  "isnpolyh p = (\<lambda>k. False)"
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   264
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   265
lemma isnpolyh_mono: "\<lbrakk>n' \<le> n ; isnpolyh p n\<rbrakk> \<Longrightarrow> isnpolyh p n'"
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   266
by (induct p rule: isnpolyh.induct, auto)
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   267
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   268
constdefs isnpoly:: "poly \<Rightarrow> bool"
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   269
  "isnpoly p \<equiv> isnpolyh p 0"
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   270
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   271
text{* polyadd preserves normal forms *}
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   272
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   273
lemma polyadd_normh: "\<lbrakk>isnpolyh p n0 ; isnpolyh q n1\<rbrakk> 
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   274
      \<Longrightarrow> isnpolyh (polyadd(p,q)) (min n0 n1)"
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proof(induct p q arbitrary: n0 n1 rule: polyadd.induct)
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  case (2 a b c' n' p' n0 n1)
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   277
  from prems have  th1: "isnpolyh (C (a,b)) (Suc n')" by simp 
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   278
  from prems(3) have th2: "isnpolyh c' (Suc n')"  and nplen1: "n' \<ge> n1" by simp_all
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   279
  with isnpolyh_mono have cp: "isnpolyh c' (Suc n')" by simp
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   280
  with prems(1)[OF th1 th2] have th3:"isnpolyh (C (a,b) +\<^sub>p c') (Suc n')" by simp
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   281
  from nplen1 have n01len1: "min n0 n1 \<le> n'" by simp 
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   282
  thus ?case using prems th3 by simp
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   283
next
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   284
  case (3 c' n' p' a b n1 n0)
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   285
  from prems have  th1: "isnpolyh (C (a,b)) (Suc n')" by simp 
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chaieb
parents:
diff changeset
   286
  from prems(2) have th2: "isnpolyh c' (Suc n')"  and nplen1: "n' \<ge> n1" by simp_all
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chaieb
parents:
diff changeset
   287
  with isnpolyh_mono have cp: "isnpolyh c' (Suc n')" by simp
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parents:
diff changeset
   288
  with prems(1)[OF th2 th1] have th3:"isnpolyh (c' +\<^sub>p C (a,b)) (Suc n')" by simp
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parents:
diff changeset
   289
  from nplen1 have n01len1: "min n0 n1 \<le> n'" by simp 
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   290
  thus ?case using prems th3 by simp
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parents:
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   291
next
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   292
  case (4 c n p c' n' p' n0 n1)
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   293
  hence nc: "isnpolyh c (Suc n)" and np: "isnpolyh p n" by simp_all
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diff changeset
   294
  from prems have nc': "isnpolyh c' (Suc n')" and np': "isnpolyh p' n'" by simp_all 
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parents:
diff changeset
   295
  from prems have ngen0: "n \<ge> n0" by simp
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diff changeset
   296
  from prems have n'gen1: "n' \<ge> n1" by simp 
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   297
  have "n < n' \<or> n' < n \<or> n = n'" by auto
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   298
  moreover {assume eq: "n = n'" hence eq': "\<not> n' < n \<and> \<not> n < n'" by simp
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diff changeset
   299
    with prems(2)[rule_format, OF eq' nc nc'] 
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   300
    have ncc':"isnpolyh (c +\<^sub>p c') (Suc n)" by auto
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
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   301
    hence ncc'n01: "isnpolyh (c +\<^sub>p c') (min n0 n1)"
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parents:
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   302
      using isnpolyh_mono[where n'="min n0 n1" and n="Suc n"] ngen0 n'gen1 by auto
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diff changeset
   303
    from eq prems(1)[rule_format, OF eq' np np'] have npp': "isnpolyh (p +\<^sub>p p') n" by simp
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   304
    have minle: "min n0 n1 \<le> n'" using ngen0 n'gen1 eq by simp
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   305
    from minle npp' ncc'n01 prems ngen0 n'gen1 ncc' have ?case by (simp add: Let_def)}
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   306
  moreover {assume lt: "n < n'"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   307
    have "min n0 n1 \<le> n0" by simp
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   308
    with prems have th1:"min n0 n1 \<le> n" by auto 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   309
    from prems have th21: "isnpolyh c (Suc n)" by simp
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   310
    from prems have th22: "isnpolyh (CN c' n' p') n'" by simp
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   311
    from lt have th23: "min (Suc n) n' = Suc n" by arith
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   312
    from prems(4)[rule_format, OF lt th21 th22]
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   313
    have "isnpolyh (polyadd (c, CN c' n' p')) (Suc n)" using th23 by simp
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   314
    with prems th1 have ?case by simp } 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   315
  moreover {assume gt: "n' < n" hence gt': "n' < n \<and> \<not> n < n'" by simp
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   316
    have "min n0 n1 \<le> n1"  by simp
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   317
    with prems have th1:"min n0 n1 \<le> n'" by auto
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   318
    from prems have th21: "isnpolyh c' (Suc n')" by simp_all
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   319
    from prems have th22: "isnpolyh (CN c n p) n" by simp
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   320
    from gt have th23: "min n (Suc n') = Suc n'" by arith
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   321
    from prems(3)[rule_format, OF  gt' th22 th21]
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   322
    have "isnpolyh (polyadd (CN c n p,c')) (Suc n')" using th23 by simp
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   323
    with prems th1 have ?case by simp}
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   324
      ultimately show ?case by blast
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   325
qed auto
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   326
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   327
lemma polyadd[simp]: "Ipoly bs (polyadd (p,q)) = (Ipoly bs p) + (Ipoly bs q)"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   328
by (induct p q rule: polyadd.induct, auto simp add: Let_def ring_simps right_distrib[symmetric] simp del: right_distrib)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   329
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   330
lemma polyadd_norm: "\<lbrakk> isnpoly p ; isnpoly q\<rbrakk> \<Longrightarrow> isnpoly (polyadd(p,q))"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   331
  using polyadd_normh[of "p" "0" "q" "0"] isnpoly_def by simp
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   332
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   333
text{* The degree of addition and other general lemmas needed for the normal form of polymul*}
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   334
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   335
lemma polyadd_different_degreen: 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   336
  "\<lbrakk>isnpolyh p n0 ; isnpolyh q n1; degreen p m \<noteq> degreen q m ; m \<le> min n0 n1\<rbrakk> \<Longrightarrow> 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   337
  degreen (polyadd(p,q)) m = max (degreen p m) (degreen q m)"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   338
proof (induct p q arbitrary: m n0 n1 rule: polyadd.induct)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   339
  case (4 c n p c' n' p' m n0 n1)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   340
  thus ?case 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   341
    apply (cases "n' < n", simp_all add: Let_def)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   342
    apply (cases "n = n'", simp_all)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   343
    apply (cases "n' = m", simp_all add: Let_def)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   344
    by (erule allE[where x="m"], erule allE[where x="Suc m"], 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   345
           erule allE[where x="m"], erule allE[where x="Suc m"], 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   346
           clarsimp,erule allE[where x="m"],erule allE[where x="Suc m"], simp)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   347
qed simp_all 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   348
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   349
lemma headnz[simp]: "\<lbrakk>isnpolyh p n ; p \<noteq> 0\<^sub>p\<rbrakk> \<Longrightarrow> headn p m \<noteq> 0\<^sub>p"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   350
  by (induct p arbitrary: n rule: headn.induct, auto)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   351
lemma degree_isnpolyh_Suc[simp]: "isnpolyh p (Suc n) \<Longrightarrow> degree p = 0"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   352
  by (induct p arbitrary: n rule: degree.induct, auto)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   353
lemma degreen_0[simp]: "isnpolyh p n \<Longrightarrow> m < n \<Longrightarrow> degreen p m = 0"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   354
  by (induct p arbitrary: n rule: degreen.induct, auto)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   355
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   356
lemma degree_isnpolyh_Suc': "n > 0 \<Longrightarrow> isnpolyh p n \<Longrightarrow> degree p = 0"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   357
  by (induct p arbitrary: n rule: degree.induct, auto)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   358
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   359
lemma degree_npolyhCN[simp]: "isnpolyh (CN c n p) n0 \<Longrightarrow> degree c = 0"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   360
  using degree_isnpolyh_Suc by auto
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   361
lemma degreen_npolyhCN[simp]: "isnpolyh (CN c n p) n0 \<Longrightarrow> degreen c n = 0"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   362
  using degreen_0 by auto
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   363
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   364
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   365
lemma degreen_polyadd:
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   366
  assumes np: "isnpolyh p n0" and nq: "isnpolyh q n1" and m: "m \<le> max n0 n1"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   367
  shows "degreen (p +\<^sub>p q) m \<le> max (degreen p m) (degreen q m)"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   368
  using np nq m
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   369
proof (induct p q arbitrary: n0 n1 m rule: polyadd.induct)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   370
  case (2 c c' n' p' n0 n1) thus ?case  by (cases n', simp_all)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   371
next
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   372
  case (3 c n p c' n0 n1) thus ?case by (cases n, auto)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   373
next
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   374
  case (4 c n p c' n' p' n0 n1 m) 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   375
  thus ?case 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   376
    apply (cases "n < n'", simp_all add: Let_def)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   377
    apply (cases "n' < n", simp_all)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   378
    apply (erule allE[where x="n"],erule allE[where x="Suc n"],clarify)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   379
    apply (erule allE[where x="n'"],erule allE[where x="Suc n'"],clarify)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   380
    by (erule allE[where x="m"],erule allE[where x="m"], auto)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   381
qed auto
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   382
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   383
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   384
lemma polyadd_eq_const_degreen: "\<lbrakk> isnpolyh p n0 ; isnpolyh q n1 ; polyadd (p,q) = C c\<rbrakk> 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   385
  \<Longrightarrow> degreen p m = degreen q m"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   386
proof (induct p q arbitrary: m n0 n1 c rule: polyadd.induct)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   387
  case (4 c n p c' n' p' m n0 n1 x) 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   388
  hence z: "CN c n p +\<^sub>p CN c' n' p' = C x" by simp
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   389
  {assume nn': "n' < n" hence ?case using prems by simp}
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   390
  moreover 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   391
  {assume nn':"\<not> n' < n" hence "n < n' \<or> n = n'" by arith
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   392
    moreover {assume "n < n'" with prems have ?case by simp }
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   393
    moreover {assume eq: "n = n'" hence ?case using prems 
33268
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   394
        by (cases "p +\<^sub>p p' = 0\<^sub>p", auto simp add: Let_def) }
33154
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   395
    ultimately have ?case by blast}
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   396
  ultimately show ?case by blast
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   397
qed simp_all
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   398
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   399
lemma polymul_properties:
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   400
  assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   401
  and np: "isnpolyh p n0" and nq: "isnpolyh q n1" and m: "m \<le> min n0 n1"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   402
  shows "isnpolyh (p *\<^sub>p q) (min n0 n1)" 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   403
  and "(p *\<^sub>p q = 0\<^sub>p) = (p = 0\<^sub>p \<or> q = 0\<^sub>p)" 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   404
  and "degreen (p *\<^sub>p q) m = (if (p = 0\<^sub>p \<or> q = 0\<^sub>p) then 0 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   405
                             else degreen p m + degreen q m)"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   406
  using np nq m
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   407
proof(induct p q arbitrary: n0 n1 m rule: polymul.induct)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   408
  case (2 a b c' n' p') 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   409
  let ?c = "(a,b)"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   410
  { case (1 n0 n1) 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   411
    hence n: "isnpolyh (C ?c) n'" "isnpolyh c' (Suc n')" "isnpolyh p' n'" "isnormNum ?c" 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   412
      "isnpolyh (CN c' n' p') n1"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   413
      by simp_all
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   414
    {assume "?c = 0\<^sub>N" hence ?case by auto}
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   415
      moreover {assume cnz: "?c \<noteq> 0\<^sub>N" 
33268
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   416
        from "2.hyps"(1)[rule_format,where xb="n'",  OF cnz n(1) n(3)] 
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   417
          "2.hyps"(2)[rule_format, where x="Suc n'" 
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   418
          and xa="Suc n'" and xb = "n'", OF cnz ] cnz n have ?case
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   419
          by (auto simp add: min_def)}
33154
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   420
      ultimately show ?case by blast
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   421
  next
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   422
    case (2 n0 n1) thus ?case by auto 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   423
  next
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   424
    case (3 n0 n1) thus ?case  using "2.hyps" by auto } 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   425
next
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   426
  case (3 c n p a b){
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   427
    let ?c' = "(a,b)"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   428
    case (1 n0 n1) 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   429
    hence n: "isnpolyh (C ?c') n" "isnpolyh c (Suc n)" "isnpolyh p n" "isnormNum ?c'" 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   430
      "isnpolyh (CN c n p) n0"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   431
      by simp_all
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   432
    {assume "?c' = 0\<^sub>N" hence ?case by auto}
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   433
      moreover {assume cnz: "?c' \<noteq> 0\<^sub>N"
33268
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   434
        from "3.hyps"(1)[rule_format,where xb="n",  OF cnz n(3) n(1)] 
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   435
          "3.hyps"(2)[rule_format, where x="Suc n" 
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   436
          and xa="Suc n" and xb = "n", OF cnz ] cnz n have ?case
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   437
          by (auto simp add: min_def)}
33154
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   438
      ultimately show ?case by blast
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   439
  next
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   440
    case (2 n0 n1) thus ?case apply auto done
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   441
  next
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   442
    case (3 n0 n1) thus ?case  using "3.hyps" by auto } 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   443
next
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   444
  case (4 c n p c' n' p')
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   445
  let ?cnp = "CN c n p" let ?cnp' = "CN c' n' p'"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   446
    {fix n0 n1
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   447
      assume "isnpolyh ?cnp n0" and "isnpolyh ?cnp' n1"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   448
      hence cnp: "isnpolyh ?cnp n" and cnp': "isnpolyh ?cnp' n'"
33268
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   449
        and np: "isnpolyh p n" and nc: "isnpolyh c (Suc n)" 
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   450
        and np': "isnpolyh p' n'" and nc': "isnpolyh c' (Suc n')"
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   451
        and nn0: "n \<ge> n0" and nn1:"n' \<ge> n1"
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   452
        by simp_all
33154
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   453
      have "n < n' \<or> n' < n \<or> n' = n" by auto
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   454
      moreover
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   455
      {assume nn': "n < n'"
33268
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   456
        with "4.hyps"(5)[rule_format, OF nn' np cnp', where xb ="n"] 
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   457
          "4.hyps"(6)[rule_format, OF nn' nc cnp', where xb="n"] nn' nn0 nn1 cnp
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   458
        have "isnpolyh (?cnp *\<^sub>p ?cnp') (min n0 n1)"
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   459
          by (simp add: min_def) }
33154
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   460
      moreover
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   461
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   462
      {assume nn': "n > n'" hence stupid: "n' < n \<and> \<not> n < n'" by arith
33268
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   463
        with "4.hyps"(3)[rule_format, OF stupid cnp np', where xb="n'"]
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   464
          "4.hyps"(4)[rule_format, OF stupid cnp nc', where xb="Suc n'"] 
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   465
          nn' nn0 nn1 cnp'
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   466
        have "isnpolyh (?cnp *\<^sub>p ?cnp') (min n0 n1)"
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   467
          by (cases "Suc n' = n", simp_all add: min_def)}
33154
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   468
      moreover
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   469
      {assume nn': "n' = n" hence stupid: "\<not> n' < n \<and> \<not> n < n'" by arith
33268
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   470
        from "4.hyps"(1)[rule_format, OF stupid cnp np', where xb="n"]
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   471
          "4.hyps"(2)[rule_format, OF stupid cnp nc', where xb="n"] nn' cnp cnp' nn1
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   472
        
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   473
        have "isnpolyh (?cnp *\<^sub>p ?cnp') (min n0 n1)"
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   474
          by simp (rule polyadd_normh,simp_all add: min_def isnpolyh_mono[OF nn0]) }
33154
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   475
      ultimately show "isnpolyh (?cnp *\<^sub>p ?cnp') (min n0 n1)" by blast }
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   476
    note th = this
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   477
    {fix n0 n1 m
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   478
      assume np: "isnpolyh ?cnp n0" and np':"isnpolyh ?cnp' n1"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   479
      and m: "m \<le> min n0 n1"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   480
      let ?d = "degreen (?cnp *\<^sub>p ?cnp') m"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   481
      let ?d1 = "degreen ?cnp m"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   482
      let ?d2 = "degreen ?cnp' m"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   483
      let ?eq = "?d = (if ?cnp = 0\<^sub>p \<or> ?cnp' = 0\<^sub>p then 0  else ?d1 + ?d2)"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   484
      have "n'<n \<or> n < n' \<or> n' = n" by auto
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   485
      moreover 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   486
      {assume "n' < n \<or> n < n'"
33268
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   487
        with "4.hyps" np np' m 
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   488
        have ?eq apply (cases "n' < n", simp_all)
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   489
        apply (erule allE[where x="n"],erule allE[where x="n"],auto) 
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   490
        done }
33154
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   491
      moreover
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   492
      {assume nn': "n' = n"  hence nn:"\<not> n' < n \<and> \<not> n < n'" by arith
33268
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   493
        from "4.hyps"(1)[rule_format, OF nn, where x="n" and xa ="n'" and xb="n"]
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   494
          "4.hyps"(2)[rule_format, OF nn, where x="n" and xa ="Suc n'" and xb="n"] 
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   495
          np np' nn'
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   496
        have norm: "isnpolyh ?cnp n" "isnpolyh c' (Suc n)" "isnpolyh (?cnp *\<^sub>p c') n"
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   497
          "isnpolyh p' n" "isnpolyh (?cnp *\<^sub>p p') n" "isnpolyh (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n"
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   498
          "(?cnp *\<^sub>p c' = 0\<^sub>p) = (c' = 0\<^sub>p)" 
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   499
          "?cnp *\<^sub>p p' \<noteq> 0\<^sub>p" by (auto simp add: min_def)
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   500
        {assume mn: "m = n" 
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   501
          from "4.hyps"(1)[rule_format, OF nn norm(1,4), where xb="n"]
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   502
            "4.hyps"(2)[rule_format, OF nn norm(1,2), where xb="n"] norm nn' mn
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   503
          have degs:  "degreen (?cnp *\<^sub>p c') n = 
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   504
            (if c'=0\<^sub>p then 0 else ?d1 + degreen c' n)"
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   505
            "degreen (?cnp *\<^sub>p p') n = ?d1  + degreen p' n" by (simp_all add: min_def)
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   506
          from degs norm
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   507
          have th1: "degreen(?cnp *\<^sub>p c') n < degreen (CN 0\<^sub>p n (?cnp *\<^sub>p p')) n" by simp
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   508
          hence neq: "degreen (?cnp *\<^sub>p c') n \<noteq> degreen (CN 0\<^sub>p n (?cnp *\<^sub>p p')) n"
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   509
            by simp
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   510
          have nmin: "n \<le> min n n" by (simp add: min_def)
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   511
          from polyadd_different_degreen[OF norm(3,6) neq nmin] th1
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   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 
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   513
          from "4.hyps"(1)[rule_format, OF nn norm(1,4), where xb="n"]
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   514
            "4.hyps"(2)[rule_format, OF nn norm(1,2), where xb="n"]
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   515
            mn norm m nn' deg
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   516
          have ?eq by simp}
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   517
        moreover
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   518
        {assume mn: "m \<noteq> n" hence mn': "m < n" using m np by auto
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   519
          from nn' m np have max1: "m \<le> max n n"  by simp 
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   520
          hence min1: "m \<le> min n n" by simp     
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   521
          hence min2: "m \<le> min n (Suc n)" by simp
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   522
          {assume "c' = 0\<^sub>p"
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   523
            from `c' = 0\<^sub>p` have ?eq
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   524
              using "4.hyps"(1)[rule_format, OF nn norm(1,4) min1]
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   525
            "4.hyps"(2)[rule_format, OF nn norm(1,2) min2] mn nn'
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   526
              apply simp
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   527
              done}
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   528
          moreover
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   529
          {assume cnz: "c' \<noteq> 0\<^sub>p"
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   530
            from "4.hyps"(1)[rule_format, OF nn norm(1,4) min1]
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   531
              "4.hyps"(2)[rule_format, OF nn norm(1,2) min2]
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   532
              degreen_polyadd[OF norm(3,6) max1]
33154
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   533
33268
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   534
            have "degreen (?cnp *\<^sub>p c' +\<^sub>p CN 0\<^sub>p n (?cnp *\<^sub>p p')) m 
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   535
              \<le> max (degreen (?cnp *\<^sub>p c') m) (degreen (CN 0\<^sub>p n (?cnp *\<^sub>p p')) m)"
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   536
              using mn nn' cnz np np' by simp
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   537
            with "4.hyps"(1)[rule_format, OF nn norm(1,4) min1]
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   538
              "4.hyps"(2)[rule_format, OF nn norm(1,2) min2]
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   539
              degreen_0[OF norm(3) mn'] have ?eq using nn' mn cnz np np' by clarsimp}
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   540
          ultimately have ?eq by blast }
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   541
        ultimately have ?eq by blast}
33154
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   542
      ultimately show ?eq by blast}
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   543
    note degth = this
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   544
    { case (2 n0 n1)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   545
      hence np: "isnpolyh ?cnp n0" and np': "isnpolyh ?cnp' n1" 
33268
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   546
        and m: "m \<le> min n0 n1" by simp_all
33154
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   547
      hence mn: "m \<le> n" by simp
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   548
      let ?c0p = "CN 0\<^sub>p n (?cnp *\<^sub>p p')"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   549
      {assume C: "?cnp *\<^sub>p c' +\<^sub>p ?c0p = 0\<^sub>p" "n' = n"
33268
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   550
        hence nn: "\<not>n' < n \<and> \<not> n<n'" by simp
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   551
        from "4.hyps"(1) [rule_format, OF nn, where x="n" and xa = "n" and xb="n"] 
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   552
          "4.hyps"(2) [rule_format, OF nn, where x="n" and xa = "Suc n" and xb="n"] 
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   553
          np np' C(2) mn
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   554
        have norm: "isnpolyh ?cnp n" "isnpolyh c' (Suc n)" "isnpolyh (?cnp *\<^sub>p c') n"
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   555
          "isnpolyh p' n" "isnpolyh (?cnp *\<^sub>p p') n" "isnpolyh (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n"
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   556
          "(?cnp *\<^sub>p c' = 0\<^sub>p) = (c' = 0\<^sub>p)" 
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   557
          "?cnp *\<^sub>p p' \<noteq> 0\<^sub>p" 
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   558
          "degreen (?cnp *\<^sub>p c') n = (if c'=0\<^sub>p then 0 else degreen ?cnp n + degreen c' n)"
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   559
            "degreen (?cnp *\<^sub>p p') n = degreen ?cnp n + degreen p' n"
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   560
          by (simp_all add: min_def)
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   561
            
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   562
          from norm have cn: "isnpolyh (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n" by simp
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   563
          have degneq: "degreen (?cnp *\<^sub>p c') n < degreen (CN 0\<^sub>p n (?cnp *\<^sub>p p')) n" 
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   564
            using norm by simp
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   565
        from polyadd_eq_const_degreen[OF norm(3) cn C(1), where m="n"]  degneq
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
   566
        have "False" by simp }
33154
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   567
      thus ?case using "4.hyps" by clarsimp}
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   568
qed auto
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   569
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   570
lemma polymul[simp]: "Ipoly bs (p *\<^sub>p q) = (Ipoly bs p) * (Ipoly bs q)"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   571
by(induct p q rule: polymul.induct, auto simp add: ring_simps)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   572
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   573
lemma polymul_normh: 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   574
    assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   575
  shows "\<lbrakk>isnpolyh p n0 ; isnpolyh q n1\<rbrakk> \<Longrightarrow> isnpolyh (p *\<^sub>p q) (min n0 n1)"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   576
  using polymul_properties(1)  by blast
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   577
lemma polymul_eq0_iff: 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   578
  assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   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) "
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   580
  using polymul_properties(2)  by blast
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   581
lemma polymul_degreen:  
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   582
  assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   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)"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   584
  using polymul_properties(3) by blast
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   585
lemma polymul_norm:   
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   586
  assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   587
  shows "\<lbrakk> isnpoly p; isnpoly q\<rbrakk> \<Longrightarrow> isnpoly (polymul (p,q))"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   588
  using polymul_normh[of "p" "0" "q" "0"] isnpoly_def by simp
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   589
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   590
lemma headconst_zero: "isnpolyh p n0 \<Longrightarrow> headconst p = 0\<^sub>N \<longleftrightarrow> p = 0\<^sub>p"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   591
  by (induct p arbitrary: n0 rule: headconst.induct, auto)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   592
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   593
lemma headconst_isnormNum: "isnpolyh p n0 \<Longrightarrow> isnormNum (headconst p)"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   594
  by (induct p arbitrary: n0, auto)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   595
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   596
lemma monic_eqI: assumes np: "isnpolyh p n0" 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   597
  shows "INum (headconst p) * Ipoly bs (fst (monic p)) = (Ipoly bs p ::'a::{ring_char_0,power,division_by_zero,field})"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   598
  unfolding monic_def Let_def
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   599
proof(cases "headconst p = 0\<^sub>N", simp_all add: headconst_zero[OF np])
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   600
  let ?h = "headconst p"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   601
  assume pz: "p \<noteq> 0\<^sub>p"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   602
  {assume hz: "INum ?h = (0::'a)"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   603
    from headconst_isnormNum[OF np] have norm: "isnormNum ?h" "isnormNum 0\<^sub>N" by simp_all
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   604
    from isnormNum_unique[where ?'a = 'a, OF norm] hz have "?h = 0\<^sub>N" by simp
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   605
    with headconst_zero[OF np] have "p =0\<^sub>p" by blast with pz have "False" by blast}
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   606
  thus "INum (headconst p) = (0::'a) \<longrightarrow> \<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> = 0" by blast
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   607
qed
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   608
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   609
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   610
 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   611
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   612
text{* polyneg is a negation and preserves normal form *}
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   613
lemma polyneg[simp]: "Ipoly bs (polyneg p) = - Ipoly bs p"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   614
by (induct p rule: polyneg.induct, auto)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   615
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   616
lemma polyneg0: "isnpolyh p n \<Longrightarrow> ((~\<^sub>p p) = 0\<^sub>p) = (p = 0\<^sub>p)"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   617
  by (induct p arbitrary: n rule: polyneg.induct, auto simp add: Nneg_def)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   618
lemma polyneg_polyneg: "isnpolyh p n0 \<Longrightarrow> ~\<^sub>p (~\<^sub>p p) = p"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   619
  by (induct p arbitrary: n0 rule: polyneg.induct, auto)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   620
lemma polyneg_normh: "\<And>n. isnpolyh p n \<Longrightarrow> isnpolyh (polyneg p) n "
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   621
by (induct p rule: polyneg.induct, auto simp add: polyneg0)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   622
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   623
lemma polyneg_norm: "isnpoly p \<Longrightarrow> isnpoly (polyneg p)"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   624
  using isnpoly_def polyneg_normh by simp
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   625
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   626
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   627
text{* polysub is a substraction and preserves normalform *}
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   628
lemma polysub[simp]: "Ipoly bs (polysub (p,q)) = (Ipoly bs p) - (Ipoly bs q)"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   629
by (simp add: polysub_def polyneg polyadd)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   630
lemma polysub_normh: "\<And> n0 n1. \<lbrakk> isnpolyh p n0 ; isnpolyh q n1\<rbrakk> \<Longrightarrow> isnpolyh (polysub(p,q)) (min n0 n1)"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   631
by (simp add: polysub_def polyneg_normh polyadd_normh)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   632
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   633
lemma polysub_norm: "\<lbrakk> isnpoly p; isnpoly q\<rbrakk> \<Longrightarrow> isnpoly (polysub(p,q))"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   634
  using polyadd_norm polyneg_norm by (simp add: polysub_def) 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   635
lemma polysub_same_0[simp]:   assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   636
  shows "isnpolyh p n0 \<Longrightarrow> polysub (p, p) = 0\<^sub>p"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   637
unfolding polysub_def split_def fst_conv snd_conv
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   638
by (induct p arbitrary: n0,auto simp add: Let_def Nsub0[simplified Nsub_def])
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   639
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   640
lemma polysub_0: 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   641
  assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   642
  shows "\<lbrakk> isnpolyh p n0 ; isnpolyh q n1\<rbrakk> \<Longrightarrow> (p -\<^sub>p q = 0\<^sub>p) = (p = q)"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   643
  unfolding polysub_def split_def fst_conv snd_conv
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   644
  apply (induct p q arbitrary: n0 n1 rule:polyadd.induct, simp_all add: Nsub0[simplified Nsub_def])
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   645
  apply (clarsimp simp add: Let_def)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   646
  apply (case_tac "n < n'", simp_all)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   647
  apply (case_tac "n' < n", simp_all)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   648
  apply (erule impE)+
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   649
  apply (rule_tac x="Suc n" in exI, simp)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   650
  apply (rule_tac x="n" in exI, simp)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   651
  apply (erule impE)+
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   652
  apply (rule_tac x="n" in exI, simp)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   653
  apply (rule_tac x="Suc n" in exI, simp)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   654
  apply (erule impE)+
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   655
  apply (rule_tac x="Suc n" in exI, simp)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   656
  apply (rule_tac x="n" in exI, simp)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   657
  apply (erule impE)+
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   658
  apply (rule_tac x="Suc n" in exI, simp)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   659
  apply clarsimp
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   660
  done
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   661
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   662
text{* polypow is a power function and preserves normal forms *}
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   663
lemma polypow[simp]: "Ipoly bs (polypow n p) = ((Ipoly bs p :: 'a::{ring_char_0,division_by_zero,field})) ^ n"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   664
proof(induct n rule: polypow.induct)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   665
  case 1 thus ?case by simp
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   666
next
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   667
  case (2 n)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   668
  let ?q = "polypow ((Suc n) div 2) p"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   669
  let ?d = "polymul(?q,?q)"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   670
  have "odd (Suc n) \<or> even (Suc n)" by simp
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   671
  moreover 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   672
  {assume odd: "odd (Suc n)"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   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
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   674
    from odd have "Ipoly bs (p ^\<^sub>p Suc n) = Ipoly bs (polymul(p, ?d))" by (simp add: Let_def)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   675
    also have "\<dots> = (Ipoly bs p) * (Ipoly bs p)^(Suc n div 2)*(Ipoly bs p)^(Suc n div 2)"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   676
      using "2.hyps" by simp
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   677
    also have "\<dots> = (Ipoly bs p) ^ (Suc n div 2 + Suc n div 2 + 1)"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   678
      apply (simp only: power_add power_one_right) by simp
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   679
    also have "\<dots> = (Ipoly bs p) ^ (Suc (Suc (Suc (0\<Colon>nat)) * (Suc n div Suc (Suc (0\<Colon>nat)))))"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   680
      by (simp only: th)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   681
    finally have ?case 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   682
    using odd_nat_div_two_times_two_plus_one[OF odd, symmetric] by simp  }
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   683
  moreover 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   684
  {assume even: "even (Suc n)"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   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
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   686
    from even have "Ipoly bs (p ^\<^sub>p Suc n) = Ipoly bs ?d" by (simp add: Let_def)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   687
    also have "\<dots> = (Ipoly bs p) ^ (Suc n div 2 + Suc n div 2)"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   688
      using "2.hyps" apply (simp only: power_add) by simp
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   689
    finally have ?case using even_nat_div_two_times_two[OF even] by (simp only: th)}
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   690
  ultimately show ?case by blast
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   691
qed
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   692
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   693
lemma polypow_normh: 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   694
    assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   695
  shows "isnpolyh p n \<Longrightarrow> isnpolyh (polypow k p) n"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   696
proof (induct k arbitrary: n rule: polypow.induct)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   697
  case (2 k n)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   698
  let ?q = "polypow (Suc k div 2) p"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   699
  let ?d = "polymul (?q,?q)"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   700
  from prems have th1:"isnpolyh ?q n" and th2: "isnpolyh p n" by blast+
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   701
  from polymul_normh[OF th1 th1] have dn: "isnpolyh ?d n" by simp
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   702
  from polymul_normh[OF th2 dn] have on: "isnpolyh (polymul(p,?d)) n" by simp
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   703
  from dn on show ?case by (simp add: Let_def)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   704
qed auto 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   705
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   706
lemma polypow_norm:   
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   707
  assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   708
  shows "isnpoly p \<Longrightarrow> isnpoly (polypow k p)"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   709
  by (simp add: polypow_normh isnpoly_def)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   710
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   711
text{* Finally the whole normalization*}
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   712
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   713
lemma polynate[simp]: "Ipoly bs (polynate p) = (Ipoly bs p :: 'a ::{ring_char_0,division_by_zero,field})"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   714
by (induct p rule:polynate.induct, auto)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   715
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   716
lemma polynate_norm[simp]: 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   717
  assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   718
  shows "isnpoly (polynate p)"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   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)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   720
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   721
text{* shift1 *}
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   722
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   723
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   724
lemma shift1: "Ipoly bs (shift1 p) = Ipoly bs (Mul (Bound 0) p)"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   725
by (simp add: shift1_def polymul)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   726
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   727
lemma shift1_isnpoly: 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   728
  assumes pn: "isnpoly p" and pnz: "p \<noteq> 0\<^sub>p" shows "isnpoly (shift1 p) "
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   729
  using pn pnz by (simp add: shift1_def isnpoly_def )
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   730
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   731
lemma shift1_nz[simp]:"shift1 p \<noteq> 0\<^sub>p"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   732
  by (simp add: shift1_def)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   733
lemma funpow_shift1_isnpoly: 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   734
  "\<lbrakk> isnpoly p ; p \<noteq> 0\<^sub>p\<rbrakk> \<Longrightarrow> isnpoly (funpow n shift1 p)"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   735
  by (induct n arbitrary: p, auto simp add: shift1_isnpoly)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   736
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   737
lemma funpow_isnpolyh: 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   738
  assumes f: "\<And> p. isnpolyh p n \<Longrightarrow> isnpolyh (f p) n "and np: "isnpolyh p n"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   739
  shows "isnpolyh (funpow k f p) n"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   740
  using f np by (induct k arbitrary: p, auto)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   741
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   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)"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   743
  by (induct n arbitrary: p, simp_all add: shift1_isnpoly shift1 power_Suc )
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   744
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   745
lemma shift1_isnpolyh: "isnpolyh p n0 \<Longrightarrow> p\<noteq> 0\<^sub>p \<Longrightarrow> isnpolyh (shift1 p) 0"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   746
  using isnpolyh_mono[where n="n0" and n'="0" and p="p"] by (simp add: shift1_def)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   747
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   748
lemma funpow_shift1_1: 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   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)"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   750
  by (simp add: funpow_shift1)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   751
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   752
lemma poly_cmul[simp]: "Ipoly bs (poly_cmul c p) = Ipoly bs (Mul (C c) p)"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   753
by (induct p  arbitrary: n0 rule: poly_cmul.induct, auto simp add: ring_simps)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   754
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   755
lemma behead:
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   756
  assumes np: "isnpolyh p n"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   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})"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   758
  using np
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   759
proof (induct p arbitrary: n rule: behead.induct)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   760
  case (1 c p n) hence pn: "isnpolyh p n" by simp
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   761
  from prems(2)[OF pn] 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   762
  have th:"Ipoly bs (Add (Mul (head p) (Pw (Bound 0) (degree p))) (behead p)) = Ipoly bs p" . 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   763
  then show ?case using "1.hyps" apply (simp add: Let_def,cases "behead p = 0\<^sub>p")
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   764
    by (simp_all add: th[symmetric] ring_simps power_Suc)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   765
qed (auto simp add: Let_def)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   766
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   767
lemma behead_isnpolyh:
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   768
  assumes np: "isnpolyh p n" shows "isnpolyh (behead p) n"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   769
  using np by (induct p rule: behead.induct, auto simp add: Let_def isnpolyh_mono)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   770
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   771
subsection{* Miscilanious lemmas about indexes, decrementation, substitution  etc ... *}
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   772
lemma isnpolyh_polybound0: "isnpolyh p (Suc n) \<Longrightarrow> polybound0 p"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   773
proof(induct p arbitrary: n rule: polybound0.induct, auto)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   774
  case (goal1 c n p n')
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   775
  hence "n = Suc (n - 1)" by simp
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   776
  hence "isnpolyh p (Suc (n - 1))"  using `isnpolyh p n` by simp
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   777
  with prems(2) show ?case by simp
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   778
qed
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   779
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   780
lemma isconstant_polybound0: "isnpolyh p n0 \<Longrightarrow> isconstant p \<longleftrightarrow> polybound0 p"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   781
by (induct p arbitrary: n0 rule: isconstant.induct, auto simp add: isnpolyh_polybound0)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   782
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   783
lemma decrpoly_zero[simp]: "decrpoly p = 0\<^sub>p \<longleftrightarrow> p = 0\<^sub>p" by (induct p, auto)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   784
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   785
lemma decrpoly_normh: "isnpolyh p n0 \<Longrightarrow> polybound0 p \<Longrightarrow> isnpolyh (decrpoly p) (n0 - 1)"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   786
  apply (induct p arbitrary: n0, auto)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   787
  apply (atomize)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   788
  apply (erule_tac x = "Suc nat" in allE)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   789
  apply auto
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   790
  done
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   791
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   792
lemma head_polybound0: "isnpolyh p n0 \<Longrightarrow> polybound0 (head p)"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   793
 by (induct p  arbitrary: n0 rule: head.induct, auto intro: isnpolyh_polybound0)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   794
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   795
lemma polybound0_I:
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   796
  assumes nb: "polybound0 a"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   797
  shows "Ipoly (b#bs) a = Ipoly (b'#bs) a"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   798
using nb
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   799
by (induct a rule: polybound0.induct) auto 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   800
lemma polysubst0_I:
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   801
  shows "Ipoly (b#bs) (polysubst0 a t) = Ipoly ((Ipoly (b#bs) a)#bs) t"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   802
  by (induct t) simp_all
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   803
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   804
lemma polysubst0_I':
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   805
  assumes nb: "polybound0 a"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   806
  shows "Ipoly (b#bs) (polysubst0 a t) = Ipoly ((Ipoly (b'#bs) a)#bs) t"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   807
  by (induct t) (simp_all add: polybound0_I[OF nb, where b="b" and b'="b'"])
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   808
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   809
lemma decrpoly: assumes nb: "polybound0 t"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   810
  shows "Ipoly (x#bs) t = Ipoly bs (decrpoly t)"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   811
  using nb by (induct t rule: decrpoly.induct, simp_all)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   812
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   813
lemma polysubst0_polybound0: assumes nb: "polybound0 t"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   814
  shows "polybound0 (polysubst0 t a)"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   815
using nb by (induct a rule: polysubst0.induct, auto)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   816
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   817
lemma degree0_polybound0: "isnpolyh p n \<Longrightarrow> degree p = 0 \<Longrightarrow> polybound0 p"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   818
  by (induct p arbitrary: n rule: degree.induct, auto simp add: isnpolyh_polybound0)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   819
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   820
fun maxindex :: "poly \<Rightarrow> nat" where
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   821
  "maxindex (Bound n) = n + 1"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   822
| "maxindex (CN c n p) = max  (n + 1) (max (maxindex c) (maxindex p))"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   823
| "maxindex (Add p q) = max (maxindex p) (maxindex q)"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   824
| "maxindex (Sub p q) = max (maxindex p) (maxindex q)"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   825
| "maxindex (Mul p q) = max (maxindex p) (maxindex q)"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   826
| "maxindex (Neg p) = maxindex p"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   827
| "maxindex (Pw p n) = maxindex p"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   828
| "maxindex (C x) = 0"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   829
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   830
definition wf_bs :: "'a list \<Rightarrow> poly \<Rightarrow> bool" where
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   831
  "wf_bs bs p = (length bs \<ge> maxindex p)"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   832
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   833
lemma wf_bs_coefficients: "wf_bs bs p \<Longrightarrow> \<forall> c \<in> set (coefficients p). wf_bs bs c"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   834
proof(induct p rule: coefficients.induct)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   835
  case (1 c p) 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   836
  show ?case 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   837
  proof
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   838
    fix x assume xc: "x \<in> set (coefficients (CN c 0 p))"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   839
    hence "x = c \<or> x \<in> set (coefficients p)" by simp
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   840
    moreover 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   841
    {assume "x = c" hence "wf_bs bs x" using "1.prems"  unfolding wf_bs_def by simp}
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   842
    moreover 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   843
    {assume H: "x \<in> set (coefficients p)" 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   844
      from "1.prems" have "wf_bs bs p" unfolding wf_bs_def by simp
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   845
      with "1.hyps" H have "wf_bs bs x" by blast }
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   846
    ultimately  show "wf_bs bs x" by blast
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   847
  qed
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   848
qed simp_all
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   849
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   850
lemma maxindex_coefficients: " \<forall>c\<in> set (coefficients p). maxindex c \<le> maxindex p"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   851
by (induct p rule: coefficients.induct, auto)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   852
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   853
lemma length_exists: "\<exists>xs. length xs = n" by (rule exI[where x="replicate n x"], simp)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   854
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   855
lemma wf_bs_I: "wf_bs bs p ==> Ipoly (bs@bs') p = Ipoly bs p"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   856
  unfolding wf_bs_def by (induct p, auto simp add: nth_append)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   857
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   858
lemma take_maxindex_wf: assumes wf: "wf_bs bs p" 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   859
  shows "Ipoly (take (maxindex p) bs) p = Ipoly bs p"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   860
proof-
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   861
  let ?ip = "maxindex p"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   862
  let ?tbs = "take ?ip bs"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   863
  from wf have "length ?tbs = ?ip" unfolding wf_bs_def by simp
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   864
  hence wf': "wf_bs ?tbs p" unfolding wf_bs_def by  simp
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   865
  have eq: "bs = ?tbs @ (drop ?ip bs)" by simp
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   866
  from wf_bs_I[OF wf', of "drop ?ip bs"] show ?thesis using eq by simp
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   867
qed
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   868
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   869
lemma decr_maxindex: "polybound0 p \<Longrightarrow> maxindex (decrpoly p) = maxindex p - 1"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   870
  by (induct p, auto)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   871
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   872
lemma wf_bs_insensitive: "length bs = length bs' \<Longrightarrow> wf_bs bs p = wf_bs bs' p"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   873
  unfolding wf_bs_def by simp
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   874
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   875
lemma wf_bs_insensitive': "wf_bs (x#bs) p = wf_bs (y#bs) p"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   876
  unfolding wf_bs_def by simp
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   877
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   878
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   879
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   880
lemma wf_bs_coefficients': "\<forall>c \<in> set (coefficients p). wf_bs bs c \<Longrightarrow> wf_bs (x#bs) p"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   881
by(induct p rule: coefficients.induct, auto simp add: wf_bs_def)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   882
lemma coefficients_Nil[simp]: "coefficients p \<noteq> []"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   883
  by (induct p rule: coefficients.induct, simp_all)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   884
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   885
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   886
lemma coefficients_head: "last (coefficients p) = head p"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   887
  by (induct p rule: coefficients.induct, auto)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   888
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   889
lemma wf_bs_decrpoly: "wf_bs bs (decrpoly p) \<Longrightarrow> wf_bs (x#bs) p"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   890
  unfolding wf_bs_def by (induct p rule: decrpoly.induct, auto)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   891
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   892
lemma length_le_list_ex: "length xs \<le> n \<Longrightarrow> \<exists> ys. length (xs @ ys) = n"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   893
  apply (rule exI[where x="replicate (n - length xs) z"])
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   894
  by simp
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   895
lemma isnpolyh_Suc_const:"isnpolyh p (Suc n) \<Longrightarrow> isconstant p"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   896
by (cases p, auto) (case_tac "nat", simp_all)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   897
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   898
lemma wf_bs_polyadd: "wf_bs bs p \<and> wf_bs bs q \<longrightarrow> wf_bs bs (p +\<^sub>p q)"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   899
  unfolding wf_bs_def 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   900
  apply (induct p q rule: polyadd.induct)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   901
  apply (auto simp add: Let_def)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   902
  done
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   903
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   904
lemma wf_bs_polyul: "wf_bs bs p \<Longrightarrow> wf_bs bs q \<Longrightarrow> wf_bs bs (p *\<^sub>p q)"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   905
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   906
 unfolding wf_bs_def 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   907
  apply (induct p q arbitrary: bs rule: polymul.induct) 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   908
  apply (simp_all add: wf_bs_polyadd)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   909
  apply clarsimp
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   910
  apply (rule wf_bs_polyadd[unfolded wf_bs_def, rule_format])
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   911
  apply auto
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   912
  done
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   913
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   914
lemma wf_bs_polyneg: "wf_bs bs p \<Longrightarrow> wf_bs bs (~\<^sub>p p)"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   915
  unfolding wf_bs_def by (induct p rule: polyneg.induct, auto)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   916
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   917
lemma wf_bs_polysub: "wf_bs bs p \<Longrightarrow> wf_bs bs q \<Longrightarrow> wf_bs bs (p -\<^sub>p q)"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   918
  unfolding polysub_def split_def fst_conv snd_conv using wf_bs_polyadd wf_bs_polyneg by blast
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   919
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   920
subsection{* Canonicity of polynomial representation, see lemma isnpolyh_unique*}
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   921
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   922
definition "polypoly bs p = map (Ipoly bs) (coefficients p)"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   923
definition "polypoly' bs p = map ((Ipoly bs o decrpoly)) (coefficients p)"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   924
definition "poly_nate bs p = map ((Ipoly bs o decrpoly)) (coefficients (polynate p))"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   925
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   926
lemma coefficients_normh: "isnpolyh p n0 \<Longrightarrow> \<forall> q \<in> set (coefficients p). isnpolyh q n0"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   927
proof (induct p arbitrary: n0 rule: coefficients.induct)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   928
  case (1 c p n0)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   929
  have cp: "isnpolyh (CN c 0 p) n0" by fact
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   930
  hence norm: "isnpolyh c 0" "isnpolyh p 0" "p \<noteq> 0\<^sub>p" "n0 = 0"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   931
    by (auto simp add: isnpolyh_mono[where n'=0])
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   932
  from "1.hyps"[OF norm(2)] norm(1) norm(4)  show ?case by simp 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   933
qed auto
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   934
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   935
lemma coefficients_isconst:
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   936
  "isnpolyh p n \<Longrightarrow> \<forall>q\<in>set (coefficients p). isconstant q"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   937
  by (induct p arbitrary: n rule: coefficients.induct, 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   938
    auto simp add: isnpolyh_Suc_const)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   939
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   940
lemma polypoly_polypoly':
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   941
  assumes np: "isnpolyh p n0"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   942
  shows "polypoly (x#bs) p = polypoly' bs p"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   943
proof-
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   944
  let ?cf = "set (coefficients p)"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   945
  from coefficients_normh[OF np] have cn_norm: "\<forall> q\<in> ?cf. isnpolyh q n0" .
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   946
  {fix q assume q: "q \<in> ?cf"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   947
    from q cn_norm have th: "isnpolyh q n0" by blast
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   948
    from coefficients_isconst[OF np] q have "isconstant q" by blast
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   949
    with isconstant_polybound0[OF th] have "polybound0 q" by blast}
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   950
  hence "\<forall>q \<in> ?cf. polybound0 q" ..
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   951
  hence "\<forall>q \<in> ?cf. Ipoly (x#bs) q = Ipoly bs (decrpoly q)"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   952
    using polybound0_I[where b=x and bs=bs and b'=y] decrpoly[where x=x and bs=bs]
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   953
    by auto
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   954
  
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   955
  thus ?thesis unfolding polypoly_def polypoly'_def by simp 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   956
qed
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   957
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   958
lemma polypoly_poly:
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   959
  assumes np: "isnpolyh p n0" shows "Ipoly (x#bs) p = poly (polypoly (x#bs) p) x"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   960
  using np 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   961
by (induct p arbitrary: n0 bs rule: coefficients.induct, auto simp add: polypoly_def)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   962
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   963
lemma polypoly'_poly: 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   964
  assumes np: "isnpolyh p n0" shows "\<lparr>p\<rparr>\<^sub>p\<^bsup>x # bs\<^esup> = poly (polypoly' bs p) x"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   965
  using polypoly_poly[OF np, simplified polypoly_polypoly'[OF np]] .
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   966
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   967
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   968
lemma polypoly_poly_polybound0:
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   969
  assumes np: "isnpolyh p n0" and nb: "polybound0 p"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   970
  shows "polypoly bs p = [Ipoly bs p]"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   971
  using np nb unfolding polypoly_def 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   972
  by (cases p, auto, case_tac nat, auto)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   973
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   974
lemma head_isnpolyh: "isnpolyh p n0 \<Longrightarrow> isnpolyh (head p) n0" 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   975
  by (induct p rule: head.induct, auto)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   976
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   977
lemma headn_nz[simp]: "isnpolyh p n0 \<Longrightarrow> (headn p m = 0\<^sub>p) = (p = 0\<^sub>p)"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   978
  by (cases p,auto)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   979
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   980
lemma head_eq_headn0: "head p = headn p 0"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   981
  by (induct p rule: head.induct, simp_all)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   982
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   983
lemma head_nz[simp]: "isnpolyh p n0 \<Longrightarrow> (head p = 0\<^sub>p) = (p = 0\<^sub>p)"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   984
  by (simp add: head_eq_headn0)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   985
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   986
lemma isnpolyh_zero_iff: 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   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})"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   988
  shows "p = 0\<^sub>p"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   989
using nq eq
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   990
proof (induct n\<equiv>"maxindex p" arbitrary: p n0 rule: nat_less_induct)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   991
  fix n p n0
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   992
  assume H: "\<forall>m<n. \<forall>p n0. isnpolyh p n0 \<longrightarrow>
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   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"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   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"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   995
  {assume nz: "n = 0"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   996
    then obtain c where "p = C c" using n np by (cases p, auto)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   997
    with zp np have "p = 0\<^sub>p" unfolding wf_bs_def by simp}
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   998
  moreover
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
   999
  {assume nz: "n \<noteq> 0"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1000
    let ?h = "head p"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1001
    let ?hd = "decrpoly ?h"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1002
    let ?ihd = "maxindex ?hd"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1003
    from head_isnpolyh[OF np] head_polybound0[OF np] have h:"isnpolyh ?h n0" "polybound0 ?h" 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1004
      by simp_all
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1005
    hence nhd: "isnpolyh ?hd (n0 - 1)" using decrpoly_normh by blast
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1006
    
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1007
    from maxindex_coefficients[of p] coefficients_head[of p, symmetric]
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1008
    have mihn: "maxindex ?h \<le> n" unfolding n by auto
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1009
    with decr_maxindex[OF h(2)] nz  have ihd_lt_n: "?ihd < n" by auto
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1010
    {fix bs:: "'a list"  assume bs: "wf_bs bs ?hd"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1011
      let ?ts = "take ?ihd bs"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1012
      let ?rs = "drop ?ihd bs"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1013
      have ts: "wf_bs ?ts ?hd" using bs unfolding wf_bs_def by simp
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1014
      have bs_ts_eq: "?ts@ ?rs = bs" by simp
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1015
      from wf_bs_decrpoly[OF ts] have tsh: " \<forall>x. wf_bs (x#?ts) ?h" by simp
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1016
      from ihd_lt_n have "ALL x. length (x#?ts) \<le> n" by simp
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1017
      with length_le_list_ex obtain xs where xs:"length ((x#?ts) @ xs) = n" by blast
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1018
      hence "\<forall> x. wf_bs ((x#?ts) @ xs) p" using n unfolding wf_bs_def by simp
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1019
      with zp have "\<forall> x. Ipoly ((x#?ts) @ xs) p = 0" by blast
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1020
      hence "\<forall> x. Ipoly (x#(?ts @ xs)) p = 0" by simp
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1021
      with polypoly_poly[OF np, where ?'a = 'a] polypoly_polypoly'[OF np, where ?'a = 'a]
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1022
      have "\<forall>x. poly (polypoly' (?ts @ xs) p) x = poly [] x"  by simp
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1023
      hence "poly (polypoly' (?ts @ xs) p) = poly []" by (auto intro: ext) 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1024
      hence "\<forall> c \<in> set (coefficients p). Ipoly (?ts @ xs) (decrpoly c) = 0"
33268
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
  1025
        thm poly_zero
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
  1026
        using poly_zero[where ?'a='a] by (simp add: polypoly'_def list_all_iff)
33154
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1027
      with coefficients_head[of p, symmetric]
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1028
      have th0: "Ipoly (?ts @ xs) ?hd = 0" by simp
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1029
      from bs have wf'': "wf_bs ?ts ?hd" unfolding wf_bs_def by simp
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1030
      with th0 wf_bs_I[of ?ts ?hd xs] have "Ipoly ?ts ?hd = 0" by simp
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  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 }
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1032
    then have hdz: "\<forall>bs. wf_bs bs ?hd \<longrightarrow> \<lparr>?hd\<rparr>\<^sub>p\<^bsup>bs\<^esup> = (0::'a)" by blast
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1033
    
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1034
    from H[rule_format, OF ihd_lt_n nhd] hdz have "?hd = 0\<^sub>p" by blast
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1035
    hence "?h = 0\<^sub>p" by simp
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1036
    with head_nz[OF np] have "p = 0\<^sub>p" by simp}
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1037
  ultimately show "p = 0\<^sub>p" by blast
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1038
qed
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1039
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1040
lemma isnpolyh_unique:  
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1041
  assumes np:"isnpolyh p n0" and nq: "isnpolyh q n1"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  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"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1043
proof(auto)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1044
  assume H: "\<forall>bs. (\<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> ::'a)= \<lparr>q\<rparr>\<^sub>p\<^bsup>bs\<^esup>"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1045
  hence "\<forall>bs.\<lparr>p -\<^sub>p q\<rparr>\<^sub>p\<^bsup>bs\<^esup>= (0::'a)" by simp
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  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)" 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1047
    using wf_bs_polysub[where p=p and q=q] by auto
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1048
  with isnpolyh_zero_iff[OF polysub_normh[OF np nq]] polysub_0[OF np nq]
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1049
  show "p = q" by blast
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1050
qed
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1051
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1052
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1053
text{* consequenses of unicity on the algorithms for polynomial normalization *}
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1054
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1055
lemma polyadd_commute:   assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1056
  and np: "isnpolyh p n0" and nq: "isnpolyh q n1" shows "p +\<^sub>p q = q +\<^sub>p p"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1057
  using isnpolyh_unique[OF polyadd_normh[OF np nq] polyadd_normh[OF nq np]] by simp
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1058
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1059
lemma zero_normh: "isnpolyh 0\<^sub>p n" by simp
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1060
lemma one_normh: "isnpolyh 1\<^sub>p n" by simp
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1061
lemma polyadd_0[simp]: 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1062
  assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1063
  and np: "isnpolyh p n0" shows "p +\<^sub>p 0\<^sub>p = p" and "0\<^sub>p +\<^sub>p p = p"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1064
  using isnpolyh_unique[OF polyadd_normh[OF np zero_normh] np] 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1065
    isnpolyh_unique[OF polyadd_normh[OF zero_normh np] np] by simp_all
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1066
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1067
lemma polymul_1[simp]: 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1068
    assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1069
  and np: "isnpolyh p n0" shows "p *\<^sub>p 1\<^sub>p = p" and "1\<^sub>p *\<^sub>p p = p"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1070
  using isnpolyh_unique[OF polymul_normh[OF np one_normh] np] 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1071
    isnpolyh_unique[OF polymul_normh[OF one_normh np] np] by simp_all
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1072
lemma polymul_0[simp]: 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1073
  assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  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"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1075
  using isnpolyh_unique[OF polymul_normh[OF np zero_normh] zero_normh] 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1076
    isnpolyh_unique[OF polymul_normh[OF zero_normh np] zero_normh] by simp_all
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1077
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1078
lemma polymul_commute: 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1079
    assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1080
  and np:"isnpolyh p n0" and nq: "isnpolyh q n1"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1081
  shows "p *\<^sub>p q = q *\<^sub>p p"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  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
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1083
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1084
declare polyneg_polyneg[simp]
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1085
  
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1086
lemma isnpolyh_polynate_id[simp]: 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1087
  assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1088
  and np:"isnpolyh p n0" shows "polynate p = p"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  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
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1090
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1091
lemma polynate_idempotent[simp]: 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1092
    assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1093
  shows "polynate (polynate p) = polynate p"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1094
  using isnpolyh_polynate_id[OF polynate_norm[of p, unfolded isnpoly_def]] .
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1095
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1096
lemma poly_nate_polypoly': "poly_nate bs p = polypoly' bs (polynate p)"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1097
  unfolding poly_nate_def polypoly'_def ..
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  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>)"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1099
  using polypoly'_poly[OF polynate_norm[unfolded isnpoly_def], symmetric, of bs p]
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1100
  unfolding poly_nate_polypoly' by (auto intro: ext)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1101
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1102
subsection{* heads, degrees and all that *}
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1103
lemma degree_eq_degreen0: "degree p = degreen p 0"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1104
  by (induct p rule: degree.induct, simp_all)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1105
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1106
lemma degree_polyneg: assumes n: "isnpolyh p n"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1107
  shows "degree (polyneg p) = degree p"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1108
  using n
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1109
  by (induct p arbitrary: n rule: polyneg.induct, simp_all) (case_tac na, auto)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1110
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1111
lemma degree_polyadd:
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1112
  assumes np: "isnpolyh p n0" and nq: "isnpolyh q n1"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1113
  shows "degree (p +\<^sub>p q) \<le> max (degree p) (degree q)"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1114
using degreen_polyadd[OF np nq, where m= "0"] degree_eq_degreen0 by simp
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1115
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1116
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1117
lemma degree_polysub: assumes np: "isnpolyh p n0" and nq: "isnpolyh q n1"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1118
  shows "degree (p -\<^sub>p q) \<le> max (degree p) (degree q)"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1119
proof-
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1120
  from nq have nq': "isnpolyh (~\<^sub>p q) n1" using polyneg_normh by simp
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1121
  from degree_polyadd[OF np nq'] show ?thesis by (simp add: polysub_def degree_polyneg[OF nq])
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1122
qed
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1123
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1124
lemma degree_polysub_samehead: 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1125
  assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1126
  and np: "isnpolyh p n0" and nq: "isnpolyh q n1" and h: "head p = head q" 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1127
  and d: "degree p = degree q"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1128
  shows "degree (p -\<^sub>p q) < degree p \<or> (p -\<^sub>p q = 0\<^sub>p)"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1129
unfolding polysub_def split_def fst_conv snd_conv
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1130
using np nq h d
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1131
proof(induct p q rule:polyadd.induct)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1132
  case (1 a b a' b') thus ?case by (simp add: Nsub_def Nsub0[simplified Nsub_def]) 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1133
next
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1134
  case (2 a b c' n' p') 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1135
  let ?c = "(a,b)"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1136
  from prems have "degree (C ?c) = degree (CN c' n' p')" by simp
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1137
  hence nz:"n' > 0" by (cases n', auto)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1138
  hence "head (CN c' n' p') = CN c' n' p'" by (cases n', auto)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1139
  with prems show ?case by simp
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1140
next
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1141
  case (3 c n p a' b') 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1142
  let ?c' = "(a',b')"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1143
  from prems have "degree (C ?c') = degree (CN c n p)" by simp
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1144
  hence nz:"n > 0" by (cases n, auto)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1145
  hence "head (CN c n p) = CN c n p" by (cases n, auto)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1146
  with prems show ?case by simp
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1147
next
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1148
  case (4 c n p c' n' p')
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1149
  hence H: "isnpolyh (CN c n p) n0" "isnpolyh (CN c' n' p') n1" 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1150
    "head (CN c n p) = head (CN c' n' p')" "degree (CN c n p) = degree (CN c' n' p')" by simp+
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1151
  hence degc: "degree c = 0" and degc': "degree c' = 0" by simp_all  
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1152
  hence degnc: "degree (~\<^sub>p c) = 0" and degnc': "degree (~\<^sub>p c') = 0" 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1153
    using H(1-2) degree_polyneg by auto
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1154
  from H have cnh: "isnpolyh c (Suc n)" and c'nh: "isnpolyh c' (Suc n')"  by simp+
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1155
  from degree_polysub[OF cnh c'nh, simplified polysub_def] degc degc' have degcmc': "degree (c +\<^sub>p  ~\<^sub>pc') = 0"  by simp
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1156
  from H have pnh: "isnpolyh p n" and p'nh: "isnpolyh p' n'" by auto
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1157
  have "n = n' \<or> n < n' \<or> n > n'" by arith
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1158
  moreover
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1159
  {assume nn': "n = n'"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1160
    have "n = 0 \<or> n >0" by arith
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1161
    moreover {assume nz: "n = 0" hence ?case using prems by (auto simp add: Let_def degcmc')}
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1162
    moreover {assume nz: "n > 0"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1163
      with nn' H(3) have  cc':"c = c'" and pp': "p = p'" by (cases n, auto)+
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  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)}
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1165
    ultimately have ?case by blast}
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1166
  moreover
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1167
  {assume nn': "n < n'" hence n'p: "n' > 0" by simp 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1168
    hence headcnp':"head (CN c' n' p') = CN c' n' p'"  by (cases n', simp_all)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  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)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1170
    hence "n > 0" by (cases n, simp_all)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1171
    hence headcnp: "head (CN c n p) = CN c n p" by (cases n, auto)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1172
    from H(3) headcnp headcnp' nn' have ?case by auto}
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1173
  moreover
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1174
  {assume nn': "n > n'"  hence np: "n > 0" by simp 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1175
    hence headcnp:"head (CN c n p) = CN c n p"  by (cases n, simp_all)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1176
    from prems have degcnpeq: "degree (CN c' n' p') = degree (CN c n p)" by simp
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1177
    from np have degcnp: "degree (CN c n p) = 0" by (cases n, simp_all)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1178
    with degcnpeq have "n' > 0" by (cases n', simp_all)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1179
    hence headcnp': "head (CN c' n' p') = CN c' n' p'" by (cases n', auto)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1180
    from H(3) headcnp headcnp' nn' have ?case by auto}
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1181
  ultimately show ?case  by blast
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1182
qed auto 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1183
 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1184
lemma shift1_head : "isnpolyh p n0 \<Longrightarrow> head (shift1 p) = head p"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1185
by (induct p arbitrary: n0 rule: head.induct, simp_all add: shift1_def)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1186
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1187
lemma funpow_shift1_head: "isnpolyh p n0 \<Longrightarrow> p\<noteq> 0\<^sub>p \<Longrightarrow> head (funpow k shift1 p) = head p"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1188
proof(induct k arbitrary: n0 p)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1189
  case (Suc k n0 p) hence "isnpolyh (shift1 p) 0" by (simp add: shift1_isnpolyh)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1190
  with prems have "head (funpow k shift1 (shift1 p)) = head (shift1 p)"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1191
    and "head (shift1 p) = head p" by (simp_all add: shift1_head) 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1192
  thus ?case by simp
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1193
qed auto  
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1194
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1195
lemma shift1_degree: "degree (shift1 p) = 1 + degree p"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1196
  by (simp add: shift1_def)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1197
lemma funpow_shift1_degree: "degree (funpow k shift1 p) = k + degree p "
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1198
  by (induct k arbitrary: p, auto simp add: shift1_degree)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1199
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1200
lemma funpow_shift1_nz: "p \<noteq> 0\<^sub>p \<Longrightarrow> funpow n shift1 p \<noteq> 0\<^sub>p"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1201
  by (induct n arbitrary: p, simp_all add: funpow_def)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1202
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1203
lemma head_isnpolyh_Suc[simp]: "isnpolyh p (Suc n) \<Longrightarrow> head p = p"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1204
  by (induct p arbitrary: n rule: degree.induct, auto)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1205
lemma headn_0[simp]: "isnpolyh p n \<Longrightarrow> m < n \<Longrightarrow> headn p m = p"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1206
  by (induct p arbitrary: n rule: degreen.induct, auto)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1207
lemma head_isnpolyh_Suc': "n > 0 \<Longrightarrow> isnpolyh p n \<Longrightarrow> head p = p"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1208
  by (induct p arbitrary: n rule: degree.induct, auto)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1209
lemma head_head[simp]: "isnpolyh p n0 \<Longrightarrow> head (head p) = head p"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1210
  by (induct p rule: head.induct, auto)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1211
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1212
lemma polyadd_eq_const_degree: 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1213
  "\<lbrakk> isnpolyh p n0 ; isnpolyh q n1 ; polyadd (p,q) = C c\<rbrakk> \<Longrightarrow> degree p = degree q" 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1214
  using polyadd_eq_const_degreen degree_eq_degreen0 by simp
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1215
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1216
lemma polyadd_head: assumes np: "isnpolyh p n0" and nq: "isnpolyh q n1"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1217
  and deg: "degree p \<noteq> degree q"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1218
  shows "head (p +\<^sub>p q) = (if degree p < degree q then head q else head p)"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1219
using np nq deg
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1220
apply(induct p q arbitrary: n0 n1 rule: polyadd.induct,simp_all)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1221
apply (case_tac n', simp, simp)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1222
apply (case_tac n, simp, simp)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1223
apply (case_tac n, case_tac n', simp add: Let_def)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1224
apply (case_tac "pa +\<^sub>p p' = 0\<^sub>p")
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1225
apply (clarsimp simp add: polyadd_eq_const_degree)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1226
apply clarsimp
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1227
apply (erule_tac impE,blast)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1228
apply (erule_tac impE,blast)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1229
apply clarsimp
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1230
apply simp
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1231
apply (case_tac n', simp_all)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1232
done
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1233
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1234
lemma polymul_head_polyeq: 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1235
   assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  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"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1237
proof (induct p q arbitrary: n0 n1 rule: polymul.induct)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1238
  case (2 a b c' n' p' n0 n1)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1239
  hence "isnpolyh (head (CN c' n' p')) n1" "isnormNum (a,b)"  by (simp_all add: head_isnpolyh)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1240
  thus ?case using prems by (cases n', auto) 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1241
next 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1242
  case (3 c n p a' b' n0 n1) 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1243
  hence "isnpolyh (head (CN c n p)) n0" "isnormNum (a',b')"  by (simp_all add: head_isnpolyh)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1244
  thus ?case using prems by (cases n, auto)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1245
next
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1246
  case (4 c n p c' n' p' n0 n1)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1247
  hence norm: "isnpolyh p n" "isnpolyh c (Suc n)" "isnpolyh p' n'" "isnpolyh c' (Suc n')"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1248
    "isnpolyh (CN c n p) n" "isnpolyh (CN c' n' p') n'"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1249
    by simp_all
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1250
  have "n < n' \<or> n' < n \<or> n = n'" by arith
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1251
  moreover 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1252
  {assume nn': "n < n'" hence ?case 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1253
      thm prems
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1254
      using norm 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1255
    prems(6)[rule_format, OF nn' norm(1,6)]
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1256
    prems(7)[rule_format, OF nn' norm(2,6)] by (simp, cases n, simp,cases n', simp_all)}
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1257
  moreover {assume nn': "n'< n"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1258
    hence stupid: "n' < n \<and> \<not> n < n'" by simp
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1259
    hence ?case using norm prems(4) [rule_format, OF stupid norm(5,3)]
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1260
      prems(5)[rule_format, OF stupid norm(5,4)] 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1261
      by (simp,cases n',simp,cases n,auto)}
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1262
  moreover {assume nn': "n' = n"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1263
    hence stupid: "\<not> n' < n \<and> \<not> n < n'" by simp
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1264
    from nn' polymul_normh[OF norm(5,4)] 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1265
    have ncnpc':"isnpolyh (CN c n p *\<^sub>p c') n" by (simp add: min_def)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1266
    from nn' polymul_normh[OF norm(5,3)] norm 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1267
    have ncnpp':"isnpolyh (CN c n p *\<^sub>p p') n" by simp
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1268
    from nn' ncnpp' polymul_eq0_iff[OF norm(5,3)] norm(6)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1269
    have ncnpp0':"isnpolyh (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n" by simp 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1270
    from polyadd_normh[OF ncnpc' ncnpp0'] 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  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" 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1272
      by (simp add: min_def)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1273
    {assume np: "n > 0"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1274
      with nn' head_isnpolyh_Suc'[OF np nth]
33268
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
  1275
        head_isnpolyh_Suc'[OF np norm(5)] head_isnpolyh_Suc'[OF np norm(6)[simplified nn']]
33154
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1276
      have ?case by simp}
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1277
    moreover
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1278
    {moreover assume nz: "n = 0"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1279
      from polymul_degreen[OF norm(5,4), where m="0"]
33268
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
  1280
        polymul_degreen[OF norm(5,3), where m="0"] nn' nz degree_eq_degreen0
33154
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1281
      norm(5,6) degree_npolyhCN[OF norm(6)]
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  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
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  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
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1284
    from polyadd_head[OF ncnpc'[simplified nz] ncnpp0'[simplified nz] dth'] dth
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1285
    have ?case   using norm prems(2)[rule_format, OF stupid norm(5,3)]
33268
02de0317f66f eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 33154
diff changeset
  1286
        prems(3)[rule_format, OF stupid norm(5,4)] nn' nz by simp }
33154
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1287
    ultimately have ?case by (cases n) auto} 
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1288
  ultimately show ?case by blast
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1289
qed simp_all
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1290
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1291
lemma degree_coefficients: "degree p = length (coefficients p) - 1"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1292
  by(induct p rule: degree.induct, auto)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1293
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1294
lemma degree_head[simp]: "degree (head p) = 0"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1295
  by (induct p rule: head.induct, auto)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1296
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1297
lemma degree_CN: "isnpolyh p n \<Longrightarrow> degree (CN c n p) \<le> 1+ degree p"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1298
  by (cases n, simp_all)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1299
lemma degree_CN': "isnpolyh p n \<Longrightarrow> degree (CN c n p) \<ge>  degree p"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1300
  by (cases n, simp_all)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1301
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  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)"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1303
  using polyadd_different_degreen degree_eq_degreen0 by simp
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1304
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1305
lemma degreen_polyneg: "isnpolyh p n0 \<Longrightarrow> degreen (~\<^sub>p p) m = degreen p m"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1306
  by (induct p arbitrary: n0 rule: polyneg.induct, auto)
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1307
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1308
lemma degree_polymul:
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1309
  assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
daa6ddece9f0 Multivariate polynomials library over fields
chaieb
parents:
diff changeset
  1310
  and np: "isnpolyh p n0" and nq: "isnpolyh q n1"