author | berghofe |
Sun, 10 Jan 2010 18:43:45 +0100 | |
changeset 34915 | 7894c7dab132 |
parent 33268 | 02de0317f66f |
child 35046 | 1266f04f42ec |
permissions | -rw-r--r-- |
33154 | 1 |
(* Title: HOL/Decision_Procs/Reflected_Multivariate_Polynomial.thy |
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Author: Amine Chaieb |
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*) |
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header {* Implementation and verification of mutivariate polynomials Library *} |
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theory Reflected_Multivariate_Polynomial |
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imports Parity Abstract_Rat Efficient_Nat List Polynomial_List |
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begin |
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(* Impelementation *) |
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subsection{* Datatype of polynomial expressions *} |
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datatype poly = C Num| Bound nat| Add poly poly|Sub poly poly |
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| Mul poly poly| Neg poly| Pw poly nat| CN poly nat poly |
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ML{* @{term "Add"}*} |
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syntax "_poly0" :: "poly" ("0\<^sub>p") |
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translations "0\<^sub>p" \<rightleftharpoons> "C (0\<^sub>N)" |
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syntax "_poly" :: "int \<Rightarrow> poly" ("_\<^sub>p") |
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translations "i\<^sub>p" \<rightleftharpoons> "C (i\<^sub>N)" |
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subsection{* Boundedness, substitution and all that *} |
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consts polysize:: "poly \<Rightarrow> nat" |
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primrec |
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"polysize (C c) = 1" |
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"polysize (Bound n) = 1" |
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"polysize (Neg p) = 1 + polysize p" |
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"polysize (Add p q) = 1 + polysize p + polysize q" |
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"polysize (Sub p q) = 1 + polysize p + polysize q" |
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"polysize (Mul p q) = 1 + polysize p + polysize q" |
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"polysize (Pw p n) = 1 + polysize p" |
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"polysize (CN c n p) = 4 + polysize c + polysize p" |
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consts |
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polybound0:: "poly \<Rightarrow> bool" (* a poly is INDEPENDENT of Bound 0 *) |
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polysubst0:: "poly \<Rightarrow> poly \<Rightarrow> poly" (* substitute a poly into a poly for Bound 0 *) |
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primrec |
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"polybound0 (C c) = True" |
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"polybound0 (Bound n) = (n>0)" |
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"polybound0 (Neg a) = polybound0 a" |
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"polybound0 (Add a b) = (polybound0 a \<and> polybound0 b)" |
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"polybound0 (Sub a b) = (polybound0 a \<and> polybound0 b)" |
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"polybound0 (Mul a b) = (polybound0 a \<and> polybound0 b)" |
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"polybound0 (Pw p n) = (polybound0 p)" |
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"polybound0 (CN c n p) = (n \<noteq> 0 \<and> polybound0 c \<and> polybound0 p)" |
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primrec |
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"polysubst0 t (C c) = (C c)" |
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"polysubst0 t (Bound n) = (if n=0 then t else Bound n)" |
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"polysubst0 t (Neg a) = Neg (polysubst0 t a)" |
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"polysubst0 t (Add a b) = Add (polysubst0 t a) (polysubst0 t b)" |
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"polysubst0 t (Sub a b) = Sub (polysubst0 t a) (polysubst0 t b)" |
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"polysubst0 t (Mul a b) = Mul (polysubst0 t a) (polysubst0 t b)" |
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"polysubst0 t (Pw p n) = Pw (polysubst0 t p) n" |
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"polysubst0 t (CN c n p) = (if n=0 then Add (polysubst0 t c) (Mul t (polysubst0 t p)) |
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else CN (polysubst0 t c) n (polysubst0 t p))" |
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consts |
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decrpoly:: "poly \<Rightarrow> poly" |
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recdef decrpoly "measure polysize" |
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"decrpoly (Bound n) = Bound (n - 1)" |
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"decrpoly (Neg a) = Neg (decrpoly a)" |
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"decrpoly (Add a b) = Add (decrpoly a) (decrpoly b)" |
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"decrpoly (Sub a b) = Sub (decrpoly a) (decrpoly b)" |
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"decrpoly (Mul a b) = Mul (decrpoly a) (decrpoly b)" |
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"decrpoly (Pw p n) = Pw (decrpoly p) n" |
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"decrpoly (CN c n p) = CN (decrpoly c) (n - 1) (decrpoly p)" |
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"decrpoly a = a" |
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subsection{* Degrees and heads and coefficients *} |
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consts degree:: "poly \<Rightarrow> nat" |
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recdef degree "measure size" |
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"degree (CN c 0 p) = 1 + degree p" |
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"degree p = 0" |
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consts head:: "poly \<Rightarrow> poly" |
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recdef head "measure size" |
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"head (CN c 0 p) = head p" |
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"head p = p" |
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(* More general notions of degree and head *) |
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consts degreen:: "poly \<Rightarrow> nat \<Rightarrow> nat" |
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recdef degreen "measure size" |
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"degreen (CN c n p) = (\<lambda>m. if n=m then 1 + degreen p n else 0)" |
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"degreen p = (\<lambda>m. 0)" |
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consts headn:: "poly \<Rightarrow> nat \<Rightarrow> poly" |
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recdef headn "measure size" |
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"headn (CN c n p) = (\<lambda>m. if n \<le> m then headn p m else CN c n p)" |
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"headn p = (\<lambda>m. p)" |
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consts coefficients:: "poly \<Rightarrow> poly list" |
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recdef coefficients "measure size" |
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"coefficients (CN c 0 p) = c#(coefficients p)" |
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"coefficients p = [p]" |
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consts isconstant:: "poly \<Rightarrow> bool" |
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recdef isconstant "measure size" |
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"isconstant (CN c 0 p) = False" |
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"isconstant p = True" |
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consts behead:: "poly \<Rightarrow> poly" |
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recdef behead "measure size" |
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"behead (CN c 0 p) = (let p' = behead p in if p' = 0\<^sub>p then c else CN c 0 p')" |
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"behead p = 0\<^sub>p" |
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consts headconst:: "poly \<Rightarrow> Num" |
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recdef headconst "measure size" |
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"headconst (CN c n p) = headconst p" |
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"headconst (C n) = n" |
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subsection{* Operations for normalization *} |
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consts |
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polyadd :: "poly\<times>poly \<Rightarrow> poly" |
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polyneg :: "poly \<Rightarrow> poly" ("~\<^sub>p") |
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polysub :: "poly\<times>poly \<Rightarrow> poly" |
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polymul :: "poly\<times>poly \<Rightarrow> poly" |
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polypow :: "nat \<Rightarrow> poly \<Rightarrow> poly" |
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syntax "_polyadd" :: "poly \<Rightarrow> poly \<Rightarrow> poly" (infixl "+\<^sub>p" 60) |
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translations "a +\<^sub>p b" \<rightleftharpoons> "polyadd (a,b)" |
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syntax "_polymul" :: "poly \<Rightarrow> poly \<Rightarrow> poly" (infixl "*\<^sub>p" 60) |
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translations "a *\<^sub>p b" \<rightleftharpoons> "polymul (a,b)" |
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syntax "_polysub" :: "poly \<Rightarrow> poly \<Rightarrow> poly" (infixl "-\<^sub>p" 60) |
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translations "a -\<^sub>p b" \<rightleftharpoons> "polysub (a,b)" |
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syntax "_polypow" :: "nat \<Rightarrow> poly \<Rightarrow> poly" (infixl "^\<^sub>p" 60) |
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translations "a ^\<^sub>p k" \<rightleftharpoons> "polypow k a" |
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recdef polyadd "measure (\<lambda> (a,b). polysize a + polysize b)" |
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"polyadd (C c, C c') = C (c+\<^sub>Nc')" |
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"polyadd (C c, CN c' n' p') = CN (polyadd (C c, c')) n' p'" |
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"polyadd (CN c n p, C c') = CN (polyadd (c, C c')) n p" |
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stupid: "polyadd (CN c n p, CN c' n' p') = |
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(if n < n' then CN (polyadd(c,CN c' n' p')) n p |
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else if n'<n then CN (polyadd(CN c n p, c')) n' p' |
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else (let cc' = polyadd (c,c') ; |
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pp' = polyadd (p,p') |
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in (if pp' = 0\<^sub>p then cc' else CN cc' n pp')))" |
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"polyadd (a, b) = Add a b" |
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(hints recdef_simp add: Let_def measure_def split_def inv_image_def) |
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(* |
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declare stupid [simp del, code del] |
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lemma [simp,code]: "polyadd (CN c n p, CN c' n' p') = |
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(if n < n' then CN (polyadd(c,CN c' n' p')) n p |
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else if n'<n then CN (polyadd(CN c n p, c')) n' p' |
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else (let cc' = polyadd (c,c') ; |
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pp' = polyadd (p,p') |
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in (if pp' = 0\<^sub>p then cc' else CN cc' n pp')))" |
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by (simp add: Let_def stupid) |
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*) |
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recdef polyneg "measure size" |
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"polyneg (C c) = C (~\<^sub>N c)" |
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"polyneg (CN c n p) = CN (polyneg c) n (polyneg p)" |
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"polyneg a = Neg a" |
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defs polysub_def[code]: "polysub \<equiv> \<lambda> (p,q). polyadd (p,polyneg q)" |
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recdef polymul "measure (\<lambda>(a,b). size a + size b)" |
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"polymul(C c, C c') = C (c*\<^sub>Nc')" |
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"polymul(C c, CN c' n' p') = |
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(if c = 0\<^sub>N then 0\<^sub>p else CN (polymul(C c,c')) n' (polymul(C c, p')))" |
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"polymul(CN c n p, C c') = |
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(if c' = 0\<^sub>N then 0\<^sub>p else CN (polymul(c,C c')) n (polymul(p, C c')))" |
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"polymul(CN c n p, CN c' n' p') = |
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(if n<n' then CN (polymul(c,CN c' n' p')) n (polymul(p,CN c' n' p')) |
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else if n' < n |
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then CN (polymul(CN c n p,c')) n' (polymul(CN c n p,p')) |
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else polyadd(polymul(CN c n p, c'),CN 0\<^sub>p n' (polymul(CN c n p, p'))))" |
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"polymul (a,b) = Mul a b" |
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recdef polypow "measure id" |
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"polypow 0 = (\<lambda>p. 1\<^sub>p)" |
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"polypow n = (\<lambda>p. let q = polypow (n div 2) p ; d = polymul(q,q) in |
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if even n then d else polymul(p,d))" |
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consts polynate :: "poly \<Rightarrow> poly" |
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recdef polynate "measure polysize" |
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"polynate (Bound n) = CN 0\<^sub>p n 1\<^sub>p" |
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"polynate (Add p q) = (polynate p +\<^sub>p polynate q)" |
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"polynate (Sub p q) = (polynate p -\<^sub>p polynate q)" |
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"polynate (Mul p q) = (polynate p *\<^sub>p polynate q)" |
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"polynate (Neg p) = (~\<^sub>p (polynate p))" |
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"polynate (Pw p n) = ((polynate p) ^\<^sub>p n)" |
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"polynate (CN c n p) = polynate (Add c (Mul (Bound n) p))" |
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"polynate (C c) = C (normNum c)" |
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fun poly_cmul :: "Num \<Rightarrow> poly \<Rightarrow> poly" where |
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"poly_cmul y (C x) = C (y *\<^sub>N x)" |
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| "poly_cmul y (CN c n p) = CN (poly_cmul y c) n (poly_cmul y p)" |
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| "poly_cmul y p = C y *\<^sub>p p" |
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constdefs monic:: "poly \<Rightarrow> (poly \<times> bool)" |
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"monic p \<equiv> (let h = headconst p in if h = 0\<^sub>N then (p,False) else ((C (Ninv h)) *\<^sub>p p, 0>\<^sub>N h))" |
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subsection{* Pseudo-division *} |
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constdefs shift1:: "poly \<Rightarrow> poly" |
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"shift1 p \<equiv> CN 0\<^sub>p 0 p" |
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consts funpow :: "nat \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a" |
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primrec |
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"funpow 0 f x = x" |
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"funpow (Suc n) f x = funpow n f (f x)" |
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function (tailrec) polydivide_aux :: "(poly \<times> nat \<times> poly \<times> nat \<times> poly) \<Rightarrow> (nat \<times> poly)" |
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where |
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"polydivide_aux (a,n,p,k,s) = |
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(if s = 0\<^sub>p then (k,s) |
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else (let b = head s; m = degree s in |
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(if m < n then (k,s) else |
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(let p'= funpow (m - n) shift1 p in |
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(if a = b then polydivide_aux (a,n,p,k,s -\<^sub>p p') |
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else polydivide_aux (a,n,p,Suc k, (a *\<^sub>p s) -\<^sub>p (b *\<^sub>p p')))))))" |
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by pat_completeness auto |
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constdefs polydivide:: "poly \<Rightarrow> poly \<Rightarrow> (nat \<times> poly)" |
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"polydivide s p \<equiv> polydivide_aux (head p,degree p,p,0, s)" |
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fun poly_deriv_aux :: "nat \<Rightarrow> poly \<Rightarrow> poly" where |
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"poly_deriv_aux n (CN c 0 p) = CN (poly_cmul ((int n)\<^sub>N) c) 0 (poly_deriv_aux (n + 1) p)" |
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| "poly_deriv_aux n p = poly_cmul ((int n)\<^sub>N) p" |
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fun poly_deriv :: "poly \<Rightarrow> poly" where |
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"poly_deriv (CN c 0 p) = poly_deriv_aux 1 p" |
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| "poly_deriv p = 0\<^sub>p" |
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(* Verification *) |
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lemma nth_pos2[simp]: "0 < n \<Longrightarrow> (x#xs) ! n = xs ! (n - 1)" |
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using Nat.gr0_conv_Suc |
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by clarsimp |
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subsection{* Semantics of the polynomial representation *} |
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consts Ipoly :: "'a list \<Rightarrow> poly \<Rightarrow> 'a::{ring_char_0,power,division_by_zero,field}" |
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primrec |
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"Ipoly bs (C c) = INum c" |
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"Ipoly bs (Bound n) = bs!n" |
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"Ipoly bs (Neg a) = - Ipoly bs a" |
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"Ipoly bs (Add a b) = Ipoly bs a + Ipoly bs b" |
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"Ipoly bs (Sub a b) = Ipoly bs a - Ipoly bs b" |
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"Ipoly bs (Mul a b) = Ipoly bs a * Ipoly bs b" |
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"Ipoly bs (Pw t n) = (Ipoly bs t) ^ n" |
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"Ipoly bs (CN c n p) = (Ipoly bs c) + (bs!n)*(Ipoly bs p)" |
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syntax "_Ipoly" :: "poly \<Rightarrow> 'a list \<Rightarrow>'a::{ring_char_0,power,division_by_zero,field}" ("\<lparr>_\<rparr>\<^sub>p\<^bsup>_\<^esup>") |
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translations "\<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup>" \<rightleftharpoons> "Ipoly bs p" |
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lemma Ipoly_CInt: "Ipoly bs (C (i,1)) = of_int i" |
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by (simp add: INum_def) |
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lemma Ipoly_CRat: "Ipoly bs (C (i, j)) = of_int i / of_int j" |
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by (simp add: INum_def) |
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lemmas RIpoly_eqs = Ipoly.simps(2-7) Ipoly_CInt Ipoly_CRat |
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subsection {* Normal form and normalization *} |
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consts isnpolyh:: "poly \<Rightarrow> nat \<Rightarrow> bool" |
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recdef isnpolyh "measure size" |
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"isnpolyh (C c) = (\<lambda>k. isnormNum c)" |
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"isnpolyh (CN c n p) = (\<lambda>k. n\<ge> k \<and> (isnpolyh c (Suc n)) \<and> (isnpolyh p n) \<and> (p \<noteq> 0\<^sub>p))" |
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"isnpolyh p = (\<lambda>k. False)" |
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lemma isnpolyh_mono: "\<lbrakk>n' \<le> n ; isnpolyh p n\<rbrakk> \<Longrightarrow> isnpolyh p n'" |
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by (induct p rule: isnpolyh.induct, auto) |
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constdefs isnpoly:: "poly \<Rightarrow> bool" |
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"isnpoly p \<equiv> isnpolyh p 0" |
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text{* polyadd preserves normal forms *} |
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lemma polyadd_normh: "\<lbrakk>isnpolyh p n0 ; isnpolyh q n1\<rbrakk> |
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\<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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from prems have th1: "isnpolyh (C (a,b)) (Suc n')" by simp |
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from prems(3) have th2: "isnpolyh c' (Suc n')" and nplen1: "n' \<ge> n1" by simp_all |
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with isnpolyh_mono have cp: "isnpolyh c' (Suc n')" by simp |
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with prems(1)[OF th1 th2] have th3:"isnpolyh (C (a,b) +\<^sub>p c') (Suc n')" by simp |
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from nplen1 have n01len1: "min n0 n1 \<le> n'" by simp |
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thus ?case using prems th3 by simp |
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next |
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case (3 c' n' p' a b n1 n0) |
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from prems have th1: "isnpolyh (C (a,b)) (Suc n')" by simp |
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from prems(2) have th2: "isnpolyh c' (Suc n')" and nplen1: "n' \<ge> n1" by simp_all |
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with isnpolyh_mono have cp: "isnpolyh c' (Suc n')" by simp |
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with prems(1)[OF th2 th1] have th3:"isnpolyh (c' +\<^sub>p C (a,b)) (Suc n')" by simp |
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from nplen1 have n01len1: "min n0 n1 \<le> n'" by simp |
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thus ?case using prems th3 by simp |
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next |
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case (4 c n p c' n' p' n0 n1) |
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hence nc: "isnpolyh c (Suc n)" and np: "isnpolyh p n" by simp_all |
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from prems have nc': "isnpolyh c' (Suc n')" and np': "isnpolyh p' n'" by simp_all |
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from prems have ngen0: "n \<ge> n0" by simp |
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from prems have n'gen1: "n' \<ge> n1" by simp |
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have "n < n' \<or> n' < n \<or> n = n'" by auto |
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moreover {assume eq: "n = n'" hence eq': "\<not> n' < n \<and> \<not> n < n'" by simp |
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with prems(2)[rule_format, OF eq' nc nc'] |
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have ncc':"isnpolyh (c +\<^sub>p c') (Suc n)" by auto |
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hence ncc'n01: "isnpolyh (c +\<^sub>p c') (min n0 n1)" |
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using isnpolyh_mono[where n'="min n0 n1" and n="Suc n"] ngen0 n'gen1 by auto |
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from eq prems(1)[rule_format, OF eq' np np'] have npp': "isnpolyh (p +\<^sub>p p') n" by simp |
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have minle: "min n0 n1 \<le> n'" using ngen0 n'gen1 eq by simp |
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from minle npp' ncc'n01 prems ngen0 n'gen1 ncc' have ?case by (simp add: Let_def)} |
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moreover {assume lt: "n < n'" |
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have "min n0 n1 \<le> n0" by simp |
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with prems have th1:"min n0 n1 \<le> n" by auto |
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from prems have th21: "isnpolyh c (Suc n)" by simp |
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from prems have th22: "isnpolyh (CN c' n' p') n'" by simp |
|
311 |
from lt have th23: "min (Suc n) n' = Suc n" by arith |
|
312 |
from prems(4)[rule_format, OF lt th21 th22] |
|
313 |
have "isnpolyh (polyadd (c, CN c' n' p')) (Suc n)" using th23 by simp |
|
314 |
with prems th1 have ?case by simp } |
|
315 |
moreover {assume gt: "n' < n" hence gt': "n' < n \<and> \<not> n < n'" by simp |
|
316 |
have "min n0 n1 \<le> n1" by simp |
|
317 |
with prems have th1:"min n0 n1 \<le> n'" by auto |
|
318 |
from prems have th21: "isnpolyh c' (Suc n')" by simp_all |
|
319 |
from prems have th22: "isnpolyh (CN c n p) n" by simp |
|
320 |
from gt have th23: "min n (Suc n') = Suc n'" by arith |
|
321 |
from prems(3)[rule_format, OF gt' th22 th21] |
|
322 |
have "isnpolyh (polyadd (CN c n p,c')) (Suc n')" using th23 by simp |
|
323 |
with prems th1 have ?case by simp} |
|
324 |
ultimately show ?case by blast |
|
325 |
qed auto |
|
326 |
||
327 |
lemma polyadd[simp]: "Ipoly bs (polyadd (p,q)) = (Ipoly bs p) + (Ipoly bs q)" |
|
328 |
by (induct p q rule: polyadd.induct, auto simp add: Let_def ring_simps right_distrib[symmetric] simp del: right_distrib) |
|
329 |
||
330 |
lemma polyadd_norm: "\<lbrakk> isnpoly p ; isnpoly q\<rbrakk> \<Longrightarrow> isnpoly (polyadd(p,q))" |
|
331 |
using polyadd_normh[of "p" "0" "q" "0"] isnpoly_def by simp |
|
332 |
||
333 |
text{* The degree of addition and other general lemmas needed for the normal form of polymul*} |
|
334 |
||
335 |
lemma polyadd_different_degreen: |
|
336 |
"\<lbrakk>isnpolyh p n0 ; isnpolyh q n1; degreen p m \<noteq> degreen q m ; m \<le> min n0 n1\<rbrakk> \<Longrightarrow> |
|
337 |
degreen (polyadd(p,q)) m = max (degreen p m) (degreen q m)" |
|
338 |
proof (induct p q arbitrary: m n0 n1 rule: polyadd.induct) |
|
339 |
case (4 c n p c' n' p' m n0 n1) |
|
340 |
thus ?case |
|
341 |
apply (cases "n' < n", simp_all add: Let_def) |
|
342 |
apply (cases "n = n'", simp_all) |
|
343 |
apply (cases "n' = m", simp_all add: Let_def) |
|
344 |
by (erule allE[where x="m"], erule allE[where x="Suc m"], |
|
345 |
erule allE[where x="m"], erule allE[where x="Suc m"], |
|
346 |
clarsimp,erule allE[where x="m"],erule allE[where x="Suc m"], simp) |
|
347 |
qed simp_all |
|
348 |
||
349 |
lemma headnz[simp]: "\<lbrakk>isnpolyh p n ; p \<noteq> 0\<^sub>p\<rbrakk> \<Longrightarrow> headn p m \<noteq> 0\<^sub>p" |
|
350 |
by (induct p arbitrary: n rule: headn.induct, auto) |
|
351 |
lemma degree_isnpolyh_Suc[simp]: "isnpolyh p (Suc n) \<Longrightarrow> degree p = 0" |
|
352 |
by (induct p arbitrary: n rule: degree.induct, auto) |
|
353 |
lemma degreen_0[simp]: "isnpolyh p n \<Longrightarrow> m < n \<Longrightarrow> degreen p m = 0" |
|
354 |
by (induct p arbitrary: n rule: degreen.induct, auto) |
|
355 |
||
356 |
lemma degree_isnpolyh_Suc': "n > 0 \<Longrightarrow> isnpolyh p n \<Longrightarrow> degree p = 0" |
|
357 |
by (induct p arbitrary: n rule: degree.induct, auto) |
|
358 |
||
359 |
lemma degree_npolyhCN[simp]: "isnpolyh (CN c n p) n0 \<Longrightarrow> degree c = 0" |
|
360 |
using degree_isnpolyh_Suc by auto |
|
361 |
lemma degreen_npolyhCN[simp]: "isnpolyh (CN c n p) n0 \<Longrightarrow> degreen c n = 0" |
|
362 |
using degreen_0 by auto |
|
363 |
||
364 |
||
365 |
lemma degreen_polyadd: |
|
366 |
assumes np: "isnpolyh p n0" and nq: "isnpolyh q n1" and m: "m \<le> max n0 n1" |
|
367 |
shows "degreen (p +\<^sub>p q) m \<le> max (degreen p m) (degreen q m)" |
|
368 |
using np nq m |
|
369 |
proof (induct p q arbitrary: n0 n1 m rule: polyadd.induct) |
|
370 |
case (2 c c' n' p' n0 n1) thus ?case by (cases n', simp_all) |
|
371 |
next |
|
372 |
case (3 c n p c' n0 n1) thus ?case by (cases n, auto) |
|
373 |
next |
|
374 |
case (4 c n p c' n' p' n0 n1 m) |
|
375 |
thus ?case |
|
376 |
apply (cases "n < n'", simp_all add: Let_def) |
|
377 |
apply (cases "n' < n", simp_all) |
|
378 |
apply (erule allE[where x="n"],erule allE[where x="Suc n"],clarify) |
|
379 |
apply (erule allE[where x="n'"],erule allE[where x="Suc n'"],clarify) |
|
380 |
by (erule allE[where x="m"],erule allE[where x="m"], auto) |
|
381 |
qed auto |
|
382 |
||
383 |
||
384 |
lemma polyadd_eq_const_degreen: "\<lbrakk> isnpolyh p n0 ; isnpolyh q n1 ; polyadd (p,q) = C c\<rbrakk> |
|
385 |
\<Longrightarrow> degreen p m = degreen q m" |
|
386 |
proof (induct p q arbitrary: m n0 n1 c rule: polyadd.induct) |
|
387 |
case (4 c n p c' n' p' m n0 n1 x) |
|
388 |
hence z: "CN c n p +\<^sub>p CN c' n' p' = C x" by simp |
|
389 |
{assume nn': "n' < n" hence ?case using prems by simp} |
|
390 |
moreover |
|
391 |
{assume nn':"\<not> n' < n" hence "n < n' \<or> n = n'" by arith |
|
392 |
moreover {assume "n < n'" with prems have ?case by simp } |
|
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 | 395 |
ultimately have ?case by blast} |
396 |
ultimately show ?case by blast |
|
397 |
qed simp_all |
|
398 |
||
399 |
lemma polymul_properties: |
|
400 |
assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})" |
|
401 |
and np: "isnpolyh p n0" and nq: "isnpolyh q n1" and m: "m \<le> min n0 n1" |
|
402 |
shows "isnpolyh (p *\<^sub>p q) (min n0 n1)" |
|
403 |
and "(p *\<^sub>p q = 0\<^sub>p) = (p = 0\<^sub>p \<or> q = 0\<^sub>p)" |
|
404 |
and "degreen (p *\<^sub>p q) m = (if (p = 0\<^sub>p \<or> q = 0\<^sub>p) then 0 |
|
405 |
else degreen p m + degreen q m)" |
|
406 |
using np nq m |
|
407 |
proof(induct p q arbitrary: n0 n1 m rule: polymul.induct) |
|
408 |
case (2 a b c' n' p') |
|
409 |
let ?c = "(a,b)" |
|
410 |
{ case (1 n0 n1) |
|
411 |
hence n: "isnpolyh (C ?c) n'" "isnpolyh c' (Suc n')" "isnpolyh p' n'" "isnormNum ?c" |
|
412 |
"isnpolyh (CN c' n' p') n1" |
|
413 |
by simp_all |
|
414 |
{assume "?c = 0\<^sub>N" hence ?case by auto} |
|
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 | 420 |
ultimately show ?case by blast |
421 |
next |
|
422 |
case (2 n0 n1) thus ?case by auto |
|
423 |
next |
|
424 |
case (3 n0 n1) thus ?case using "2.hyps" by auto } |
|
425 |
next |
|
426 |
case (3 c n p a b){ |
|
427 |
let ?c' = "(a,b)" |
|
428 |
case (1 n0 n1) |
|
429 |
hence n: "isnpolyh (C ?c') n" "isnpolyh c (Suc n)" "isnpolyh p n" "isnormNum ?c'" |
|
430 |
"isnpolyh (CN c n p) n0" |
|
431 |
by simp_all |
|
432 |
{assume "?c' = 0\<^sub>N" hence ?case by auto} |
|
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 | 438 |
ultimately show ?case by blast |
439 |
next |
|
440 |
case (2 n0 n1) thus ?case apply auto done |
|
441 |
next |
|
442 |
case (3 n0 n1) thus ?case using "3.hyps" by auto } |
|
443 |
next |
|
444 |
case (4 c n p c' n' p') |
|
445 |
let ?cnp = "CN c n p" let ?cnp' = "CN c' n' p'" |
|
446 |
{fix n0 n1 |
|
447 |
assume "isnpolyh ?cnp n0" and "isnpolyh ?cnp' n1" |
|
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 | 453 |
have "n < n' \<or> n' < n \<or> n' = n" by auto |
454 |
moreover |
|
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 | 460 |
moreover |
461 |
||
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 | 468 |
moreover |
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 | 475 |
ultimately show "isnpolyh (?cnp *\<^sub>p ?cnp') (min n0 n1)" by blast } |
476 |
note th = this |
|
477 |
{fix n0 n1 m |
|
478 |
assume np: "isnpolyh ?cnp n0" and np':"isnpolyh ?cnp' n1" |
|
479 |
and m: "m \<le> min n0 n1" |
|
480 |
let ?d = "degreen (?cnp *\<^sub>p ?cnp') m" |
|
481 |
let ?d1 = "degreen ?cnp m" |
|
482 |
let ?d2 = "degreen ?cnp' m" |
|
483 |
let ?eq = "?d = (if ?cnp = 0\<^sub>p \<or> ?cnp' = 0\<^sub>p then 0 else ?d1 + ?d2)" |
|
484 |
have "n'<n \<or> n < n' \<or> n' = n" by auto |
|
485 |
moreover |
|
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 | 491 |
moreover |
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 | 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 | 542 |
ultimately show ?eq by blast} |
543 |
note degth = this |
|
544 |
{ case (2 n0 n1) |
|
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 | 547 |
hence mn: "m \<le> n" by simp |
548 |
let ?c0p = "CN 0\<^sub>p n (?cnp *\<^sub>p p')" |
|
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 | 567 |
thus ?case using "4.hyps" by clarsimp} |
568 |
qed auto |
|
569 |
||
570 |
lemma polymul[simp]: "Ipoly bs (p *\<^sub>p q) = (Ipoly bs p) * (Ipoly bs q)" |
|
571 |
by(induct p q rule: polymul.induct, auto simp add: ring_simps) |
|
572 |
||
573 |
lemma polymul_normh: |
|
574 |
assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})" |
|
575 |
shows "\<lbrakk>isnpolyh p n0 ; isnpolyh q n1\<rbrakk> \<Longrightarrow> isnpolyh (p *\<^sub>p q) (min n0 n1)" |
|
576 |
using polymul_properties(1) by blast |
|
577 |
lemma polymul_eq0_iff: |
|
578 |
assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})" |
|
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) " |
|
580 |
using polymul_properties(2) by blast |
|
581 |
lemma polymul_degreen: |
|
582 |
assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})" |
|
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)" |
|
584 |
using polymul_properties(3) by blast |
|
585 |
lemma polymul_norm: |
|
586 |
assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})" |
|
587 |
shows "\<lbrakk> isnpoly p; isnpoly q\<rbrakk> \<Longrightarrow> isnpoly (polymul (p,q))" |
|
588 |
using polymul_normh[of "p" "0" "q" "0"] isnpoly_def by simp |
|
589 |
||
590 |
lemma headconst_zero: "isnpolyh p n0 \<Longrightarrow> headconst p = 0\<^sub>N \<longleftrightarrow> p = 0\<^sub>p" |
|
591 |
by (induct p arbitrary: n0 rule: headconst.induct, auto) |
|
592 |
||
593 |
lemma headconst_isnormNum: "isnpolyh p n0 \<Longrightarrow> isnormNum (headconst p)" |
|
594 |
by (induct p arbitrary: n0, auto) |
|
595 |
||
596 |
lemma monic_eqI: assumes np: "isnpolyh p n0" |
|
597 |
shows "INum (headconst p) * Ipoly bs (fst (monic p)) = (Ipoly bs p ::'a::{ring_char_0,power,division_by_zero,field})" |
|
598 |
unfolding monic_def Let_def |
|
599 |
proof(cases "headconst p = 0\<^sub>N", simp_all add: headconst_zero[OF np]) |
|
600 |
let ?h = "headconst p" |
|
601 |
assume pz: "p \<noteq> 0\<^sub>p" |
|
602 |
{assume hz: "INum ?h = (0::'a)" |
|
603 |
from headconst_isnormNum[OF np] have norm: "isnormNum ?h" "isnormNum 0\<^sub>N" by simp_all |
|
604 |
from isnormNum_unique[where ?'a = 'a, OF norm] hz have "?h = 0\<^sub>N" by simp |
|
605 |
with headconst_zero[OF np] have "p =0\<^sub>p" by blast with pz have "False" by blast} |
|
606 |
thus "INum (headconst p) = (0::'a) \<longrightarrow> \<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> = 0" by blast |
|
607 |
qed |
|
608 |
||
609 |
||
610 |
||
611 |
||
612 |
text{* polyneg is a negation and preserves normal form *} |
|
613 |
lemma polyneg[simp]: "Ipoly bs (polyneg p) = - Ipoly bs p" |
|
614 |
by (induct p rule: polyneg.induct, auto) |
|
615 |
||
616 |
lemma polyneg0: "isnpolyh p n \<Longrightarrow> ((~\<^sub>p p) = 0\<^sub>p) = (p = 0\<^sub>p)" |
|
617 |
by (induct p arbitrary: n rule: polyneg.induct, auto simp add: Nneg_def) |
|
618 |
lemma polyneg_polyneg: "isnpolyh p n0 \<Longrightarrow> ~\<^sub>p (~\<^sub>p p) = p" |
|
619 |
by (induct p arbitrary: n0 rule: polyneg.induct, auto) |
|
620 |
lemma polyneg_normh: "\<And>n. isnpolyh p n \<Longrightarrow> isnpolyh (polyneg p) n " |
|
621 |
by (induct p rule: polyneg.induct, auto simp add: polyneg0) |
|
622 |
||
623 |
lemma polyneg_norm: "isnpoly p \<Longrightarrow> isnpoly (polyneg p)" |
|
624 |
using isnpoly_def polyneg_normh by simp |
|
625 |
||
626 |
||
627 |
text{* polysub is a substraction and preserves normalform *} |
|
628 |
lemma polysub[simp]: "Ipoly bs (polysub (p,q)) = (Ipoly bs p) - (Ipoly bs q)" |
|
629 |
by (simp add: polysub_def polyneg polyadd) |
|
630 |
lemma polysub_normh: "\<And> n0 n1. \<lbrakk> isnpolyh p n0 ; isnpolyh q n1\<rbrakk> \<Longrightarrow> isnpolyh (polysub(p,q)) (min n0 n1)" |
|
631 |
by (simp add: polysub_def polyneg_normh polyadd_normh) |
|
632 |
||
633 |
lemma polysub_norm: "\<lbrakk> isnpoly p; isnpoly q\<rbrakk> \<Longrightarrow> isnpoly (polysub(p,q))" |
|
634 |
using polyadd_norm polyneg_norm by (simp add: polysub_def) |
|
635 |
lemma polysub_same_0[simp]: assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})" |
|
636 |
shows "isnpolyh p n0 \<Longrightarrow> polysub (p, p) = 0\<^sub>p" |
|
637 |
unfolding polysub_def split_def fst_conv snd_conv |
|
638 |
by (induct p arbitrary: n0,auto simp add: Let_def Nsub0[simplified Nsub_def]) |
|
639 |
||
640 |
lemma polysub_0: |
|
641 |
assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})" |
|
642 |
shows "\<lbrakk> isnpolyh p n0 ; isnpolyh q n1\<rbrakk> \<Longrightarrow> (p -\<^sub>p q = 0\<^sub>p) = (p = q)" |
|
643 |
unfolding polysub_def split_def fst_conv snd_conv |
|
644 |
apply (induct p q arbitrary: n0 n1 rule:polyadd.induct, simp_all add: Nsub0[simplified Nsub_def]) |
|
645 |
apply (clarsimp simp add: Let_def) |
|
646 |
apply (case_tac "n < n'", simp_all) |
|
647 |
apply (case_tac "n' < n", simp_all) |
|
648 |
apply (erule impE)+ |
|
649 |
apply (rule_tac x="Suc n" in exI, simp) |
|
650 |
apply (rule_tac x="n" in exI, simp) |
|
651 |
apply (erule impE)+ |
|
652 |
apply (rule_tac x="n" in exI, simp) |
|
653 |
apply (rule_tac x="Suc n" in exI, simp) |
|
654 |
apply (erule impE)+ |
|
655 |
apply (rule_tac x="Suc n" in exI, simp) |
|
656 |
apply (rule_tac x="n" in exI, simp) |
|
657 |
apply (erule impE)+ |
|
658 |
apply (rule_tac x="Suc n" in exI, simp) |
|
659 |
apply clarsimp |
|
660 |
done |
|
661 |
||
662 |
text{* polypow is a power function and preserves normal forms *} |
|
663 |
lemma polypow[simp]: "Ipoly bs (polypow n p) = ((Ipoly bs p :: 'a::{ring_char_0,division_by_zero,field})) ^ n" |
|
664 |
proof(induct n rule: polypow.induct) |
|
665 |
case 1 thus ?case by simp |
|
666 |
next |
|
667 |
case (2 n) |
|
668 |
let ?q = "polypow ((Suc n) div 2) p" |
|
669 |
let ?d = "polymul(?q,?q)" |
|
670 |
have "odd (Suc n) \<or> even (Suc n)" by simp |
|
671 |
moreover |
|
672 |
{assume odd: "odd (Suc n)" |
|
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 |
|
674 |
from odd have "Ipoly bs (p ^\<^sub>p Suc n) = Ipoly bs (polymul(p, ?d))" by (simp add: Let_def) |
|
675 |
also have "\<dots> = (Ipoly bs p) * (Ipoly bs p)^(Suc n div 2)*(Ipoly bs p)^(Suc n div 2)" |
|
676 |
using "2.hyps" by simp |
|
677 |
also have "\<dots> = (Ipoly bs p) ^ (Suc n div 2 + Suc n div 2 + 1)" |
|
678 |
apply (simp only: power_add power_one_right) by simp |
|
679 |
also have "\<dots> = (Ipoly bs p) ^ (Suc (Suc (Suc (0\<Colon>nat)) * (Suc n div Suc (Suc (0\<Colon>nat)))))" |
|
680 |
by (simp only: th) |
|
681 |
finally have ?case |
|
682 |
using odd_nat_div_two_times_two_plus_one[OF odd, symmetric] by simp } |
|
683 |
moreover |
|
684 |
{assume even: "even (Suc n)" |
|
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 |
|
686 |
from even have "Ipoly bs (p ^\<^sub>p Suc n) = Ipoly bs ?d" by (simp add: Let_def) |
|
687 |
also have "\<dots> = (Ipoly bs p) ^ (Suc n div 2 + Suc n div 2)" |
|
688 |
using "2.hyps" apply (simp only: power_add) by simp |
|
689 |
finally have ?case using even_nat_div_two_times_two[OF even] by (simp only: th)} |
|
690 |
ultimately show ?case by blast |
|
691 |
qed |
|
692 |
||
693 |
lemma polypow_normh: |
|
694 |
assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})" |
|
695 |
shows "isnpolyh p n \<Longrightarrow> isnpolyh (polypow k p) n" |
|
696 |
proof (induct k arbitrary: n rule: polypow.induct) |
|
697 |
case (2 k n) |
|
698 |
let ?q = "polypow (Suc k div 2) p" |
|
699 |
let ?d = "polymul (?q,?q)" |
|
700 |
from prems have th1:"isnpolyh ?q n" and th2: "isnpolyh p n" by blast+ |
|
701 |
from polymul_normh[OF th1 th1] have dn: "isnpolyh ?d n" by simp |
|
702 |
from polymul_normh[OF th2 dn] have on: "isnpolyh (polymul(p,?d)) n" by simp |
|
703 |
from dn on show ?case by (simp add: Let_def) |
|
704 |
qed auto |
|
705 |
||
706 |
lemma polypow_norm: |
|
707 |
assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})" |
|
708 |
shows "isnpoly p \<Longrightarrow> isnpoly (polypow k p)" |
|
709 |
by (simp add: polypow_normh isnpoly_def) |
|
710 |
||
711 |
text{* Finally the whole normalization*} |
|
712 |
||
713 |
lemma polynate[simp]: "Ipoly bs (polynate p) = (Ipoly bs p :: 'a ::{ring_char_0,division_by_zero,field})" |
|
714 |
by (induct p rule:polynate.induct, auto) |
|
715 |
||
716 |
lemma polynate_norm[simp]: |
|
717 |
assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})" |
|
718 |
shows "isnpoly (polynate p)" |
|
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) |
|
720 |
||
721 |
text{* shift1 *} |
|
722 |
||
723 |
||
724 |
lemma shift1: "Ipoly bs (shift1 p) = Ipoly bs (Mul (Bound 0) p)" |
|
725 |
by (simp add: shift1_def polymul) |
|
726 |
||
727 |
lemma shift1_isnpoly: |
|
728 |
assumes pn: "isnpoly p" and pnz: "p \<noteq> 0\<^sub>p" shows "isnpoly (shift1 p) " |
|
729 |
using pn pnz by (simp add: shift1_def isnpoly_def ) |
|
730 |
||
731 |
lemma shift1_nz[simp]:"shift1 p \<noteq> 0\<^sub>p" |
|
732 |
by (simp add: shift1_def) |
|
733 |
lemma funpow_shift1_isnpoly: |
|
734 |
"\<lbrakk> isnpoly p ; p \<noteq> 0\<^sub>p\<rbrakk> \<Longrightarrow> isnpoly (funpow n shift1 p)" |
|
735 |
by (induct n arbitrary: p, auto simp add: shift1_isnpoly) |
|
736 |
||
737 |
lemma funpow_isnpolyh: |
|
738 |
assumes f: "\<And> p. isnpolyh p n \<Longrightarrow> isnpolyh (f p) n "and np: "isnpolyh p n" |
|
739 |
shows "isnpolyh (funpow k f p) n" |
|
740 |
using f np by (induct k arbitrary: p, auto) |
|
741 |
||
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)" |
|
743 |
by (induct n arbitrary: p, simp_all add: shift1_isnpoly shift1 power_Suc ) |
|
744 |
||
745 |
lemma shift1_isnpolyh: "isnpolyh p n0 \<Longrightarrow> p\<noteq> 0\<^sub>p \<Longrightarrow> isnpolyh (shift1 p) 0" |
|
746 |
using isnpolyh_mono[where n="n0" and n'="0" and p="p"] by (simp add: shift1_def) |
|
747 |
||
748 |
lemma funpow_shift1_1: |
|
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)" |
|
750 |
by (simp add: funpow_shift1) |
|
751 |
||
752 |
lemma poly_cmul[simp]: "Ipoly bs (poly_cmul c p) = Ipoly bs (Mul (C c) p)" |
|
753 |
by (induct p arbitrary: n0 rule: poly_cmul.induct, auto simp add: ring_simps) |
|
754 |
||
755 |
lemma behead: |
|
756 |
assumes np: "isnpolyh p n" |
|
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})" |
|
758 |
using np |
|
759 |
proof (induct p arbitrary: n rule: behead.induct) |
|
760 |
case (1 c p n) hence pn: "isnpolyh p n" by simp |
|
761 |
from prems(2)[OF pn] |
|
762 |
have th:"Ipoly bs (Add (Mul (head p) (Pw (Bound 0) (degree p))) (behead p)) = Ipoly bs p" . |
|
763 |
then show ?case using "1.hyps" apply (simp add: Let_def,cases "behead p = 0\<^sub>p") |
|
764 |
by (simp_all add: th[symmetric] ring_simps power_Suc) |
|
765 |
qed (auto simp add: Let_def) |
|
766 |
||
767 |
lemma behead_isnpolyh: |
|
768 |
assumes np: "isnpolyh p n" shows "isnpolyh (behead p) n" |
|
769 |
using np by (induct p rule: behead.induct, auto simp add: Let_def isnpolyh_mono) |
|
770 |
||
771 |
subsection{* Miscilanious lemmas about indexes, decrementation, substitution etc ... *} |
|
772 |
lemma isnpolyh_polybound0: "isnpolyh p (Suc n) \<Longrightarrow> polybound0 p" |
|
773 |
proof(induct p arbitrary: n rule: polybound0.induct, auto) |
|
774 |
case (goal1 c n p n') |
|
775 |
hence "n = Suc (n - 1)" by simp |
|
776 |
hence "isnpolyh p (Suc (n - 1))" using `isnpolyh p n` by simp |
|
777 |
with prems(2) show ?case by simp |
|
778 |
qed |
|
779 |
||
780 |
lemma isconstant_polybound0: "isnpolyh p n0 \<Longrightarrow> isconstant p \<longleftrightarrow> polybound0 p" |
|
781 |
by (induct p arbitrary: n0 rule: isconstant.induct, auto simp add: isnpolyh_polybound0) |
|
782 |
||
783 |
lemma decrpoly_zero[simp]: "decrpoly p = 0\<^sub>p \<longleftrightarrow> p = 0\<^sub>p" by (induct p, auto) |
|
784 |
||
785 |
lemma decrpoly_normh: "isnpolyh p n0 \<Longrightarrow> polybound0 p \<Longrightarrow> isnpolyh (decrpoly p) (n0 - 1)" |
|
786 |
apply (induct p arbitrary: n0, auto) |
|
787 |
apply (atomize) |
|
788 |
apply (erule_tac x = "Suc nat" in allE) |
|
789 |
apply auto |
|
790 |
done |
|
791 |
||
792 |
lemma head_polybound0: "isnpolyh p n0 \<Longrightarrow> polybound0 (head p)" |
|
793 |
by (induct p arbitrary: n0 rule: head.induct, auto intro: isnpolyh_polybound0) |
|
794 |
||
795 |
lemma polybound0_I: |
|
796 |
assumes nb: "polybound0 a" |
|
797 |
shows "Ipoly (b#bs) a = Ipoly (b'#bs) a" |
|
798 |
using nb |
|
799 |
by (induct a rule: polybound0.induct) auto |
|
800 |
lemma polysubst0_I: |
|
801 |
shows "Ipoly (b#bs) (polysubst0 a t) = Ipoly ((Ipoly (b#bs) a)#bs) t" |
|
802 |
by (induct t) simp_all |
|
803 |
||
804 |
lemma polysubst0_I': |
|
805 |
assumes nb: "polybound0 a" |
|
806 |
shows "Ipoly (b#bs) (polysubst0 a t) = Ipoly ((Ipoly (b'#bs) a)#bs) t" |
|
807 |
by (induct t) (simp_all add: polybound0_I[OF nb, where b="b" and b'="b'"]) |
|
808 |
||
809 |
lemma decrpoly: assumes nb: "polybound0 t" |
|
810 |
shows "Ipoly (x#bs) t = Ipoly bs (decrpoly t)" |
|
811 |
using nb by (induct t rule: decrpoly.induct, simp_all) |
|
812 |
||
813 |
lemma polysubst0_polybound0: assumes nb: "polybound0 t" |
|
814 |
shows "polybound0 (polysubst0 t a)" |
|
815 |
using nb by (induct a rule: polysubst0.induct, auto) |
|
816 |
||
817 |
lemma degree0_polybound0: "isnpolyh p n \<Longrightarrow> degree p = 0 \<Longrightarrow> polybound0 p" |
|
818 |
by (induct p arbitrary: n rule: degree.induct, auto simp add: isnpolyh_polybound0) |
|
819 |
||
820 |
fun maxindex :: "poly \<Rightarrow> nat" where |
|
821 |
"maxindex (Bound n) = n + 1" |
|
822 |
| "maxindex (CN c n p) = max (n + 1) (max (maxindex c) (maxindex p))" |
|
823 |
| "maxindex (Add p q) = max (maxindex p) (maxindex q)" |
|
824 |
| "maxindex (Sub p q) = max (maxindex p) (maxindex q)" |
|
825 |
| "maxindex (Mul p q) = max (maxindex p) (maxindex q)" |
|
826 |
| "maxindex (Neg p) = maxindex p" |
|
827 |
| "maxindex (Pw p n) = maxindex p" |
|
828 |
| "maxindex (C x) = 0" |
|
829 |
||
830 |
definition wf_bs :: "'a list \<Rightarrow> poly \<Rightarrow> bool" where |
|
831 |
"wf_bs bs p = (length bs \<ge> maxindex p)" |
|
832 |
||
833 |
lemma wf_bs_coefficients: "wf_bs bs p \<Longrightarrow> \<forall> c \<in> set (coefficients p). wf_bs bs c" |
|
834 |
proof(induct p rule: coefficients.induct) |
|
835 |
case (1 c p) |
|
836 |
show ?case |
|
837 |
proof |
|
838 |
fix x assume xc: "x \<in> set (coefficients (CN c 0 p))" |
|
839 |
hence "x = c \<or> x \<in> set (coefficients p)" by simp |
|
840 |
moreover |
|
841 |
{assume "x = c" hence "wf_bs bs x" using "1.prems" unfolding wf_bs_def by simp} |
|
842 |
moreover |
|
843 |
{assume H: "x \<in> set (coefficients p)" |
|
844 |
from "1.prems" have "wf_bs bs p" unfolding wf_bs_def by simp |
|
845 |
with "1.hyps" H have "wf_bs bs x" by blast } |
|
846 |
ultimately show "wf_bs bs x" by blast |
|
847 |
qed |
|
848 |
qed simp_all |
|
849 |
||
850 |
lemma maxindex_coefficients: " \<forall>c\<in> set (coefficients p). maxindex c \<le> maxindex p" |
|
851 |
by (induct p rule: coefficients.induct, auto) |
|
852 |
||
853 |
lemma length_exists: "\<exists>xs. length xs = n" by (rule exI[where x="replicate n x"], simp) |
|
854 |
||
855 |
lemma wf_bs_I: "wf_bs bs p ==> Ipoly (bs@bs') p = Ipoly bs p" |
|
856 |
unfolding wf_bs_def by (induct p, auto simp add: nth_append) |
|
857 |
||
858 |
lemma take_maxindex_wf: assumes wf: "wf_bs bs p" |
|
859 |
shows "Ipoly (take (maxindex p) bs) p = Ipoly bs p" |
|
860 |
proof- |
|
861 |
let ?ip = "maxindex p" |
|
862 |
let ?tbs = "take ?ip bs" |
|
863 |
from wf have "length ?tbs = ?ip" unfolding wf_bs_def by simp |
|
864 |
hence wf': "wf_bs ?tbs p" unfolding wf_bs_def by simp |
|
865 |
have eq: "bs = ?tbs @ (drop ?ip bs)" by simp |
|
866 |
from wf_bs_I[OF wf', of "drop ?ip bs"] show ?thesis using eq by simp |
|
867 |
qed |
|
868 |
||
869 |
lemma decr_maxindex: "polybound0 p \<Longrightarrow> maxindex (decrpoly p) = maxindex p - 1" |
|
870 |
by (induct p, auto) |
|
871 |
||
872 |
lemma wf_bs_insensitive: "length bs = length bs' \<Longrightarrow> wf_bs bs p = wf_bs bs' p" |
|
873 |
unfolding wf_bs_def by simp |
|
874 |
||
875 |
lemma wf_bs_insensitive': "wf_bs (x#bs) p = wf_bs (y#bs) p" |
|
876 |
unfolding wf_bs_def by simp |
|
877 |
||
878 |
||
879 |
||
880 |
lemma wf_bs_coefficients': "\<forall>c \<in> set (coefficients p). wf_bs bs c \<Longrightarrow> wf_bs (x#bs) p" |
|
881 |
by(induct p rule: coefficients.induct, auto simp add: wf_bs_def) |
|
882 |
lemma coefficients_Nil[simp]: "coefficients p \<noteq> []" |
|
883 |
by (induct p rule: coefficients.induct, simp_all) |
|
884 |
||
885 |
||
886 |
lemma coefficients_head: "last (coefficients p) = head p" |
|
887 |
by (induct p rule: coefficients.induct, auto) |
|
888 |
||
889 |
lemma wf_bs_decrpoly: "wf_bs bs (decrpoly p) \<Longrightarrow> wf_bs (x#bs) p" |
|
890 |
unfolding wf_bs_def by (induct p rule: decrpoly.induct, auto) |
|
891 |
||
892 |
lemma length_le_list_ex: "length xs \<le> n \<Longrightarrow> \<exists> ys. length (xs @ ys) = n" |
|
893 |
apply (rule exI[where x="replicate (n - length xs) z"]) |
|
894 |
by simp |
|
895 |
lemma isnpolyh_Suc_const:"isnpolyh p (Suc n) \<Longrightarrow> isconstant p" |
|
896 |
by (cases p, auto) (case_tac "nat", simp_all) |
|
897 |
||
898 |
lemma wf_bs_polyadd: "wf_bs bs p \<and> wf_bs bs q \<longrightarrow> wf_bs bs (p +\<^sub>p q)" |
|
899 |
unfolding wf_bs_def |
|
900 |
apply (induct p q rule: polyadd.induct) |
|
901 |
apply (auto simp add: Let_def) |
|
902 |
done |
|
903 |
||
904 |
lemma wf_bs_polyul: "wf_bs bs p \<Longrightarrow> wf_bs bs q \<Longrightarrow> wf_bs bs (p *\<^sub>p q)" |
|
905 |
||
906 |
unfolding wf_bs_def |
|
907 |
apply (induct p q arbitrary: bs rule: polymul.induct) |
|
908 |
apply (simp_all add: wf_bs_polyadd) |
|
909 |
apply clarsimp |
|
910 |
apply (rule wf_bs_polyadd[unfolded wf_bs_def, rule_format]) |
|
911 |
apply auto |
|
912 |
done |
|
913 |
||
914 |
lemma wf_bs_polyneg: "wf_bs bs p \<Longrightarrow> wf_bs bs (~\<^sub>p p)" |
|
915 |
unfolding wf_bs_def by (induct p rule: polyneg.induct, auto) |
|
916 |
||
917 |
lemma wf_bs_polysub: "wf_bs bs p \<Longrightarrow> wf_bs bs q \<Longrightarrow> wf_bs bs (p -\<^sub>p q)" |
|
918 |
unfolding polysub_def split_def fst_conv snd_conv using wf_bs_polyadd wf_bs_polyneg by blast |
|
919 |
||
920 |
subsection{* Canonicity of polynomial representation, see lemma isnpolyh_unique*} |
|
921 |
||
922 |
definition "polypoly bs p = map (Ipoly bs) (coefficients p)" |
|
923 |
definition "polypoly' bs p = map ((Ipoly bs o decrpoly)) (coefficients p)" |
|
924 |
definition "poly_nate bs p = map ((Ipoly bs o decrpoly)) (coefficients (polynate p))" |
|
925 |
||
926 |
lemma coefficients_normh: "isnpolyh p n0 \<Longrightarrow> \<forall> q \<in> set (coefficients p). isnpolyh q n0" |
|
927 |
proof (induct p arbitrary: n0 rule: coefficients.induct) |
|
928 |
case (1 c p n0) |
|
929 |
have cp: "isnpolyh (CN c 0 p) n0" by fact |
|
930 |
hence norm: "isnpolyh c 0" "isnpolyh p 0" "p \<noteq> 0\<^sub>p" "n0 = 0" |
|
931 |
by (auto simp add: isnpolyh_mono[where n'=0]) |
|
932 |
from "1.hyps"[OF norm(2)] norm(1) norm(4) show ?case by simp |
|
933 |
qed auto |
|
934 |
||
935 |
lemma coefficients_isconst: |
|
936 |
"isnpolyh p n \<Longrightarrow> \<forall>q\<in>set (coefficients p). isconstant q" |
|
937 |
by (induct p arbitrary: n rule: coefficients.induct, |
|
938 |
auto simp add: isnpolyh_Suc_const) |
|
939 |
||
940 |
lemma polypoly_polypoly': |
|
941 |
assumes np: "isnpolyh p n0" |
|
942 |
shows "polypoly (x#bs) p = polypoly' bs p" |
|
943 |
proof- |
|
944 |
let ?cf = "set (coefficients p)" |
|
945 |
from coefficients_normh[OF np] have cn_norm: "\<forall> q\<in> ?cf. isnpolyh q n0" . |
|
946 |
{fix q assume q: "q \<in> ?cf" |
|
947 |
from q cn_norm have th: "isnpolyh q n0" by blast |
|
948 |
from coefficients_isconst[OF np] q have "isconstant q" by blast |
|
949 |
with isconstant_polybound0[OF th] have "polybound0 q" by blast} |
|
950 |
hence "\<forall>q \<in> ?cf. polybound0 q" .. |
|
951 |
hence "\<forall>q \<in> ?cf. Ipoly (x#bs) q = Ipoly bs (decrpoly q)" |
|
952 |
using polybound0_I[where b=x and bs=bs and b'=y] decrpoly[where x=x and bs=bs] |
|
953 |
by auto |
|
954 |
||
955 |
thus ?thesis unfolding polypoly_def polypoly'_def by simp |
|
956 |
qed |
|
957 |
||
958 |
lemma polypoly_poly: |
|
959 |
assumes np: "isnpolyh p n0" shows "Ipoly (x#bs) p = poly (polypoly (x#bs) p) x" |
|
960 |
using np |
|
961 |
by (induct p arbitrary: n0 bs rule: coefficients.induct, auto simp add: polypoly_def) |
|
962 |
||
963 |
lemma polypoly'_poly: |
|
964 |
assumes np: "isnpolyh p n0" shows "\<lparr>p\<rparr>\<^sub>p\<^bsup>x # bs\<^esup> = poly (polypoly' bs p) x" |
|
965 |
using polypoly_poly[OF np, simplified polypoly_polypoly'[OF np]] . |
|
966 |
||
967 |
||
968 |
lemma polypoly_poly_polybound0: |
|
969 |
assumes np: "isnpolyh p n0" and nb: "polybound0 p" |
|
970 |
shows "polypoly bs p = [Ipoly bs p]" |
|
971 |
using np nb unfolding polypoly_def |
|
972 |
by (cases p, auto, case_tac nat, auto) |
|
973 |
||
974 |
lemma head_isnpolyh: "isnpolyh p n0 \<Longrightarrow> isnpolyh (head p) n0" |
|
975 |
by (induct p rule: head.induct, auto) |
|
976 |
||
977 |
lemma headn_nz[simp]: "isnpolyh p n0 \<Longrightarrow> (headn p m = 0\<^sub>p) = (p = 0\<^sub>p)" |
|
978 |
by (cases p,auto) |
|
979 |
||
980 |
lemma head_eq_headn0: "head p = headn p 0" |
|
981 |
by (induct p rule: head.induct, simp_all) |
|
982 |
||
983 |
lemma head_nz[simp]: "isnpolyh p n0 \<Longrightarrow> (head p = 0\<^sub>p) = (p = 0\<^sub>p)" |
|
984 |
by (simp add: head_eq_headn0) |
|
985 |
||
986 |
lemma isnpolyh_zero_iff: |
|
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})" |
|
988 |
shows "p = 0\<^sub>p" |
|
989 |
using nq eq |
|
34915 | 990 |
proof (induct "maxindex p" arbitrary: p n0 rule: less_induct) |
991 |
case less |
|
992 |
note np = `isnpolyh p n0` and zp = `\<forall>bs. wf_bs bs p \<longrightarrow> \<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> = (0::'a)` |
|
993 |
{assume nz: "maxindex p = 0" |
|
994 |
then obtain c where "p = C c" using np by (cases p, auto) |
|
33154 | 995 |
with zp np have "p = 0\<^sub>p" unfolding wf_bs_def by simp} |
996 |
moreover |
|
34915 | 997 |
{assume nz: "maxindex p \<noteq> 0" |
33154 | 998 |
let ?h = "head p" |
999 |
let ?hd = "decrpoly ?h" |
|
1000 |
let ?ihd = "maxindex ?hd" |
|
1001 |
from head_isnpolyh[OF np] head_polybound0[OF np] have h:"isnpolyh ?h n0" "polybound0 ?h" |
|
1002 |
by simp_all |
|
1003 |
hence nhd: "isnpolyh ?hd (n0 - 1)" using decrpoly_normh by blast |
|
1004 |
||
1005 |
from maxindex_coefficients[of p] coefficients_head[of p, symmetric] |
|
34915 | 1006 |
have mihn: "maxindex ?h \<le> maxindex p" by auto |
1007 |
with decr_maxindex[OF h(2)] nz have ihd_lt_n: "?ihd < maxindex p" by auto |
|
33154 | 1008 |
{fix bs:: "'a list" assume bs: "wf_bs bs ?hd" |
1009 |
let ?ts = "take ?ihd bs" |
|
1010 |
let ?rs = "drop ?ihd bs" |
|
1011 |
have ts: "wf_bs ?ts ?hd" using bs unfolding wf_bs_def by simp |
|
1012 |
have bs_ts_eq: "?ts@ ?rs = bs" by simp |
|
1013 |
from wf_bs_decrpoly[OF ts] have tsh: " \<forall>x. wf_bs (x#?ts) ?h" by simp |
|
34915 | 1014 |
from ihd_lt_n have "ALL x. length (x#?ts) \<le> maxindex p" by simp |
1015 |
with length_le_list_ex obtain xs where xs:"length ((x#?ts) @ xs) = maxindex p" by blast |
|
1016 |
hence "\<forall> x. wf_bs ((x#?ts) @ xs) p" unfolding wf_bs_def by simp |
|
33154 | 1017 |
with zp have "\<forall> x. Ipoly ((x#?ts) @ xs) p = 0" by blast |
1018 |
hence "\<forall> x. Ipoly (x#(?ts @ xs)) p = 0" by simp |
|
1019 |
with polypoly_poly[OF np, where ?'a = 'a] polypoly_polypoly'[OF np, where ?'a = 'a] |
|
1020 |
have "\<forall>x. poly (polypoly' (?ts @ xs) p) x = poly [] x" by simp |
|
1021 |
hence "poly (polypoly' (?ts @ xs) p) = poly []" by (auto intro: ext) |
|
1022 |
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
|
1023 |
using poly_zero[where ?'a='a] by (simp add: polypoly'_def list_all_iff) |
33154 | 1024 |
with coefficients_head[of p, symmetric] |
1025 |
have th0: "Ipoly (?ts @ xs) ?hd = 0" by simp |
|
1026 |
from bs have wf'': "wf_bs ?ts ?hd" unfolding wf_bs_def by simp |
|
1027 |
with th0 wf_bs_I[of ?ts ?hd xs] have "Ipoly ?ts ?hd = 0" by simp |
|
1028 |
with wf'' wf_bs_I[of ?ts ?hd ?rs] bs_ts_eq have "\<lparr>?hd\<rparr>\<^sub>p\<^bsup>bs\<^esup> = 0" by simp } |
|
1029 |
then have hdz: "\<forall>bs. wf_bs bs ?hd \<longrightarrow> \<lparr>?hd\<rparr>\<^sub>p\<^bsup>bs\<^esup> = (0::'a)" by blast |
|
1030 |
||
34915 | 1031 |
from less(1)[OF ihd_lt_n nhd] hdz have "?hd = 0\<^sub>p" by blast |
33154 | 1032 |
hence "?h = 0\<^sub>p" by simp |
1033 |
with head_nz[OF np] have "p = 0\<^sub>p" by simp} |
|
1034 |
ultimately show "p = 0\<^sub>p" by blast |
|
1035 |
qed |
|
1036 |
||
1037 |
lemma isnpolyh_unique: |
|
1038 |
assumes np:"isnpolyh p n0" and nq: "isnpolyh q n1" |
|
1039 |
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" |
|
1040 |
proof(auto) |
|
1041 |
assume H: "\<forall>bs. (\<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> ::'a)= \<lparr>q\<rparr>\<^sub>p\<^bsup>bs\<^esup>" |
|
1042 |
hence "\<forall>bs.\<lparr>p -\<^sub>p q\<rparr>\<^sub>p\<^bsup>bs\<^esup>= (0::'a)" by simp |
|
1043 |
hence "\<forall>bs. wf_bs bs (p -\<^sub>p q) \<longrightarrow> \<lparr>p -\<^sub>p q\<rparr>\<^sub>p\<^bsup>bs\<^esup> = (0::'a)" |
|
1044 |
using wf_bs_polysub[where p=p and q=q] by auto |
|
1045 |
with isnpolyh_zero_iff[OF polysub_normh[OF np nq]] polysub_0[OF np nq] |
|
1046 |
show "p = q" by blast |
|
1047 |
qed |
|
1048 |
||
1049 |
||
1050 |
text{* consequenses of unicity on the algorithms for polynomial normalization *} |
|
1051 |
||
1052 |
lemma polyadd_commute: assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})" |
|
1053 |
and np: "isnpolyh p n0" and nq: "isnpolyh q n1" shows "p +\<^sub>p q = q +\<^sub>p p" |
|
1054 |
using isnpolyh_unique[OF polyadd_normh[OF np nq] polyadd_normh[OF nq np]] by simp |
|
1055 |
||
1056 |
lemma zero_normh: "isnpolyh 0\<^sub>p n" by simp |
|
1057 |
lemma one_normh: "isnpolyh 1\<^sub>p n" by simp |
|
1058 |
lemma polyadd_0[simp]: |
|
1059 |
assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})" |
|
1060 |
and np: "isnpolyh p n0" shows "p +\<^sub>p 0\<^sub>p = p" and "0\<^sub>p +\<^sub>p p = p" |
|
1061 |
using isnpolyh_unique[OF polyadd_normh[OF np zero_normh] np] |
|
1062 |
isnpolyh_unique[OF polyadd_normh[OF zero_normh np] np] by simp_all |
|
1063 |
||
1064 |
lemma polymul_1[simp]: |
|
1065 |
assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})" |
|
1066 |
and np: "isnpolyh p n0" shows "p *\<^sub>p 1\<^sub>p = p" and "1\<^sub>p *\<^sub>p p = p" |
|
1067 |
using isnpolyh_unique[OF polymul_normh[OF np one_normh] np] |
|
1068 |
isnpolyh_unique[OF polymul_normh[OF one_normh np] np] by simp_all |
|
1069 |
lemma polymul_0[simp]: |
|
1070 |
assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})" |
|
1071 |
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" |
|
1072 |
using isnpolyh_unique[OF polymul_normh[OF np zero_normh] zero_normh] |
|
1073 |
isnpolyh_unique[OF polymul_normh[OF zero_normh np] zero_normh] by simp_all |
|
1074 |
||
1075 |
lemma polymul_commute: |
|
1076 |
assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})" |
|
1077 |
and np:"isnpolyh p n0" and nq: "isnpolyh q n1" |
|
1078 |
shows "p *\<^sub>p q = q *\<^sub>p p" |
|
1079 |
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 |
|
1080 |
||
1081 |
declare polyneg_polyneg[simp] |
|
1082 |
||
1083 |
lemma isnpolyh_polynate_id[simp]: |
|
1084 |
assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})" |
|
1085 |
and np:"isnpolyh p n0" shows "polynate p = p" |
|
1086 |
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 |
|
1087 |
||
1088 |
lemma polynate_idempotent[simp]: |
|
1089 |
assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})" |
|
1090 |
shows "polynate (polynate p) = polynate p" |
|
1091 |
using isnpolyh_polynate_id[OF polynate_norm[of p, unfolded isnpoly_def]] . |
|
1092 |
||
1093 |
lemma poly_nate_polypoly': "poly_nate bs p = polypoly' bs (polynate p)" |
|
1094 |
unfolding poly_nate_def polypoly'_def .. |
|
1095 |
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>)" |
|
1096 |
using polypoly'_poly[OF polynate_norm[unfolded isnpoly_def], symmetric, of bs p] |
|
1097 |
unfolding poly_nate_polypoly' by (auto intro: ext) |
|
1098 |
||
1099 |
subsection{* heads, degrees and all that *} |
|
1100 |
lemma degree_eq_degreen0: "degree p = degreen p 0" |
|
1101 |
by (induct p rule: degree.induct, simp_all) |
|
1102 |
||
1103 |
lemma degree_polyneg: assumes n: "isnpolyh p n" |
|
1104 |
shows "degree (polyneg p) = degree p" |
|
1105 |
using n |
|
1106 |
by (induct p arbitrary: n rule: polyneg.induct, simp_all) (case_tac na, auto) |
|
1107 |
||
1108 |
lemma degree_polyadd: |
|
1109 |
assumes np: "isnpolyh p n0" and nq: "isnpolyh q n1" |
|
1110 |
shows "degree (p +\<^sub>p q) \<le> max (degree p) (degree q)" |
|
1111 |
using degreen_polyadd[OF np nq, where m= "0"] degree_eq_degreen0 by simp |
|
1112 |
||
1113 |
||
1114 |
lemma degree_polysub: assumes np: "isnpolyh p n0" and nq: "isnpolyh q n1" |
|
1115 |
shows "degree (p -\<^sub>p q) \<le> max (degree p) (degree q)" |
|
1116 |
proof- |
|
1117 |
from nq have nq': "isnpolyh (~\<^sub>p q) n1" using polyneg_normh by simp |
|
1118 |
from degree_polyadd[OF np nq'] show ?thesis by (simp add: polysub_def degree_polyneg[OF nq]) |
|
1119 |
qed |
|
1120 |
||
1121 |
lemma degree_polysub_samehead: |
|
1122 |
assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})" |
|
1123 |
and np: "isnpolyh p n0" and nq: "isnpolyh q n1" and h: "head p = head q" |
|
1124 |
and d: "degree p = degree q" |
|
1125 |
shows "degree (p -\<^sub>p q) < degree p \<or> (p -\<^sub>p q = 0\<^sub>p)" |
|
1126 |
unfolding polysub_def split_def fst_conv snd_conv |
|
1127 |
using np nq h d |
|
1128 |
proof(induct p q rule:polyadd.induct) |
|
1129 |
case (1 a b a' b') thus ?case by (simp add: Nsub_def Nsub0[simplified Nsub_def]) |
|
1130 |
next |
|
1131 |
case (2 a b c' n' p') |
|
1132 |
let ?c = "(a,b)" |
|
1133 |
from prems have "degree (C ?c) = degree (CN c' n' p')" by simp |
|
1134 |
hence nz:"n' > 0" by (cases n', auto) |
|
1135 |
hence "head (CN c' n' p') = CN c' n' p'" by (cases n', auto) |
|
1136 |
with prems show ?case by simp |
|
1137 |
next |
|
1138 |
case (3 c n p a' b') |
|
1139 |
let ?c' = "(a',b')" |
|
1140 |
from prems have "degree (C ?c') = degree (CN c n p)" by simp |
|
1141 |
hence nz:"n > 0" by (cases n, auto) |
|
1142 |
hence "head (CN c n p) = CN c n p" by (cases n, auto) |
|
1143 |
with prems show ?case by simp |
|
1144 |
next |
|
1145 |
case (4 c n p c' n' p') |
|
1146 |
hence H: "isnpolyh (CN c n p) n0" "isnpolyh (CN c' n' p') n1" |
|
1147 |
"head (CN c n p) = head (CN c' n' p')" "degree (CN c n p) = degree (CN c' n' p')" by simp+ |
|
1148 |
hence degc: "degree c = 0" and degc': "degree c' = 0" by simp_all |
|
1149 |
hence degnc: "degree (~\<^sub>p c) = 0" and degnc': "degree (~\<^sub>p c') = 0" |
|
1150 |
using H(1-2) degree_polyneg by auto |
|
1151 |
from H have cnh: "isnpolyh c (Suc n)" and c'nh: "isnpolyh c' (Suc n')" by simp+ |
|
1152 |
from degree_polysub[OF cnh c'nh, simplified polysub_def] degc degc' have degcmc': "degree (c +\<^sub>p ~\<^sub>pc') = 0" by simp |
|
1153 |
from H have pnh: "isnpolyh p n" and p'nh: "isnpolyh p' n'" by auto |
|
1154 |
have "n = n' \<or> n < n' \<or> n > n'" by arith |
|
1155 |
moreover |
|
1156 |
{assume nn': "n = n'" |
|
1157 |
have "n = 0 \<or> n >0" by arith |
|
1158 |
moreover {assume nz: "n = 0" hence ?case using prems by (auto simp add: Let_def degcmc')} |
|
1159 |
moreover {assume nz: "n > 0" |
|
1160 |
with nn' H(3) have cc':"c = c'" and pp': "p = p'" by (cases n, auto)+ |
|
1161 |
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)} |
|
1162 |
ultimately have ?case by blast} |
|
1163 |
moreover |
|
1164 |
{assume nn': "n < n'" hence n'p: "n' > 0" by simp |
|
1165 |
hence headcnp':"head (CN c' n' p') = CN c' n' p'" by (cases n', simp_all) |
|
1166 |
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) |
|
1167 |
hence "n > 0" by (cases n, simp_all) |
|
1168 |
hence headcnp: "head (CN c n p) = CN c n p" by (cases n, auto) |
|
1169 |
from H(3) headcnp headcnp' nn' have ?case by auto} |
|
1170 |
moreover |
|
1171 |
{assume nn': "n > n'" hence np: "n > 0" by simp |
|
1172 |
hence headcnp:"head (CN c n p) = CN c n p" by (cases n, simp_all) |
|
1173 |
from prems have degcnpeq: "degree (CN c' n' p') = degree (CN c n p)" by simp |
|
1174 |
from np have degcnp: "degree (CN c n p) = 0" by (cases n, simp_all) |
|
1175 |
with degcnpeq have "n' > 0" by (cases n', simp_all) |
|
1176 |
hence headcnp': "head (CN c' n' p') = CN c' n' p'" by (cases n', auto) |
|
1177 |
from H(3) headcnp headcnp' nn' have ?case by auto} |
|
1178 |
ultimately show ?case by blast |
|
1179 |
qed auto |
|
1180 |
||
1181 |
lemma shift1_head : "isnpolyh p n0 \<Longrightarrow> head (shift1 p) = head p" |
|
1182 |
by (induct p arbitrary: n0 rule: head.induct, simp_all add: shift1_def) |
|
1183 |
||
1184 |
lemma funpow_shift1_head: "isnpolyh p n0 \<Longrightarrow> p\<noteq> 0\<^sub>p \<Longrightarrow> head (funpow k shift1 p) = head p" |
|
1185 |
proof(induct k arbitrary: n0 p) |
|
1186 |
case (Suc k n0 p) hence "isnpolyh (shift1 p) 0" by (simp add: shift1_isnpolyh) |
|
1187 |
with prems have "head (funpow k shift1 (shift1 p)) = head (shift1 p)" |
|
1188 |
and "head (shift1 p) = head p" by (simp_all add: shift1_head) |
|
1189 |
thus ?case by simp |
|
1190 |
qed auto |
|
1191 |
||
1192 |
lemma shift1_degree: "degree (shift1 p) = 1 + degree p" |
|
1193 |
by (simp add: shift1_def) |
|
1194 |
lemma funpow_shift1_degree: "degree (funpow k shift1 p) = k + degree p " |
|
1195 |
by (induct k arbitrary: p, auto simp add: shift1_degree) |
|
1196 |
||
1197 |
lemma funpow_shift1_nz: "p \<noteq> 0\<^sub>p \<Longrightarrow> funpow n shift1 p \<noteq> 0\<^sub>p" |
|
1198 |
by (induct n arbitrary: p, simp_all add: funpow_def) |
|
1199 |
||
1200 |
lemma head_isnpolyh_Suc[simp]: "isnpolyh p (Suc n) \<Longrightarrow> head p = p" |
|
1201 |
by (induct p arbitrary: n rule: degree.induct, auto) |
|
1202 |
lemma headn_0[simp]: "isnpolyh p n \<Longrightarrow> m < n \<Longrightarrow> headn p m = p" |
|
1203 |
by (induct p arbitrary: n rule: degreen.induct, auto) |
|
1204 |
lemma head_isnpolyh_Suc': "n > 0 \<Longrightarrow> isnpolyh p n \<Longrightarrow> head p = p" |
|
1205 |
by (induct p arbitrary: n rule: degree.induct, auto) |
|
1206 |
lemma head_head[simp]: "isnpolyh p n0 \<Longrightarrow> head (head p) = head p" |
|
1207 |
by (induct p rule: head.induct, auto) |
|
1208 |
||
1209 |
lemma polyadd_eq_const_degree: |
|
1210 |
"\<lbrakk> isnpolyh p n0 ; isnpolyh q n1 ; polyadd (p,q) = C c\<rbrakk> \<Longrightarrow> degree p = degree q" |
|
1211 |
using polyadd_eq_const_degreen degree_eq_degreen0 by simp |
|
1212 |
||
1213 |
lemma polyadd_head: assumes np: "isnpolyh p n0" and nq: "isnpolyh q n1" |
|
1214 |
and deg: "degree p \<noteq> degree q" |
|
1215 |
shows "head (p +\<^sub>p q) = (if degree p < degree q then head q else head p)" |
|
1216 |
using np nq deg |
|
1217 |
apply(induct p q arbitrary: n0 n1 rule: polyadd.induct,simp_all) |
|
1218 |
apply (case_tac n', simp, simp) |
|
1219 |
apply (case_tac n, simp, simp) |
|
1220 |
apply (case_tac n, case_tac n', simp add: Let_def) |
|
1221 |
apply (case_tac "pa +\<^sub>p p' = 0\<^sub>p") |
|
1222 |
apply (clarsimp simp add: polyadd_eq_const_degree) |
|
1223 |
apply clarsimp |
|
1224 |
apply (erule_tac impE,blast) |
|
1225 |
apply (erule_tac impE,blast) |
|
1226 |
apply clarsimp |
|
1227 |
apply simp |
|
1228 |
apply (case_tac n', simp_all) |
|
1229 |
done |
|
1230 |
||
1231 |
lemma polymul_head_polyeq: |
|
1232 |
assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})" |
|
1233 |
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" |
|
1234 |
proof (induct p q arbitrary: n0 n1 rule: polymul.induct) |
|
1235 |
case (2 a b c' n' p' n0 n1) |
|
1236 |
hence "isnpolyh (head (CN c' n' p')) n1" "isnormNum (a,b)" by (simp_all add: head_isnpolyh) |
|
1237 |
thus ?case using prems by (cases n', auto) |
|
1238 |
next |
|
1239 |
case (3 c n p a' b' n0 n1) |
|
1240 |
hence "isnpolyh (head (CN c n p)) n0" "isnormNum (a',b')" by (simp_all add: head_isnpolyh) |
|
1241 |
thus ?case using prems by (cases n, auto) |
|
1242 |
next |
|
1243 |
case (4 c n p c' n' p' n0 n1) |
|
1244 |
hence norm: "isnpolyh p n" "isnpolyh c (Suc n)" "isnpolyh p' n'" "isnpolyh c' (Suc n')" |
|
1245 |
"isnpolyh (CN c n p) n" "isnpolyh (CN c' n' p') n'" |
|
1246 |
by simp_all |
|
1247 |
have "n < n' \<or> n' < n \<or> n = n'" by arith |
|
1248 |
moreover |
|
1249 |
{assume nn': "n < n'" hence ?case |
|
1250 |
thm prems |
|
1251 |
using norm |
|
1252 |
prems(6)[rule_format, OF nn' norm(1,6)] |
|
1253 |
prems(7)[rule_format, OF nn' norm(2,6)] by (simp, cases n, simp,cases n', simp_all)} |
|
1254 |
moreover {assume nn': "n'< n" |
|
1255 |
hence stupid: "n' < n \<and> \<not> n < n'" by simp |
|
1256 |
hence ?case using norm prems(4) [rule_format, OF stupid norm(5,3)] |
|
1257 |
prems(5)[rule_format, OF stupid norm(5,4)] |
|
1258 |
by (simp,cases n',simp,cases n,auto)} |
|
1259 |
moreover {assume nn': "n' = n" |
|
1260 |
hence stupid: "\<not> n' < n \<and> \<not> n < n'" by simp |
|
1261 |
from nn' polymul_normh[OF norm(5,4)] |
|
1262 |
have ncnpc':"isnpolyh (CN c n p *\<^sub>p c') n" by (simp add: min_def) |
|
1263 |
from nn' polymul_normh[OF norm(5,3)] norm |
|
1264 |
have ncnpp':"isnpolyh (CN c n p *\<^sub>p p') n" by simp |
|
1265 |
from nn' ncnpp' polymul_eq0_iff[OF norm(5,3)] norm(6) |
|
1266 |
have ncnpp0':"isnpolyh (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n" by simp |
|
1267 |
from polyadd_normh[OF ncnpc' ncnpp0'] |
|
1268 |
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" |
|
1269 |
by (simp add: min_def) |
|
1270 |
{assume np: "n > 0" |
|
1271 |
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
|
1272 |
head_isnpolyh_Suc'[OF np norm(5)] head_isnpolyh_Suc'[OF np norm(6)[simplified nn']] |
33154 | 1273 |
have ?case by simp} |
1274 |
moreover |
|
1275 |
{moreover assume nz: "n = 0" |
|
1276 |
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
|
1277 |
polymul_degreen[OF norm(5,3), where m="0"] nn' nz degree_eq_degreen0 |
33154 | 1278 |
norm(5,6) degree_npolyhCN[OF norm(6)] |
1279 |
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 |
|
1280 |
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 |
|
1281 |
from polyadd_head[OF ncnpc'[simplified nz] ncnpp0'[simplified nz] dth'] dth |
|
1282 |
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
|
1283 |
prems(3)[rule_format, OF stupid norm(5,4)] nn' nz by simp } |
33154 | 1284 |
ultimately have ?case by (cases n) auto} |
1285 |
ultimately show ?case by blast |
|
1286 |
qed simp_all |
|
1287 |
||
1288 |
lemma degree_coefficients: "degree p = length (coefficients p) - 1" |
|
1289 |
by(induct p rule: degree.induct, auto) |
|
1290 |
||
1291 |
lemma degree_head[simp]: "degree (head p) = 0" |
|
1292 |
by (induct p rule: head.induct, auto) |
|
1293 |
||
1294 |
lemma degree_CN: "isnpolyh p n \<Longrightarrow> degree (CN c n p) \<le> 1+ degree p" |
|
1295 |
by (cases n, simp_all) |
|
1296 |
lemma degree_CN': "isnpolyh p n \<Longrightarrow> degree (CN c n p) \<ge> degree p" |
|
1297 |
by (cases n, simp_all) |
|
1298 |
||
1299 |
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)" |
|
1300 |
using polyadd_different_degreen degree_eq_degreen0 by simp |
|
1301 |
||
1302 |
lemma degreen_polyneg: "isnpolyh p n0 \<Longrightarrow> degreen (~\<^sub>p p) m = degreen p m" |
|
1303 |
by (induct p arbitrary: n0 rule: polyneg.induct, auto) |
|
1304 |
||
1305 |
lemma degree_polymul: |
|
1306 |
assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})" |
|
1307 |
and np: "isnpolyh p n0" and nq: "isnpolyh q n1" |
|
1308 |
shows "degree (p *\<^sub>p q) \<le> degree p + degree q" |
|
1309 |
using polymul_degreen[OF np nq, where m="0"] degree_eq_degreen0 by simp |
|
1310 |
||
1311 |
lemma polyneg_degree: "isnpolyh p n \<Longrightarrow> degree (polyneg p) = degree p" |
|
1312 |
by (induct p arbitrary: n rule: degree.induct, auto) |
|
1313 |
||
1314 |
lemma polyneg_head: "isnpolyh p n \<Longrightarrow> head(polyneg p) = polyneg (head p)" |
|
1315 |
by (induct p arbitrary: n rule: degree.induct, auto) |
|
1316 |
||
1317 |
subsection {* Correctness of polynomial pseudo division *} |
|
1318 |
||
1319 |
lemma polydivide_aux_real_domintros: |
|
1320 |
assumes call1: "\<lbrakk>s \<noteq> 0\<^sub>p; \<not> degree s < n; a = head s\<rbrakk> |
|
1321 |
\<Longrightarrow> polydivide_aux_dom (a, n, p, k, s -\<^sub>p funpow (degree s - n) shift1 p)" |
|
1322 |
and call2 : "\<lbrakk>s \<noteq> 0\<^sub>p; \<not> degree s < n; a \<noteq> head s\<rbrakk> |
|
1323 |
\<Longrightarrow> polydivide_aux_dom(a, n, p,Suc k, a *\<^sub>p s -\<^sub>p (head s *\<^sub>p funpow (degree s - n) shift1 p))" |
|
1324 |
shows "polydivide_aux_dom (a, n, p, k, s)" |
|
1325 |
proof (rule accpI, erule polydivide_aux_rel.cases) |
|
1326 |
fix y aa ka na pa sa x xa xb |
|
1327 |
assume eqs: "y = (aa, na, pa, ka, sa -\<^sub>p xb)" "(a, n, p, k, s) = (aa, na, pa, ka, sa)" |
|
1328 |
and \<Gamma>1': "sa \<noteq> 0\<^sub>p" "x = head sa" "xa = degree sa" "\<not> xa < na" |
|
1329 |
"xb = funpow (xa - na) shift1 pa" "aa = x" |
|
1330 |
||
1331 |
hence \<Gamma>1: "s \<noteq> 0\<^sub>p" "a = head s" "xa = degree s" "\<not> degree s < n" "\<not> xa < na" |
|
1332 |
"xb = funpow (xa - na) shift1 pa" "aa = x" by auto |
|
1333 |
||
1334 |
with call1 have "polydivide_aux_dom (a, n, p, k, s -\<^sub>p funpow (degree s - n) shift1 p)" |
|
1335 |
by auto |
|
1336 |
with eqs \<Gamma>1 show "polydivide_aux_dom y" by auto |
|
1337 |
next |
|
1338 |
fix y aa ka na pa sa x xa xb |
|
1339 |
assume eqs: "y = (aa, na, pa, Suc ka, aa *\<^sub>p sa -\<^sub>p (x *\<^sub>p xb))" |
|
1340 |
"(a, n, p, k, s) =(aa, na, pa, ka, sa)" |
|
1341 |
and \<Gamma>2': "sa \<noteq> 0\<^sub>p" "x = head sa" "xa = degree sa" "\<not> xa < na" |
|
1342 |
"xb = funpow (xa - na) shift1 pa" "aa \<noteq> x" |
|
1343 |
hence \<Gamma>2: "s \<noteq> 0\<^sub>p" "a \<noteq> head s" "xa = degree s" "\<not> degree s < n" |
|
1344 |
"xb = funpow (xa - na) shift1 pa" "aa \<noteq> x" by auto |
|
1345 |
with call2 have "polydivide_aux_dom (a, n, p, Suc k, a *\<^sub>p s -\<^sub>p (head s *\<^sub>p funpow (degree s - n) shift1 p))" by auto |
|
1346 |
with eqs \<Gamma>2' show "polydivide_aux_dom y" by auto |
|
1347 |
qed |
|
1348 |
||
1349 |
lemma polydivide_aux_properties: |
|
1350 |
assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})" |
|
1351 |
and np: "isnpolyh p n0" and ns: "isnpolyh s n1" |
|
1352 |
and ap: "head p = a" and ndp: "degree p = n" and pnz: "p \<noteq> 0\<^sub>p" |
|
1353 |
shows "polydivide_aux_dom (a,n,p,k,s) \<and> |
|
1354 |
(polydivide_aux (a,n,p,k,s) = (k',r) \<longrightarrow> (k' \<ge> k) \<and> (degree r = 0 \<or> degree r < degree p) |
|
1355 |
\<and> (\<exists>nr. isnpolyh r nr) \<and> (\<exists>q n1. isnpolyh q n1 \<and> ((polypow (k' - k) a) *\<^sub>p s = p *\<^sub>p q +\<^sub>p r)))" |
|
1356 |
using ns |
|
34915 | 1357 |
proof(induct "degree s" arbitrary: s k k' r n1 rule: less_induct) |
1358 |
case less |
|
33154 | 1359 |
let ?D = "polydivide_aux_dom" |
1360 |
let ?dths = "?D (a, n, p, k, s)" |
|
1361 |
let ?qths = "\<exists>q n1. isnpolyh q n1 \<and> (a ^\<^sub>p (k' - k) *\<^sub>p s = p *\<^sub>p q +\<^sub>p r)" |
|
1362 |
let ?qrths = "polydivide_aux (a, n, p, k, s) = (k', r) \<longrightarrow> k \<le> k' \<and> (degree r = 0 \<or> degree r < degree p) |
|
1363 |
\<and> (\<exists>nr. isnpolyh r nr) \<and> ?qths" |
|
1364 |
let ?ths = "?dths \<and> ?qrths" |
|
1365 |
let ?b = "head s" |
|
34915 | 1366 |
let ?p' = "funpow (degree s - n) shift1 p" |
1367 |
let ?xdn = "funpow (degree s - n) shift1 1\<^sub>p" |
|
33154 | 1368 |
let ?akk' = "a ^\<^sub>p (k' - k)" |
34915 | 1369 |
note ns = `isnpolyh s n1` |
33154 | 1370 |
from np have np0: "isnpolyh p 0" |
1371 |
using isnpolyh_mono[where n="n0" and n'="0" and p="p"] by simp |
|
34915 | 1372 |
have np': "isnpolyh ?p' 0" using funpow_shift1_isnpoly[OF np0[simplified isnpoly_def[symmetric]] pnz, where n="degree s - n"] isnpoly_def by simp |
33154 | 1373 |
have headp': "head ?p' = head p" using funpow_shift1_head[OF np pnz] by simp |
1374 |
from funpow_shift1_isnpoly[where p="1\<^sub>p"] have nxdn: "isnpolyh ?xdn 0" by (simp add: isnpoly_def) |
|
1375 |
from polypow_normh [OF head_isnpolyh[OF np0], where k="k' - k"] ap |
|
1376 |
have nakk':"isnpolyh ?akk' 0" by blast |
|
1377 |
{assume sz: "s = 0\<^sub>p" |
|
1378 |
hence dom: ?dths apply - apply (rule polydivide_aux_real_domintros) apply simp_all done |
|
1379 |
from polydivide_aux.psimps[OF dom] sz |
|
1380 |
have ?qrths using np apply clarsimp by (rule exI[where x="0\<^sub>p"], simp) |
|
1381 |
hence ?ths using dom by blast} |
|
1382 |
moreover |
|
1383 |
{assume sz: "s \<noteq> 0\<^sub>p" |
|
34915 | 1384 |
{assume dn: "degree s < n" |
1385 |
with sz have dom:"?dths" by - (rule polydivide_aux_real_domintros,simp_all) |
|
1386 |
from polydivide_aux.psimps[OF dom] sz dn |
|
33154 | 1387 |
have "?qrths" using ns ndp np by auto (rule exI[where x="0\<^sub>p"],simp) |
1388 |
with dom have ?ths by blast} |
|
1389 |
moreover |
|
34915 | 1390 |
{assume dn': "\<not> degree s < n" hence dn: "degree s \<ge> n" by arith |
33154 | 1391 |
have degsp': "degree s = degree ?p'" |
34915 | 1392 |
using dn ndp funpow_shift1_degree[where k = "degree s - n" and p="p"] by simp |
33154 | 1393 |
{assume ba: "?b = a" |
33268
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1394 |
hence headsp': "head s = head ?p'" using ap headp' by simp |
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1395 |
have nr: "isnpolyh (s -\<^sub>p ?p') 0" using polysub_normh[OF ns np'] by simp |
34915 | 1396 |
from degree_polysub_samehead[OF ns np' headsp' degsp'] |
1397 |
have "degree (s -\<^sub>p ?p') < degree s \<or> s -\<^sub>p ?p' = 0\<^sub>p" by simp |
|
33268
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1398 |
moreover |
34915 | 1399 |
{assume deglt:"degree (s -\<^sub>p ?p') < degree s" |
1400 |
from less(1)[OF deglt nr] |
|
33268
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1401 |
have domsp: "?D (a, n, p, k, s -\<^sub>p ?p')" by blast |
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1402 |
have dom: ?dths apply (rule polydivide_aux_real_domintros) |
34915 | 1403 |
using ba dn' domsp by simp_all |
1404 |
from polydivide_aux.psimps[OF dom] sz dn' ba |
|
33268
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1405 |
have eq: "polydivide_aux (a,n,p,k,s) = polydivide_aux (a,n,p,k, s -\<^sub>p ?p')" |
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1406 |
by (simp add: Let_def) |
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1407 |
{assume h1: "polydivide_aux (a, n, p, k, s) = (k', r)" |
34915 | 1408 |
from less(1)[OF deglt nr, of k k' r] |
33268
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1409 |
trans[OF eq[symmetric] h1] |
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1410 |
have kk': "k \<le> k'" and nr:"\<exists>nr. isnpolyh r nr" and dr: "degree r = 0 \<or> degree r < degree p" |
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1411 |
and q1:"\<exists>q nq. isnpolyh q nq \<and> (a ^\<^sub>p k' - k *\<^sub>p (s -\<^sub>p ?p') = p *\<^sub>p q +\<^sub>p r)" by auto |
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1412 |
from q1 obtain q n1 where nq: "isnpolyh q n1" |
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1413 |
and asp:"a^\<^sub>p (k' - k) *\<^sub>p (s -\<^sub>p ?p') = p *\<^sub>p q +\<^sub>p r" by blast |
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1414 |
from nr obtain nr where nr': "isnpolyh r nr" by blast |
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1415 |
from polymul_normh[OF nakk' ns] have nakks': "isnpolyh (a ^\<^sub>p (k' - k) *\<^sub>p s) 0" by simp |
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1416 |
from polyadd_normh[OF polymul_normh[OF nakk' nxdn] nq] |
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1417 |
have nq': "isnpolyh (?akk' *\<^sub>p ?xdn +\<^sub>p q) 0" by simp |
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1418 |
from polyadd_normh[OF polymul_normh[OF np |
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1419 |
polyadd_normh[OF polymul_normh[OF nakk' nxdn] nq]] nr'] |
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1420 |
have nqr': "isnpolyh (p *\<^sub>p (?akk' *\<^sub>p ?xdn +\<^sub>p q) +\<^sub>p r) 0" by simp |
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1421 |
from asp have "\<forall> (bs:: 'a::{ring_char_0,division_by_zero,field} list). Ipoly bs (a^\<^sub>p (k' - k) *\<^sub>p (s -\<^sub>p ?p')) = |
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1422 |
Ipoly bs (p *\<^sub>p q +\<^sub>p r)" by simp |
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1423 |
hence " \<forall>(bs:: 'a::{ring_char_0,division_by_zero,field} list). Ipoly bs (a^\<^sub>p (k' - k)*\<^sub>p s) = |
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1424 |
Ipoly bs (a^\<^sub>p (k' - k)) * Ipoly bs ?p' + Ipoly bs p * Ipoly bs q + Ipoly bs r" |
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1425 |
by (simp add: ring_simps) |
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1426 |
hence " \<forall>(bs:: 'a::{ring_char_0,division_by_zero,field} list). Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) = |
34915 | 1427 |
Ipoly bs (a^\<^sub>p (k' - k)) * Ipoly bs (funpow (degree s - n) shift1 1\<^sub>p *\<^sub>p p) |
33268
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1428 |
+ Ipoly bs p * Ipoly bs q + Ipoly bs r" |
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1429 |
by (auto simp only: funpow_shift1_1) |
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1430 |
hence "\<forall>(bs:: 'a::{ring_char_0,division_by_zero,field} list). Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) = |
34915 | 1431 |
Ipoly bs p * (Ipoly bs (a^\<^sub>p (k' - k)) * Ipoly bs (funpow (degree s - n) shift1 1\<^sub>p) |
33268
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1432 |
+ Ipoly bs q) + Ipoly bs r" by (simp add: ring_simps) |
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1433 |
hence "\<forall>(bs:: 'a::{ring_char_0,division_by_zero,field} list). Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) = |
34915 | 1434 |
Ipoly bs (p *\<^sub>p ((a^\<^sub>p (k' - k)) *\<^sub>p (funpow (degree s - n) shift1 1\<^sub>p) +\<^sub>p q) +\<^sub>p r)" by simp |
33268
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1435 |
with isnpolyh_unique[OF nakks' nqr'] |
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1436 |
have "a ^\<^sub>p (k' - k) *\<^sub>p s = |
34915 | 1437 |
p *\<^sub>p ((a^\<^sub>p (k' - k)) *\<^sub>p (funpow (degree s - n) shift1 1\<^sub>p) +\<^sub>p q) +\<^sub>p r" by blast |
33268
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1438 |
hence ?qths using nq' |
34915 | 1439 |
apply (rule_tac x="(a^\<^sub>p (k' - k)) *\<^sub>p (funpow (degree s - n) shift1 1\<^sub>p) +\<^sub>p q" in exI) |
33268
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1440 |
apply (rule_tac x="0" in exI) by simp |
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1441 |
with kk' nr dr have "k \<le> k' \<and> (degree r = 0 \<or> degree r < degree p) \<and> (\<exists>nr. isnpolyh r nr) \<and> ?qths" |
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1442 |
by blast } hence ?qrths by blast |
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1443 |
with dom have ?ths by blast} |
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1444 |
moreover |
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1445 |
{assume spz:"s -\<^sub>p ?p' = 0\<^sub>p" |
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1446 |
hence domsp: "?D (a, n, p, k, s -\<^sub>p ?p')" |
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1447 |
apply (simp) by (rule polydivide_aux_real_domintros, simp_all) |
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1448 |
have dom: ?dths apply (rule polydivide_aux_real_domintros) |
34915 | 1449 |
using ba dn' domsp by simp_all |
33268
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1450 |
from spz isnpolyh_unique[OF polysub_normh[OF ns np'], where q="0\<^sub>p", symmetric, where ?'a = "'a::{ring_char_0,division_by_zero,field}"] |
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1451 |
have " \<forall>(bs:: 'a::{ring_char_0,division_by_zero,field} list). Ipoly bs s = Ipoly bs ?p'" by simp |
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1452 |
hence "\<forall>(bs:: 'a::{ring_char_0,division_by_zero,field} list). Ipoly bs s = Ipoly bs (?xdn *\<^sub>p p)" using np nxdn apply simp |
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1453 |
by (simp only: funpow_shift1_1) simp |
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1454 |
hence sp': "s = ?xdn *\<^sub>p p" using isnpolyh_unique[OF ns polymul_normh[OF nxdn np]] by blast |
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1455 |
{assume h1: "polydivide_aux (a,n,p,k,s) = (k',r)" |
34915 | 1456 |
from polydivide_aux.psimps[OF dom] sz dn' ba |
33268
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1457 |
have eq: "polydivide_aux (a,n,p,k,s) = polydivide_aux (a,n,p,k, s -\<^sub>p ?p')" |
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1458 |
by (simp add: Let_def) |
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1459 |
also have "\<dots> = (k,0\<^sub>p)" using polydivide_aux.psimps[OF domsp] spz by simp |
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1460 |
finally have "(k',r) = (k,0\<^sub>p)" using h1 by simp |
34915 | 1461 |
with sp'[symmetric] ns np nxdn polyadd_0(1)[OF polymul_normh[OF np nxdn]] |
33268
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1462 |
polyadd_0(2)[OF polymul_normh[OF np nxdn]] have ?qrths |
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1463 |
apply auto |
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1464 |
apply (rule exI[where x="?xdn"]) |
34915 | 1465 |
apply (auto simp add: polymul_commute[of p]) |
33268
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1466 |
done} |
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1467 |
with dom have ?ths by blast} |
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1468 |
ultimately have ?ths by blast } |
33154 | 1469 |
moreover |
1470 |
{assume ba: "?b \<noteq> a" |
|
33268
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1471 |
from polysub_normh[OF polymul_normh[OF head_isnpolyh[OF np0, simplified ap] ns] |
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1472 |
polymul_normh[OF head_isnpolyh[OF ns] np']] |
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1473 |
have nth: "isnpolyh ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) 0" by(simp add: min_def) |
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1474 |
have nzths: "a *\<^sub>p s \<noteq> 0\<^sub>p" "?b *\<^sub>p ?p' \<noteq> 0\<^sub>p" |
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1475 |
using polymul_eq0_iff[OF head_isnpolyh[OF np0, simplified ap] ns] |
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1476 |
polymul_eq0_iff[OF head_isnpolyh[OF ns] np']head_nz[OF np0] ap pnz sz head_nz[OF ns] |
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1477 |
funpow_shift1_nz[OF pnz] by simp_all |
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1478 |
from polymul_head_polyeq[OF head_isnpolyh[OF np] ns] head_nz[OF np] sz ap head_head[OF np] pnz |
34915 | 1479 |
polymul_head_polyeq[OF head_isnpolyh[OF ns] np'] head_nz [OF ns] sz funpow_shift1_nz[OF pnz, where n="degree s - n"] |
33268
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1480 |
have hdth: "head (a *\<^sub>p s) = head (?b *\<^sub>p ?p')" |
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1481 |
using head_head[OF ns] funpow_shift1_head[OF np pnz] |
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1482 |
polymul_commute[OF head_isnpolyh[OF np] head_isnpolyh[OF ns]] |
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1483 |
by (simp add: ap) |
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1484 |
from polymul_degreen[OF head_isnpolyh[OF np] ns, where m="0"] |
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1485 |
head_nz[OF np] pnz sz ap[symmetric] |
34915 | 1486 |
funpow_shift1_nz[OF pnz, where n="degree s - n"] |
33268
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1487 |
polymul_degreen[OF head_isnpolyh[OF ns] np', where m="0"] head_nz[OF ns] |
34915 | 1488 |
ndp dn |
33268
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1489 |
have degth: "degree (a *\<^sub>p s) = degree (?b *\<^sub>p ?p') " |
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1490 |
by (simp add: degree_eq_degreen0[symmetric] funpow_shift1_degree) |
34915 | 1491 |
{assume dth: "degree ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) < degree s" |
33268
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1492 |
have th: "?D (a, n, p, Suc k, (a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p'))" |
34915 | 1493 |
using less(1)[OF dth nth] by blast |
1494 |
have dom: ?dths using ba dn' th |
|
1495 |
by - (rule polydivide_aux_real_domintros, simp_all) |
|
33268
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1496 |
from polysub_normh[OF polymul_normh[OF head_isnpolyh[OF np] ns] polymul_normh[OF head_isnpolyh[OF ns]np']] |
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1497 |
ap have nasbp': "isnpolyh ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) 0" by simp |
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1498 |
{assume h1:"polydivide_aux (a,n,p,k,s) = (k', r)" |
34915 | 1499 |
from h1 polydivide_aux.psimps[OF dom] sz dn' ba |
33268
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1500 |
have eq:"polydivide_aux (a,n,p,Suc k,(a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) = (k',r)" |
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1501 |
by (simp add: Let_def) |
34915 | 1502 |
with less(1)[OF dth nasbp', of "Suc k" k' r] |
33268
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1503 |
obtain q nq nr where kk': "Suc k \<le> k'" and nr: "isnpolyh r nr" and nq: "isnpolyh q nq" |
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1504 |
and dr: "degree r = 0 \<or> degree r < degree p" |
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1505 |
and qr: "a ^\<^sub>p (k' - Suc k) *\<^sub>p ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) = p *\<^sub>p q +\<^sub>p r" by auto |
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1506 |
from kk' have kk'':"Suc (k' - Suc k) = k' - k" by arith |
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1507 |
{fix bs:: "'a::{ring_char_0,division_by_zero,field} list" |
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1508 |
|
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1509 |
from qr isnpolyh_unique[OF polypow_normh[OF head_isnpolyh[OF np], where k="k' - Suc k", simplified ap] nasbp', symmetric] |
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1510 |
have "Ipoly bs (a ^\<^sub>p (k' - Suc k) *\<^sub>p ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p'))) = Ipoly bs (p *\<^sub>p q +\<^sub>p r)" by simp |
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1511 |
hence "Ipoly bs a ^ (Suc (k' - Suc k)) * Ipoly bs s = Ipoly bs p * Ipoly bs q + Ipoly bs a ^ (k' - Suc k) * Ipoly bs ?b * Ipoly bs ?p' + Ipoly bs r" |
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1512 |
by (simp add: ring_simps power_Suc) |
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1513 |
hence "Ipoly bs a ^ (k' - k) * Ipoly bs s = Ipoly bs p * Ipoly bs q + Ipoly bs a ^ (k' - Suc k) * Ipoly bs ?b * Ipoly bs ?xdn * Ipoly bs p + Ipoly bs r" |
34915 | 1514 |
by (simp add:kk'' funpow_shift1_1[where n="degree s - n" and p="p"]) |
33268
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1515 |
hence "Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) = Ipoly bs p * (Ipoly bs q + Ipoly bs a ^ (k' - Suc k) * Ipoly bs ?b * Ipoly bs ?xdn) + Ipoly bs r" |
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1516 |
by (simp add: ring_simps)} |
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1517 |
hence ieq:"\<forall>(bs :: 'a::{ring_char_0,division_by_zero,field} list). Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) = |
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1518 |
Ipoly bs (p *\<^sub>p (q +\<^sub>p (a ^\<^sub>p (k' - Suc k) *\<^sub>p ?b *\<^sub>p ?xdn)) +\<^sub>p r)" by auto |
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1519 |
let ?q = "q +\<^sub>p (a ^\<^sub>p (k' - Suc k) *\<^sub>p ?b *\<^sub>p ?xdn)" |
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1520 |
from polyadd_normh[OF nq polymul_normh[OF polymul_normh[OF polypow_normh[OF head_isnpolyh[OF np], where k="k' - Suc k"] head_isnpolyh[OF ns], simplified ap ] nxdn]] |
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1521 |
have nqw: "isnpolyh ?q 0" by simp |
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1522 |
from ieq isnpolyh_unique[OF polymul_normh[OF polypow_normh[OF head_isnpolyh[OF np], where k="k' - k"] ns, simplified ap] polyadd_normh[OF polymul_normh[OF np nqw] nr]] |
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1523 |
have asth: "(a ^\<^sub>p (k' - k) *\<^sub>p s) = p *\<^sub>p (q +\<^sub>p (a ^\<^sub>p (k' - Suc k) *\<^sub>p ?b *\<^sub>p ?xdn)) +\<^sub>p r" by blast |
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1524 |
from dr kk' nr h1 asth nqw have ?qrths apply simp |
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1525 |
apply (rule conjI) |
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1526 |
apply (rule exI[where x="nr"], simp) |
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1527 |
apply (rule exI[where x="(q +\<^sub>p (a ^\<^sub>p (k' - Suc k) *\<^sub>p ?b *\<^sub>p ?xdn))"], simp) |
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1528 |
apply (rule exI[where x="0"], simp) |
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1529 |
done} |
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1530 |
hence ?qrths by blast |
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1531 |
with dom have ?ths by blast} |
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1532 |
moreover |
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1533 |
{assume spz: "a *\<^sub>p s -\<^sub>p (?b *\<^sub>p ?p') = 0\<^sub>p" |
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1534 |
hence domsp: "?D (a, n, p, Suc k, a *\<^sub>p s -\<^sub>p (?b *\<^sub>p ?p'))" |
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1535 |
apply (simp) by (rule polydivide_aux_real_domintros, simp_all) |
34915 | 1536 |
have dom: ?dths using sz ba dn' domsp |
33268
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1537 |
by - (rule polydivide_aux_real_domintros, simp_all) |
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1538 |
{fix bs :: "'a::{ring_char_0,division_by_zero,field} list" |
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1539 |
from isnpolyh_unique[OF nth, where ?'a="'a" and q="0\<^sub>p",simplified,symmetric] spz |
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1540 |
have "Ipoly bs (a*\<^sub>p s) = Ipoly bs ?b * Ipoly bs ?p'" by simp |
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1541 |
hence "Ipoly bs (a*\<^sub>p s) = Ipoly bs (?b *\<^sub>p ?xdn) * Ipoly bs p" |
34915 | 1542 |
by (simp add: funpow_shift1_1[where n="degree s - n" and p="p"]) |
33268
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1543 |
hence "Ipoly bs (a*\<^sub>p s) = Ipoly bs (p *\<^sub>p (?b *\<^sub>p ?xdn))" by simp |
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1544 |
} |
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1545 |
hence hth: "\<forall> (bs:: 'a::{ring_char_0,division_by_zero,field} list). Ipoly bs (a*\<^sub>p s) = Ipoly bs (p *\<^sub>p (?b *\<^sub>p ?xdn))" .. |
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1546 |
from hth |
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1547 |
have asq: "a *\<^sub>p s = p *\<^sub>p (?b *\<^sub>p ?xdn)" |
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1548 |
using isnpolyh_unique[where ?'a = "'a::{ring_char_0,division_by_zero,field}", OF polymul_normh[OF head_isnpolyh[OF np] ns] |
33154 | 1549 |
polymul_normh[OF np polymul_normh[OF head_isnpolyh[OF ns] nxdn]], |
33268
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1550 |
simplified ap] by simp |
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1551 |
{assume h1: "polydivide_aux (a,n,p,k,s) = (k', r)" |
34915 | 1552 |
from h1 sz ba dn' spz polydivide_aux.psimps[OF dom] polydivide_aux.psimps[OF domsp] |
33268
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1553 |
have "(k',r) = (Suc k, 0\<^sub>p)" by (simp add: Let_def) |
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1554 |
with h1 np head_isnpolyh[OF np, simplified ap] ns polymul_normh[OF head_isnpolyh[OF ns] nxdn] |
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1555 |
polymul_normh[OF np polymul_normh[OF head_isnpolyh[OF ns] nxdn]] asq |
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1556 |
have ?qrths apply (clarsimp simp add: Let_def) |
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1557 |
apply (rule exI[where x="?b *\<^sub>p ?xdn"]) apply simp |
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1558 |
apply (rule exI[where x="0"], simp) |
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1559 |
done} |
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1560 |
hence ?qrths by blast |
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1561 |
with dom have ?ths by blast} |
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1562 |
ultimately have ?ths using degree_polysub_samehead[OF polymul_normh[OF head_isnpolyh[OF np0, simplified ap] ns] polymul_normh[OF head_isnpolyh[OF ns] np'] hdth degth] polymul_degreen[OF head_isnpolyh[OF np] ns, where m="0"] |
34915 | 1563 |
head_nz[OF np] pnz sz ap[symmetric] |
33268
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1564 |
by (simp add: degree_eq_degreen0[symmetric]) blast } |
33154 | 1565 |
ultimately have ?ths by blast |
1566 |
} |
|
1567 |
ultimately have ?ths by blast} |
|
1568 |
ultimately show ?ths by blast |
|
1569 |
qed |
|
1570 |
||
1571 |
lemma polydivide_properties: |
|
1572 |
assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})" |
|
1573 |
and np: "isnpolyh p n0" and ns: "isnpolyh s n1" and pnz: "p \<noteq> 0\<^sub>p" |
|
1574 |
shows "(\<exists> k r. polydivide s p = (k,r) \<and> (\<exists>nr. isnpolyh r nr) \<and> (degree r = 0 \<or> degree r < degree p) |
|
1575 |
\<and> (\<exists>q n1. isnpolyh q n1 \<and> ((polypow k (head p)) *\<^sub>p s = p *\<^sub>p q +\<^sub>p r)))" |
|
1576 |
proof- |
|
1577 |
have trv: "head p = head p" "degree p = degree p" by simp_all |
|
1578 |
from polydivide_aux_properties[OF np ns trv pnz, where k="0"] |
|
1579 |
have d: "polydivide_aux_dom (head p, degree p, p, 0, s)" by blast |
|
1580 |
from polydivide_def[where s="s" and p="p"] polydivide_aux.psimps[OF d] |
|
1581 |
have ex: "\<exists> k r. polydivide s p = (k,r)" by auto |
|
1582 |
then obtain k r where kr: "polydivide s p = (k,r)" by blast |
|
1583 |
from trans[OF meta_eq_to_obj_eq[OF polydivide_def[where s="s" and p="p"], symmetric] kr] |
|
1584 |
polydivide_aux_properties[OF np ns trv pnz, where k="0" and k'="k" and r="r"] |
|
1585 |
have "(degree r = 0 \<or> degree r < degree p) \<and> |
|
1586 |
(\<exists>nr. isnpolyh r nr) \<and> (\<exists>q n1. isnpolyh q n1 \<and> head p ^\<^sub>p k - 0 *\<^sub>p s = p *\<^sub>p q +\<^sub>p r)" by blast |
|
1587 |
with kr show ?thesis |
|
1588 |
apply - |
|
1589 |
apply (rule exI[where x="k"]) |
|
1590 |
apply (rule exI[where x="r"]) |
|
1591 |
apply simp |
|
1592 |
done |
|
1593 |
qed |
|
1594 |
||
1595 |
subsection{* More about polypoly and pnormal etc *} |
|
1596 |
||
1597 |
definition "isnonconstant p = (\<not> isconstant p)" |
|
1598 |
||
1599 |
lemma last_map: "xs \<noteq> [] ==> last (map f xs) = f (last xs)" by (induct xs, auto) |
|
1600 |
||
1601 |
lemma isnonconstant_pnormal_iff: assumes nc: "isnonconstant p" |
|
1602 |
shows "pnormal (polypoly bs p) \<longleftrightarrow> Ipoly bs (head p) \<noteq> 0" |
|
1603 |
proof |
|
1604 |
let ?p = "polypoly bs p" |
|
1605 |
assume H: "pnormal ?p" |
|
1606 |
have csz: "coefficients p \<noteq> []" using nc by (cases p, auto) |
|
1607 |
||
1608 |
from coefficients_head[of p] last_map[OF csz, of "Ipoly bs"] |
|
1609 |
pnormal_last_nonzero[OF H] |
|
1610 |
show "Ipoly bs (head p) \<noteq> 0" by (simp add: polypoly_def) |
|
1611 |
next |
|
1612 |
assume h: "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0" |
|
1613 |
let ?p = "polypoly bs p" |
|
1614 |
have csz: "coefficients p \<noteq> []" using nc by (cases p, auto) |
|
1615 |
hence pz: "?p \<noteq> []" by (simp add: polypoly_def) |
|
1616 |
hence lg: "length ?p > 0" by simp |
|
1617 |
from h coefficients_head[of p] last_map[OF csz, of "Ipoly bs"] |
|
1618 |
have lz: "last ?p \<noteq> 0" by (simp add: polypoly_def) |
|
1619 |
from pnormal_last_length[OF lg lz] show "pnormal ?p" . |
|
1620 |
qed |
|
1621 |
||
1622 |
lemma isnonconstant_coefficients_length: "isnonconstant p \<Longrightarrow> length (coefficients p) > 1" |
|
1623 |
unfolding isnonconstant_def |
|
1624 |
apply (cases p, simp_all) |
|
1625 |
apply (case_tac nat, auto) |
|
1626 |
done |
|
1627 |
lemma isnonconstant_nonconstant: assumes inc: "isnonconstant p" |
|
1628 |
shows "nonconstant (polypoly bs p) \<longleftrightarrow> Ipoly bs (head p) \<noteq> 0" |
|
1629 |
proof |
|
1630 |
let ?p = "polypoly bs p" |
|
1631 |
assume nc: "nonconstant ?p" |
|
1632 |
from isnonconstant_pnormal_iff[OF inc, of bs] nc |
|
1633 |
show "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0" unfolding nonconstant_def by blast |
|
1634 |
next |
|
1635 |
let ?p = "polypoly bs p" |
|
1636 |
assume h: "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0" |
|
1637 |
from isnonconstant_pnormal_iff[OF inc, of bs] h |
|
1638 |
have pn: "pnormal ?p" by blast |
|
1639 |
{fix x assume H: "?p = [x]" |
|
1640 |
from H have "length (coefficients p) = 1" unfolding polypoly_def by auto |
|
1641 |
with isnonconstant_coefficients_length[OF inc] have False by arith} |
|
1642 |
thus "nonconstant ?p" using pn unfolding nonconstant_def by blast |
|
1643 |
qed |
|
1644 |
||
1645 |
lemma pnormal_length: "p\<noteq>[] \<Longrightarrow> pnormal p \<longleftrightarrow> length (pnormalize p) = length p" |
|
1646 |
unfolding pnormal_def |
|
1647 |
apply (induct p rule: pnormalize.induct, simp_all) |
|
1648 |
apply (case_tac "p=[]", simp_all) |
|
1649 |
done |
|
1650 |
||
1651 |
lemma degree_degree: assumes inc: "isnonconstant p" |
|
1652 |
shows "degree p = Polynomial_List.degree (polypoly bs p) \<longleftrightarrow> \<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0" |
|
1653 |
proof |
|
1654 |
let ?p = "polypoly bs p" |
|
1655 |
assume H: "degree p = Polynomial_List.degree ?p" |
|
1656 |
from isnonconstant_coefficients_length[OF inc] have pz: "?p \<noteq> []" |
|
1657 |
unfolding polypoly_def by auto |
|
1658 |
from H degree_coefficients[of p] isnonconstant_coefficients_length[OF inc] |
|
1659 |
have lg:"length (pnormalize ?p) = length ?p" |
|
1660 |
unfolding Polynomial_List.degree_def polypoly_def by simp |
|
1661 |
hence "pnormal ?p" using pnormal_length[OF pz] by blast |
|
1662 |
with isnonconstant_pnormal_iff[OF inc] |
|
1663 |
show "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0" by blast |
|
1664 |
next |
|
1665 |
let ?p = "polypoly bs p" |
|
1666 |
assume H: "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0" |
|
1667 |
with isnonconstant_pnormal_iff[OF inc] have "pnormal ?p" by blast |
|
1668 |
with degree_coefficients[of p] isnonconstant_coefficients_length[OF inc] |
|
1669 |
show "degree p = Polynomial_List.degree ?p" |
|
1670 |
unfolding polypoly_def pnormal_def Polynomial_List.degree_def by auto |
|
1671 |
qed |
|
1672 |
||
1673 |
section{* Swaps ; Division by a certain variable *} |
|
1674 |
consts swap:: "nat \<Rightarrow> nat \<Rightarrow> poly \<Rightarrow> poly" |
|
1675 |
primrec |
|
1676 |
"swap n m (C x) = C x" |
|
1677 |
"swap n m (Bound k) = Bound (if k = n then m else if k=m then n else k)" |
|
1678 |
"swap n m (Neg t) = Neg (swap n m t)" |
|
1679 |
"swap n m (Add s t) = Add (swap n m s) (swap n m t)" |
|
1680 |
"swap n m (Sub s t) = Sub (swap n m s) (swap n m t)" |
|
1681 |
"swap n m (Mul s t) = Mul (swap n m s) (swap n m t)" |
|
1682 |
"swap n m (Pw t k) = Pw (swap n m t) k" |
|
1683 |
"swap n m (CN c k p) = CN (swap n m c) (if k = n then m else if k=m then n else k) |
|
1684 |
(swap n m p)" |
|
1685 |
||
1686 |
lemma swap: assumes nbs: "n < length bs" and mbs: "m < length bs" |
|
1687 |
shows "Ipoly bs (swap n m t) = Ipoly ((bs[n:= bs!m])[m:= bs!n]) t" |
|
1688 |
proof (induct t) |
|
1689 |
case (Bound k) thus ?case using nbs mbs by simp |
|
1690 |
next |
|
1691 |
case (CN c k p) thus ?case using nbs mbs by simp |
|
1692 |
qed simp_all |
|
1693 |
lemma swap_swap_id[simp]: "swap n m (swap m n t) = t" |
|
1694 |
by (induct t,simp_all) |
|
1695 |
||
1696 |
lemma swap_commute: "swap n m p = swap m n p" by (induct p, simp_all) |
|
1697 |
||
1698 |
lemma swap_same_id[simp]: "swap n n t = t" |
|
1699 |
by (induct t, simp_all) |
|
1700 |
||
1701 |
definition "swapnorm n m t = polynate (swap n m t)" |
|
1702 |
||
1703 |
lemma swapnorm: assumes nbs: "n < length bs" and mbs: "m < length bs" |
|
1704 |
shows "((Ipoly bs (swapnorm n m t) :: 'a\<Colon>{ring_char_0,division_by_zero,field})) = Ipoly ((bs[n:= bs!m])[m:= bs!n]) t" |
|
1705 |
using swap[OF prems] swapnorm_def by simp |
|
1706 |
||
1707 |
lemma swapnorm_isnpoly[simp]: |
|
1708 |
assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})" |
|
1709 |
shows "isnpoly (swapnorm n m p)" |
|
1710 |
unfolding swapnorm_def by simp |
|
1711 |
||
1712 |
definition "polydivideby n s p = |
|
1713 |
(let ss = swapnorm 0 n s ; sp = swapnorm 0 n p ; h = head sp; (k,r) = polydivide ss sp |
|
1714 |
in (k,swapnorm 0 n h,swapnorm 0 n r))" |
|
1715 |
||
1716 |
lemma swap_nz [simp]: " (swap n m p = 0\<^sub>p) = (p = 0\<^sub>p)" by (induct p, simp_all) |
|
1717 |
||
1718 |
consts isweaknpoly :: "poly \<Rightarrow> bool" |
|
1719 |
recdef isweaknpoly "measure size" |
|
1720 |
"isweaknpoly (C c) = True" |
|
1721 |
"isweaknpoly (CN c n p) \<longleftrightarrow> isweaknpoly c \<and> isweaknpoly p" |
|
33268
02de0317f66f
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
33154
diff
changeset
|
1722 |
"isweaknpoly p = False" |
33154 | 1723 |
|
1724 |
lemma isnpolyh_isweaknpoly: "isnpolyh p n0 \<Longrightarrow> isweaknpoly p" |
|
1725 |
by (induct p arbitrary: n0, auto) |
|
1726 |
||
1727 |
lemma swap_isweanpoly: "isweaknpoly p \<Longrightarrow> isweaknpoly (swap n m p)" |
|
1728 |
by (induct p, auto) |
|
1729 |
||
1730 |
end |