author | hoelzl |
Thu, 17 Jan 2013 11:59:12 +0100 | |
changeset 50936 | b28f258ebc1a |
parent 49962 | a8cc904a6820 |
child 53191 | 14ab2f821e1d |
permissions | -rw-r--r-- |
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(* Title: HOL/Library/Univ_Poly.thy |
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Author: Amine Chaieb |
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A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
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*) |
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A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
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|
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header {* Univariate Polynomials *} |
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A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
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A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
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theory Univ_Poly |
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imports Main |
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A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
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begin |
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A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
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|
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A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
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text{* Application of polynomial as a function. *} |
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A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
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primrec (in semiring_0) poly :: "'a list => 'a => 'a" where |
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A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
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poly_Nil: "poly [] x = 0" |
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A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
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| poly_Cons: "poly (h#t) x = h + x * poly t x" |
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A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
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A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
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A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
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subsection{*Arithmetic Operations on Polynomials*} |
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A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
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A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
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text{*addition*} |
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A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
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primrec (in semiring_0) padd :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list" (infixl "+++" 65) |
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A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
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where |
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A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
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padd_Nil: "[] +++ l2 = l2" |
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A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
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| padd_Cons: "(h#t) +++ l2 = (if l2 = [] then h#t |
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A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
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else (h + hd l2)#(t +++ tl l2))" |
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A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
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A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
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text{*Multiplication by a constant*} |
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primrec (in semiring_0) cmult :: "'a \<Rightarrow> 'a list \<Rightarrow> 'a list" (infixl "%*" 70) where |
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A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
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cmult_Nil: "c %* [] = []" |
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A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
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| cmult_Cons: "c %* (h#t) = (c * h)#(c %* t)" |
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A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
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|
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A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
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text{*Multiplication by a polynomial*} |
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primrec (in semiring_0) pmult :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list" (infixl "***" 70) |
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A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
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where |
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A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
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pmult_Nil: "[] *** l2 = []" |
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A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
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| pmult_Cons: "(h#t) *** l2 = (if t = [] then h %* l2 |
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A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
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else (h %* l2) +++ ((0) # (t *** l2)))" |
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A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
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|
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A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
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text{*Repeated multiplication by a polynomial*} |
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primrec (in semiring_0) mulexp :: "nat \<Rightarrow> 'a list \<Rightarrow> 'a list \<Rightarrow> 'a list" where |
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A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
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mulexp_zero: "mulexp 0 p q = q" |
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A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
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| mulexp_Suc: "mulexp (Suc n) p q = p *** mulexp n p q" |
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A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
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A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
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text{*Exponential*} |
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primrec (in semiring_1) pexp :: "'a list \<Rightarrow> nat \<Rightarrow> 'a list" (infixl "%^" 80) where |
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A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
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pexp_0: "p %^ 0 = [1]" |
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A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
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| pexp_Suc: "p %^ (Suc n) = p *** (p %^ n)" |
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A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
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|
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A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
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text{*Quotient related value of dividing a polynomial by x + a*} |
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A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
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(* Useful for divisor properties in inductive proofs *) |
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primrec (in field) "pquot" :: "'a list \<Rightarrow> 'a \<Rightarrow> 'a list" where |
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A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
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pquot_Nil: "pquot [] a= []" |
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A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
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| pquot_Cons: "pquot (h#t) a = (if t = [] then [h] |
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A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
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else (inverse(a) * (h - hd( pquot t a)))#(pquot t a))" |
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A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
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A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
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text{*normalization of polynomials (remove extra 0 coeff)*} |
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primrec (in semiring_0) pnormalize :: "'a list \<Rightarrow> 'a list" where |
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A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
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pnormalize_Nil: "pnormalize [] = []" |
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A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
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| pnormalize_Cons: "pnormalize (h#p) = (if ( (pnormalize p) = []) |
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A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
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then (if (h = 0) then [] else [h]) |
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A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
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else (h#(pnormalize p)))" |
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A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
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A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
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definition (in semiring_0) "pnormal p = ((pnormalize p = p) \<and> p \<noteq> [])" |
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A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
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definition (in semiring_0) "nonconstant p = (pnormal p \<and> (\<forall>x. p \<noteq> [x]))" |
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A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
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text{*Other definitions*} |
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A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
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|
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A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
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definition (in ring_1) |
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A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
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poly_minus :: "'a list => 'a list" ("-- _" [80] 80) where |
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A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
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"-- p = (- 1) %* p" |
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A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
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|
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A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
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definition (in semiring_0) |
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A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
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divides :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool" (infixl "divides" 70) where |
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"p1 divides p2 = (\<exists>q. poly p2 = poly(p1 *** q))" |
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A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
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|
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A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
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--{*order of a polynomial*} |
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A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
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definition (in ring_1) order :: "'a => 'a list => nat" where |
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A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
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"order a p = (SOME n. ([-a, 1] %^ n) divides p & |
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A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
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~ (([-a, 1] %^ (Suc n)) divides p))" |
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A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
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|
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A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
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--{*degree of a polynomial*} |
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definition (in semiring_0) degree :: "'a list => nat" where |
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A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
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"degree p = length (pnormalize p) - 1" |
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A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
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|
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A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
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--{*squarefree polynomials --- NB with respect to real roots only.*} |
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A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
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definition (in ring_1) |
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A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
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rsquarefree :: "'a list => bool" where |
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A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
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"rsquarefree p = (poly p \<noteq> poly [] & |
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A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
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(\<forall>a. (order a p = 0) | (order a p = 1)))" |
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A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
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|
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A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
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context semiring_0 |
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A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
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begin |
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A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
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A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
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lemma padd_Nil2[simp]: "p +++ [] = p" |
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A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
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by (induct p) auto |
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A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
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A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
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lemma padd_Cons_Cons: "(h1 # p1) +++ (h2 # p2) = (h1 + h2) # (p1 +++ p2)" |
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A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
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by auto |
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A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
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|
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A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
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lemma pminus_Nil[simp]: "-- [] = []" |
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A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
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by (simp add: poly_minus_def) |
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A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
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|
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A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
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lemma pmult_singleton: "[h1] *** p1 = h1 %* p1" by simp |
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A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
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end |
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A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
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|
46730 | 106 |
lemma (in semiring_1) poly_ident_mult[simp]: "1 %* t = t" by (induct t) auto |
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A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
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|
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A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
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lemma (in semiring_0) poly_simple_add_Cons[simp]: "[a] +++ ((0)#t) = (a#t)" |
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A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
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by simp |
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A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
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|
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A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
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111 |
text{*Handy general properties*} |
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A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
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112 |
|
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A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
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lemma (in comm_semiring_0) padd_commut: "b +++ a = a +++ b" |
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A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
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proof(induct b arbitrary: a) |
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A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
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case Nil thus ?case by auto |
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A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
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next |
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A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
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case (Cons b bs a) thus ?case by (cases a, simp_all add: add_commute) |
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A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
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118 |
qed |
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A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
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119 |
|
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A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
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120 |
lemma (in comm_semiring_0) padd_assoc: "\<forall>b c. (a +++ b) +++ c = a +++ (b +++ c)" |
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tuned proofs -- eliminated vacuous "induct arbitrary: ..." situations;
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121 |
apply (induct a) |
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A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
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122 |
apply (simp, clarify) |
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A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
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123 |
apply (case_tac b, simp_all add: add_ac) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
124 |
done |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
125 |
|
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
126 |
lemma (in semiring_0) poly_cmult_distr: "a %* ( p +++ q) = (a %* p +++ a %* q)" |
45129
1fce03e3e8ad
tuned proofs -- eliminated vacuous "induct arbitrary: ..." situations;
wenzelm
parents:
39302
diff
changeset
|
127 |
apply (induct p arbitrary: q, simp) |
49962
a8cc904a6820
Renamed {left,right}_distrib to distrib_{right,left}.
webertj
parents:
49751
diff
changeset
|
128 |
apply (case_tac q, simp_all add: distrib_left) |
26124
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
129 |
done |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
130 |
|
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
131 |
lemma (in ring_1) pmult_by_x[simp]: "[0, 1] *** t = ((0)#t)" |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
132 |
apply (induct "t", simp) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
133 |
apply (auto simp add: mult_zero_left poly_ident_mult padd_commut) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
134 |
apply (case_tac t, auto) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
135 |
done |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
136 |
|
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
137 |
text{*properties of evaluation of polynomials.*} |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
138 |
|
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
139 |
lemma (in semiring_0) poly_add: "poly (p1 +++ p2) x = poly p1 x + poly p2 x" |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
140 |
proof(induct p1 arbitrary: p2) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
141 |
case Nil thus ?case by simp |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
142 |
next |
30488 | 143 |
case (Cons a as p2) thus ?case |
49962
a8cc904a6820
Renamed {left,right}_distrib to distrib_{right,left}.
webertj
parents:
49751
diff
changeset
|
144 |
by (cases p2, simp_all add: add_ac distrib_left) |
26124
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
145 |
qed |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
146 |
|
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
147 |
lemma (in comm_semiring_0) poly_cmult: "poly (c %* p) x = c * poly p x" |
30488 | 148 |
apply (induct "p") |
26124
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
149 |
apply (case_tac [2] "x=zero") |
49962
a8cc904a6820
Renamed {left,right}_distrib to distrib_{right,left}.
webertj
parents:
49751
diff
changeset
|
150 |
apply (auto simp add: distrib_left mult_ac) |
26124
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
151 |
done |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
152 |
|
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
153 |
lemma (in comm_semiring_0) poly_cmult_map: "poly (map (op * c) p) x = c*poly p x" |
49962
a8cc904a6820
Renamed {left,right}_distrib to distrib_{right,left}.
webertj
parents:
49751
diff
changeset
|
154 |
by (induct p, auto simp add: distrib_left mult_ac) |
26124
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
155 |
|
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
156 |
lemma (in comm_ring_1) poly_minus: "poly (-- p) x = - (poly p x)" |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
157 |
apply (simp add: poly_minus_def) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
158 |
apply (auto simp add: poly_cmult minus_mult_left[symmetric]) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
159 |
done |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
160 |
|
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
161 |
lemma (in comm_semiring_0) poly_mult: "poly (p1 *** p2) x = poly p1 x * poly p2 x" |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
162 |
proof(induct p1 arbitrary: p2) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
163 |
case Nil thus ?case by simp |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
164 |
next |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
165 |
case (Cons a as p2) |
30488 | 166 |
thus ?case by (cases as, |
49962
a8cc904a6820
Renamed {left,right}_distrib to distrib_{right,left}.
webertj
parents:
49751
diff
changeset
|
167 |
simp_all add: poly_cmult poly_add distrib_right distrib_left mult_ac) |
26124
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
168 |
qed |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
169 |
|
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
170 |
class idom_char_0 = idom + ring_char_0 |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
171 |
|
31021 | 172 |
lemma (in comm_ring_1) poly_exp: "poly (p %^ n) x = (poly p x) ^ n" |
26124
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
173 |
apply (induct "n") |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
174 |
apply (auto simp add: poly_cmult poly_mult power_Suc) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
175 |
done |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
176 |
|
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
177 |
text{*More Polynomial Evaluation Lemmas*} |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
178 |
|
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
179 |
lemma (in semiring_0) poly_add_rzero[simp]: "poly (a +++ []) x = poly a x" |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
180 |
by simp |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
181 |
|
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
182 |
lemma (in comm_semiring_0) poly_mult_assoc: "poly ((a *** b) *** c) x = poly (a *** (b *** c)) x" |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
183 |
by (simp add: poly_mult mult_assoc) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
184 |
|
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
185 |
lemma (in semiring_0) poly_mult_Nil2[simp]: "poly (p *** []) x = 0" |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
186 |
by (induct "p", auto) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
187 |
|
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
188 |
lemma (in comm_semiring_1) poly_exp_add: "poly (p %^ (n + d)) x = poly( p %^ n *** p %^ d) x" |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
189 |
apply (induct "n") |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
190 |
apply (auto simp add: poly_mult mult_assoc) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
191 |
done |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
192 |
|
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
193 |
subsection{*Key Property: if @{term "f(a) = 0"} then @{term "(x - a)"} divides |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
194 |
@{term "p(x)"} *} |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
195 |
|
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
196 |
lemma (in comm_ring_1) lemma_poly_linear_rem: "\<forall>h. \<exists>q r. h#t = [r] +++ [-a, 1] *** q" |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
197 |
proof(induct t) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
198 |
case Nil |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
199 |
{fix h have "[h] = [h] +++ [- a, 1] *** []" by simp} |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
200 |
thus ?case by blast |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
201 |
next |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
202 |
case (Cons x xs) |
30488 | 203 |
{fix h |
204 |
from Cons.hyps[rule_format, of x] |
|
26124
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
205 |
obtain q r where qr: "x#xs = [r] +++ [- a, 1] *** q" by blast |
30488 | 206 |
have "h#x#xs = [a*r + h] +++ [-a, 1] *** (r#q)" |
37887 | 207 |
using qr by(cases q, simp_all add: algebra_simps diff_minus[symmetric] |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32456
diff
changeset
|
208 |
minus_mult_left[symmetric] right_minus) |
26124
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
209 |
hence "\<exists>q r. h#x#xs = [r] +++ [-a, 1] *** q" by blast} |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
210 |
thus ?case by blast |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
211 |
qed |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
212 |
|
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
213 |
lemma (in comm_ring_1) poly_linear_rem: "\<exists>q r. h#t = [r] +++ [-a, 1] *** q" |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
214 |
by (cut_tac t = t and a = a in lemma_poly_linear_rem, auto) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
215 |
|
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
216 |
|
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
217 |
lemma (in comm_ring_1) poly_linear_divides: "(poly p a = 0) = ((p = []) | (\<exists>q. p = [-a, 1] *** q))" |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
218 |
proof- |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
219 |
{assume p: "p = []" hence ?thesis by simp} |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
220 |
moreover |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
221 |
{fix x xs assume p: "p = x#xs" |
30488 | 222 |
{fix q assume "p = [-a, 1] *** q" hence "poly p a = 0" |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32456
diff
changeset
|
223 |
by (simp add: poly_add poly_cmult minus_mult_left[symmetric])} |
26124
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
224 |
moreover |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
225 |
{assume p0: "poly p a = 0" |
30488 | 226 |
from poly_linear_rem[of x xs a] obtain q r |
26124
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
227 |
where qr: "x#xs = [r] +++ [- a, 1] *** q" by blast |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
228 |
have "r = 0" using p0 by (simp only: p qr poly_mult poly_add) simp |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
229 |
hence "\<exists>q. p = [- a, 1] *** q" using p qr apply - apply (rule exI[where x=q])apply auto apply (cases q) apply auto done} |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
230 |
ultimately have ?thesis using p by blast} |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
231 |
ultimately show ?thesis by (cases p, auto) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
232 |
qed |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
233 |
|
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
234 |
lemma (in semiring_0) lemma_poly_length_mult[simp]: "\<forall>h k a. length (k %* p +++ (h # (a %* p))) = Suc (length p)" |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
235 |
by (induct "p", auto) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
236 |
|
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
237 |
lemma (in semiring_0) lemma_poly_length_mult2[simp]: "\<forall>h k. length (k %* p +++ (h # p)) = Suc (length p)" |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
238 |
by (induct "p", auto) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
239 |
|
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
240 |
lemma (in ring_1) poly_length_mult[simp]: "length([-a,1] *** q) = Suc (length q)" |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
241 |
by auto |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
242 |
|
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
243 |
subsection{*Polynomial length*} |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
244 |
|
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
245 |
lemma (in semiring_0) poly_cmult_length[simp]: "length (a %* p) = length p" |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
246 |
by (induct "p", auto) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
247 |
|
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
248 |
lemma (in semiring_0) poly_add_length: "length (p1 +++ p2) = max (length p1) (length p2)" |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
249 |
apply (induct p1 arbitrary: p2, simp_all) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
250 |
apply arith |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
251 |
done |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
252 |
|
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
253 |
lemma (in semiring_0) poly_root_mult_length[simp]: "length([a,b] *** p) = Suc (length p)" |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
254 |
by (simp add: poly_add_length) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
255 |
|
30488 | 256 |
lemma (in idom) poly_mult_not_eq_poly_Nil[simp]: |
26124
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
257 |
"poly (p *** q) x \<noteq> poly [] x \<longleftrightarrow> poly p x \<noteq> poly [] x \<and> poly q x \<noteq> poly [] x" |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
258 |
by (auto simp add: poly_mult) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
259 |
|
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
260 |
lemma (in idom) poly_mult_eq_zero_disj: "poly (p *** q) x = 0 \<longleftrightarrow> poly p x = 0 \<or> poly q x = 0" |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
261 |
by (auto simp add: poly_mult) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
262 |
|
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
263 |
text{*Normalisation Properties*} |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
264 |
|
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
265 |
lemma (in semiring_0) poly_normalized_nil: "(pnormalize p = []) --> (poly p x = 0)" |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
266 |
by (induct "p", auto) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
267 |
|
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
268 |
text{*A nontrivial polynomial of degree n has no more than n roots*} |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
269 |
lemma (in idom) poly_roots_index_lemma: |
30488 | 270 |
assumes p: "poly p x \<noteq> poly [] x" and n: "length p = n" |
26124
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
271 |
shows "\<exists>i. \<forall>x. poly p x = 0 \<longrightarrow> (\<exists>m\<le>n. x = i m)" |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
272 |
using p n |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
273 |
proof(induct n arbitrary: p x) |
30488 | 274 |
case 0 thus ?case by simp |
26124
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
275 |
next |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
276 |
case (Suc n p x) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
277 |
{assume C: "\<And>i. \<exists>x. poly p x = 0 \<and> (\<forall>m\<le>Suc n. x \<noteq> i m)" |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
278 |
from Suc.prems have p0: "poly p x \<noteq> 0" "p\<noteq> []" by auto |
30488 | 279 |
from p0(1)[unfolded poly_linear_divides[of p x]] |
26124
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
280 |
have "\<forall>q. p \<noteq> [- x, 1] *** q" by blast |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
281 |
from C obtain a where a: "poly p a = 0" by blast |
30488 | 282 |
from a[unfolded poly_linear_divides[of p a]] p0(2) |
26124
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
283 |
obtain q where q: "p = [-a, 1] *** q" by blast |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
284 |
have lg: "length q = n" using q Suc.prems(2) by simp |
30488 | 285 |
from q p0 have qx: "poly q x \<noteq> poly [] x" |
26124
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
286 |
by (auto simp add: poly_mult poly_add poly_cmult) |
30488 | 287 |
from Suc.hyps[OF qx lg] obtain i where |
26124
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
288 |
i: "\<forall>x. poly q x = 0 \<longrightarrow> (\<exists>m\<le>n. x = i m)" by blast |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
289 |
let ?i = "\<lambda>m. if m = Suc n then a else i m" |
30488 | 290 |
from C[of ?i] obtain y where y: "poly p y = 0" "\<forall>m\<le> Suc n. y \<noteq> ?i m" |
26124
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
291 |
by blast |
30488 | 292 |
from y have "y = a \<or> poly q y = 0" |
29667 | 293 |
by (simp only: q poly_mult_eq_zero_disj poly_add) (simp add: algebra_simps) |
30488 | 294 |
with i[rule_format, of y] y(1) y(2) have False apply auto |
26124
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
295 |
apply (erule_tac x="m" in allE) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
296 |
apply auto |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
297 |
done} |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
298 |
thus ?case by blast |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
299 |
qed |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
300 |
|
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
301 |
|
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
302 |
lemma (in idom) poly_roots_index_length: "poly p x \<noteq> poly [] x ==> |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
303 |
\<exists>i. \<forall>x. (poly p x = 0) --> (\<exists>n. n \<le> length p & x = i n)" |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
304 |
by (blast intro: poly_roots_index_lemma) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
305 |
|
26313 | 306 |
lemma (in idom) poly_roots_finite_lemma1: "poly p x \<noteq> poly [] x ==> |
26124
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
307 |
\<exists>N i. \<forall>x. (poly p x = 0) --> (\<exists>n. (n::nat) < N & x = i n)" |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
308 |
apply (drule poly_roots_index_length, safe) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
309 |
apply (rule_tac x = "Suc (length p)" in exI) |
30488 | 310 |
apply (rule_tac x = i in exI) |
26124
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
311 |
apply (simp add: less_Suc_eq_le) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
312 |
done |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
313 |
|
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
314 |
|
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
315 |
lemma (in idom) idom_finite_lemma: |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
316 |
assumes P: "\<forall>x. P x --> (\<exists>n. n < length j & x = j!n)" |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
317 |
shows "finite {x. P x}" |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
318 |
proof- |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
319 |
let ?M = "{x. P x}" |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
320 |
let ?N = "set j" |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
321 |
have "?M \<subseteq> ?N" using P by auto |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
322 |
thus ?thesis using finite_subset by auto |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
323 |
qed |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
324 |
|
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
325 |
|
26313 | 326 |
lemma (in idom) poly_roots_finite_lemma2: "poly p x \<noteq> poly [] x ==> |
26124
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
327 |
\<exists>i. \<forall>x. (poly p x = 0) --> x \<in> set i" |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
328 |
apply (drule poly_roots_index_length, safe) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
329 |
apply (rule_tac x="map (\<lambda>n. i n) [0 ..< Suc (length p)]" in exI) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
330 |
apply (auto simp add: image_iff) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
331 |
apply (erule_tac x="x" in allE, clarsimp) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
332 |
by (case_tac "n=length p", auto simp add: order_le_less) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
333 |
|
30488 | 334 |
lemma (in ring_char_0) UNIV_ring_char_0_infinte: |
335 |
"\<not> (finite (UNIV:: 'a set))" |
|
26124
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
336 |
proof |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
337 |
assume F: "finite (UNIV :: 'a set)" |
29292 | 338 |
have "finite (UNIV :: nat set)" |
339 |
proof (rule finite_imageD) |
|
340 |
have "of_nat ` UNIV \<subseteq> UNIV" by simp |
|
341 |
then show "finite (of_nat ` UNIV :: 'a set)" using F by (rule finite_subset) |
|
342 |
show "inj (of_nat :: nat \<Rightarrow> 'a)" by (simp add: inj_on_def) |
|
343 |
qed |
|
29879 | 344 |
with infinite_UNIV_nat show False .. |
26124
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
345 |
qed |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
346 |
|
30488 | 347 |
lemma (in idom_char_0) poly_roots_finite: "(poly p \<noteq> poly []) = |
26124
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
348 |
finite {x. poly p x = 0}" |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
349 |
proof |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
350 |
assume H: "poly p \<noteq> poly []" |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
351 |
show "finite {x. poly p x = (0::'a)}" |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
352 |
using H |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
353 |
apply - |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
354 |
apply (erule contrapos_np, rule ext) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
355 |
apply (rule ccontr) |
26313 | 356 |
apply (clarify dest!: poly_roots_finite_lemma2) |
26124
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
357 |
using finite_subset |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
358 |
proof- |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
359 |
fix x i |
30488 | 360 |
assume F: "\<not> finite {x. poly p x = (0\<Colon>'a)}" |
26124
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
361 |
and P: "\<forall>x. poly p x = (0\<Colon>'a) \<longrightarrow> x \<in> set i" |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
362 |
let ?M= "{x. poly p x = (0\<Colon>'a)}" |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
363 |
from P have "?M \<subseteq> set i" by auto |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
364 |
with finite_subset F show False by auto |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
365 |
qed |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
366 |
next |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
367 |
assume F: "finite {x. poly p x = (0\<Colon>'a)}" |
30488 | 368 |
show "poly p \<noteq> poly []" using F UNIV_ring_char_0_infinte by auto |
26124
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
369 |
qed |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
370 |
|
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
371 |
text{*Entirety and Cancellation for polynomials*} |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
372 |
|
30488 | 373 |
lemma (in idom_char_0) poly_entire_lemma2: |
26313 | 374 |
assumes p0: "poly p \<noteq> poly []" and q0: "poly q \<noteq> poly []" |
375 |
shows "poly (p***q) \<noteq> poly []" |
|
376 |
proof- |
|
377 |
let ?S = "\<lambda>p. {x. poly p x = 0}" |
|
378 |
have "?S (p *** q) = ?S p \<union> ?S q" by (auto simp add: poly_mult) |
|
379 |
with p0 q0 show ?thesis unfolding poly_roots_finite by auto |
|
380 |
qed |
|
26124
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
381 |
|
30488 | 382 |
lemma (in idom_char_0) poly_entire: |
26313 | 383 |
"poly (p *** q) = poly [] \<longleftrightarrow> poly p = poly [] \<or> poly q = poly []" |
29667 | 384 |
using poly_entire_lemma2[of p q] |
39302
d7728f65b353
renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents:
39198
diff
changeset
|
385 |
by (auto simp add: fun_eq_iff poly_mult) |
26124
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
386 |
|
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
387 |
lemma (in idom_char_0) poly_entire_neg: "(poly (p *** q) \<noteq> poly []) = ((poly p \<noteq> poly []) & (poly q \<noteq> poly []))" |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
388 |
by (simp add: poly_entire) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
389 |
|
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
390 |
lemma fun_eq: " (f = g) = (\<forall>x. f x = g x)" |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
391 |
by (auto intro!: ext) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
392 |
|
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
393 |
lemma (in comm_ring_1) poly_add_minus_zero_iff: "(poly (p +++ -- q) = poly []) = (poly p = poly q)" |
29667 | 394 |
by (auto simp add: algebra_simps poly_add poly_minus_def fun_eq poly_cmult minus_mult_left[symmetric]) |
26124
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
395 |
|
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
396 |
lemma (in comm_ring_1) poly_add_minus_mult_eq: "poly (p *** q +++ --(p *** r)) = poly (p *** (q +++ -- r))" |
49962
a8cc904a6820
Renamed {left,right}_distrib to distrib_{right,left}.
webertj
parents:
49751
diff
changeset
|
397 |
by (auto simp add: poly_add poly_minus_def fun_eq poly_mult poly_cmult distrib_left minus_mult_left[symmetric] minus_mult_right[symmetric]) |
26124
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
398 |
|
28823 | 399 |
subclass (in idom_char_0) comm_ring_1 .. |
26124
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
400 |
lemma (in idom_char_0) poly_mult_left_cancel: "(poly (p *** q) = poly (p *** r)) = (poly p = poly [] | poly q = poly r)" |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
401 |
proof- |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
402 |
have "poly (p *** q) = poly (p *** r) \<longleftrightarrow> poly (p *** q +++ -- (p *** r)) = poly []" by (simp only: poly_add_minus_zero_iff) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
403 |
also have "\<dots> \<longleftrightarrow> poly p = poly [] | poly q = poly r" |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
404 |
by (auto intro: ext simp add: poly_add_minus_mult_eq poly_entire poly_add_minus_zero_iff) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
405 |
finally show ?thesis . |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
406 |
qed |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
407 |
|
31021 | 408 |
lemma (in idom) poly_exp_eq_zero[simp]: |
26124
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
409 |
"(poly (p %^ n) = poly []) = (poly p = poly [] & n \<noteq> 0)" |
37598 | 410 |
apply (simp only: fun_eq add: HOL.all_simps [symmetric]) |
30488 | 411 |
apply (rule arg_cong [where f = All]) |
26124
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
412 |
apply (rule ext) |
26194 | 413 |
apply (induct n) |
26124
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
414 |
apply (auto simp add: poly_exp poly_mult) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
415 |
done |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
416 |
|
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
417 |
lemma (in comm_ring_1) poly_prime_eq_zero[simp]: "poly [a,1] \<noteq> poly []" |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
418 |
apply (simp add: fun_eq) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
419 |
apply (rule_tac x = "minus one a" in exI) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
420 |
apply (unfold diff_minus) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
421 |
apply (subst add_commute) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
422 |
apply (subst add_assoc) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
423 |
apply simp |
30488 | 424 |
done |
26124
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
425 |
|
31021 | 426 |
lemma (in idom) poly_exp_prime_eq_zero: "(poly ([a, 1] %^ n) \<noteq> poly [])" |
26124
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
427 |
by auto |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
428 |
|
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
429 |
text{*A more constructive notion of polynomials being trivial*} |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
430 |
|
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
431 |
lemma (in idom_char_0) poly_zero_lemma': "poly (h # t) = poly [] ==> h = 0 & poly t = poly []" |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
432 |
apply(simp add: fun_eq) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
433 |
apply (case_tac "h = zero") |
30488 | 434 |
apply (drule_tac [2] x = zero in spec, auto) |
435 |
apply (cases "poly t = poly []", simp) |
|
26124
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
436 |
proof- |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
437 |
fix x |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
438 |
assume H: "\<forall>x. x = (0\<Colon>'a) \<or> poly t x = (0\<Colon>'a)" and pnz: "poly t \<noteq> poly []" |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
439 |
let ?S = "{x. poly t x = 0}" |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
440 |
from H have "\<forall>x. x \<noteq>0 \<longrightarrow> poly t x = 0" by blast |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
441 |
hence th: "?S \<supseteq> UNIV - {0}" by auto |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
442 |
from poly_roots_finite pnz have th': "finite ?S" by blast |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
443 |
from finite_subset[OF th th'] UNIV_ring_char_0_infinte |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
444 |
show "poly t x = (0\<Colon>'a)" by simp |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
445 |
qed |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
446 |
|
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
447 |
lemma (in idom_char_0) poly_zero: "(poly p = poly []) = list_all (%c. c = 0) p" |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
448 |
apply (induct "p", simp) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
449 |
apply (rule iffI) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
450 |
apply (drule poly_zero_lemma', auto) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
451 |
done |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
452 |
|
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
453 |
lemma (in idom_char_0) poly_0: "list_all (\<lambda>c. c = 0) p \<Longrightarrow> poly p x = 0" |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
454 |
unfolding poly_zero[symmetric] by simp |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
455 |
|
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
456 |
|
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
457 |
|
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
458 |
text{*Basics of divisibility.*} |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
459 |
|
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
460 |
lemma (in idom) poly_primes: "([a, 1] divides (p *** q)) = ([a, 1] divides p | [a, 1] divides q)" |
49962
a8cc904a6820
Renamed {left,right}_distrib to distrib_{right,left}.
webertj
parents:
49751
diff
changeset
|
461 |
apply (auto simp add: divides_def fun_eq poly_mult poly_add poly_cmult distrib_right [symmetric]) |
26124
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
462 |
apply (drule_tac x = "uminus a" in spec) |
49962
a8cc904a6820
Renamed {left,right}_distrib to distrib_{right,left}.
webertj
parents:
49751
diff
changeset
|
463 |
apply (simp add: poly_linear_divides poly_add poly_cmult distrib_right [symmetric]) |
26124
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
464 |
apply (cases "p = []") |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
465 |
apply (rule exI[where x="[]"]) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
466 |
apply simp |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
467 |
apply (cases "q = []") |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
468 |
apply (erule allE[where x="[]"], simp) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
469 |
|
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
470 |
apply clarsimp |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
471 |
apply (cases "\<exists>q\<Colon>'a list. p = a %* q +++ ((0\<Colon>'a) # q)") |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
472 |
apply (clarsimp simp add: poly_add poly_cmult) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
473 |
apply (rule_tac x="qa" in exI) |
49962
a8cc904a6820
Renamed {left,right}_distrib to distrib_{right,left}.
webertj
parents:
49751
diff
changeset
|
474 |
apply (simp add: distrib_right [symmetric]) |
26124
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
475 |
apply clarsimp |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
476 |
|
49962
a8cc904a6820
Renamed {left,right}_distrib to distrib_{right,left}.
webertj
parents:
49751
diff
changeset
|
477 |
apply (auto simp add: right_minus poly_linear_divides poly_add poly_cmult distrib_right [symmetric]) |
26124
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
478 |
apply (rule_tac x = "pmult qa q" in exI) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
479 |
apply (rule_tac [2] x = "pmult p qa" in exI) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
480 |
apply (auto simp add: poly_add poly_mult poly_cmult mult_ac) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
481 |
done |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
482 |
|
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
483 |
lemma (in comm_semiring_1) poly_divides_refl[simp]: "p divides p" |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
484 |
apply (simp add: divides_def) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
485 |
apply (rule_tac x = "[one]" in exI) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
486 |
apply (auto simp add: poly_mult fun_eq) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
487 |
done |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
488 |
|
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
489 |
lemma (in comm_semiring_1) poly_divides_trans: "[| p divides q; q divides r |] ==> p divides r" |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
490 |
apply (simp add: divides_def, safe) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
491 |
apply (rule_tac x = "pmult qa qaa" in exI) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
492 |
apply (auto simp add: poly_mult fun_eq mult_assoc) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
493 |
done |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
494 |
|
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
495 |
|
31021 | 496 |
lemma (in comm_semiring_1) poly_divides_exp: "m \<le> n ==> (p %^ m) divides (p %^ n)" |
26124
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
497 |
apply (auto simp add: le_iff_add) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
498 |
apply (induct_tac k) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
499 |
apply (rule_tac [2] poly_divides_trans) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
500 |
apply (auto simp add: divides_def) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
501 |
apply (rule_tac x = p in exI) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
502 |
apply (auto simp add: poly_mult fun_eq mult_ac) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
503 |
done |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
504 |
|
31021 | 505 |
lemma (in comm_semiring_1) poly_exp_divides: "[| (p %^ n) divides q; m\<le>n |] ==> (p %^ m) divides q" |
26124
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
506 |
by (blast intro: poly_divides_exp poly_divides_trans) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
507 |
|
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
508 |
lemma (in comm_semiring_0) poly_divides_add: |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
509 |
"[| p divides q; p divides r |] ==> p divides (q +++ r)" |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
510 |
apply (simp add: divides_def, auto) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
511 |
apply (rule_tac x = "padd qa qaa" in exI) |
49962
a8cc904a6820
Renamed {left,right}_distrib to distrib_{right,left}.
webertj
parents:
49751
diff
changeset
|
512 |
apply (auto simp add: poly_add fun_eq poly_mult distrib_left) |
26124
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
513 |
done |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
514 |
|
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
515 |
lemma (in comm_ring_1) poly_divides_diff: |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
516 |
"[| p divides q; p divides (q +++ r) |] ==> p divides r" |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
517 |
apply (simp add: divides_def, auto) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
518 |
apply (rule_tac x = "padd qaa (poly_minus qa)" in exI) |
29667 | 519 |
apply (auto simp add: poly_add fun_eq poly_mult poly_minus algebra_simps) |
26124
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
520 |
done |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
521 |
|
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
522 |
lemma (in comm_ring_1) poly_divides_diff2: "[| p divides r; p divides (q +++ r) |] ==> p divides q" |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
523 |
apply (erule poly_divides_diff) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
524 |
apply (auto simp add: poly_add fun_eq poly_mult divides_def add_ac) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
525 |
done |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
526 |
|
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
527 |
lemma (in semiring_0) poly_divides_zero: "poly p = poly [] ==> q divides p" |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
528 |
apply (simp add: divides_def) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
529 |
apply (rule exI[where x="[]"]) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
530 |
apply (auto simp add: fun_eq poly_mult) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
531 |
done |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
532 |
|
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
533 |
lemma (in semiring_0) poly_divides_zero2[simp]: "q divides []" |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
534 |
apply (simp add: divides_def) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
535 |
apply (rule_tac x = "[]" in exI) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
536 |
apply (auto simp add: fun_eq) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
537 |
done |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
538 |
|
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
539 |
text{*At last, we can consider the order of a root.*} |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
540 |
|
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
541 |
lemma (in idom_char_0) poly_order_exists_lemma: |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
542 |
assumes lp: "length p = d" and p: "poly p \<noteq> poly []" |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
543 |
shows "\<exists>n q. p = mulexp n [-a, 1] q \<and> poly q a \<noteq> 0" |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
544 |
using lp p |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
545 |
proof(induct d arbitrary: p) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
546 |
case 0 thus ?case by simp |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
547 |
next |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
548 |
case (Suc n p) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
549 |
{assume p0: "poly p a = 0" |
29292 | 550 |
from Suc.prems have h: "length p = Suc n" "poly p \<noteq> poly []" by auto |
551 |
hence pN: "p \<noteq> []" by auto |
|
30488 | 552 |
from p0[unfolded poly_linear_divides] pN obtain q where |
26124
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
553 |
q: "p = [-a, 1] *** q" by blast |
30488 | 554 |
from q h p0 have qh: "length q = n" "poly q \<noteq> poly []" |
26124
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
555 |
apply - |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
556 |
apply simp |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
557 |
apply (simp only: fun_eq) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
558 |
apply (rule ccontr) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
559 |
apply (simp add: fun_eq poly_add poly_cmult minus_mult_left[symmetric]) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
560 |
done |
30488 | 561 |
from Suc.hyps[OF qh] obtain m r where |
562 |
mr: "q = mulexp m [-a,1] r" "poly r a \<noteq> 0" by blast |
|
26124
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
563 |
from mr q have "p = mulexp (Suc m) [-a,1] r \<and> poly r a \<noteq> 0" by simp |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
564 |
hence ?case by blast} |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
565 |
moreover |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
566 |
{assume p0: "poly p a \<noteq> 0" |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
567 |
hence ?case using Suc.prems apply simp by (rule exI[where x="0::nat"], simp)} |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
568 |
ultimately show ?case by blast |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
569 |
qed |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
570 |
|
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
571 |
|
31021 | 572 |
lemma (in comm_semiring_1) poly_mulexp: "poly (mulexp n p q) x = (poly p x) ^ n * poly q x" |
26124
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
573 |
by(induct n, auto simp add: poly_mult power_Suc mult_ac) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
574 |
|
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
575 |
lemma (in comm_semiring_1) divides_left_mult: |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
576 |
assumes d:"(p***q) divides r" shows "p divides r \<and> q divides r" |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
577 |
proof- |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
578 |
from d obtain t where r:"poly r = poly (p***q *** t)" |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
579 |
unfolding divides_def by blast |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
580 |
hence "poly r = poly (p *** (q *** t))" |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
581 |
"poly r = poly (q *** (p***t))" by(auto simp add: fun_eq poly_mult mult_ac) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
582 |
thus ?thesis unfolding divides_def by blast |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
583 |
qed |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
584 |
|
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
585 |
|
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
586 |
|
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
587 |
(* FIXME: Tidy up *) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
588 |
|
31021 | 589 |
lemma (in semiring_1) |
26124
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
590 |
zero_power_iff: "0 ^ n = (if n = 0 then 1 else 0)" |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
591 |
by (induct n, simp_all add: power_Suc) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
592 |
|
31021 | 593 |
lemma (in idom_char_0) poly_order_exists: |
26124
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
594 |
assumes lp: "length p = d" and p0: "poly p \<noteq> poly []" |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
595 |
shows "\<exists>n. ([-a, 1] %^ n) divides p & ~(([-a, 1] %^ (Suc n)) divides p)" |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
596 |
proof- |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
597 |
let ?poly = poly |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
598 |
let ?mulexp = mulexp |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
599 |
let ?pexp = pexp |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
600 |
from lp p0 |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
601 |
show ?thesis |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
602 |
apply - |
30488 | 603 |
apply (drule poly_order_exists_lemma [where a=a], assumption, clarify) |
26124
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
604 |
apply (rule_tac x = n in exI, safe) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
605 |
apply (unfold divides_def) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
606 |
apply (rule_tac x = q in exI) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
607 |
apply (induct_tac "n", simp) |
49962
a8cc904a6820
Renamed {left,right}_distrib to distrib_{right,left}.
webertj
parents:
49751
diff
changeset
|
608 |
apply (simp (no_asm_simp) add: poly_add poly_cmult poly_mult distrib_left mult_ac) |
26124
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
609 |
apply safe |
30488 | 610 |
apply (subgoal_tac "?poly (?mulexp n [uminus a, one] q) \<noteq> ?poly (pmult (?pexp [uminus a, one] (Suc n)) qa)") |
611 |
apply simp |
|
26124
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
612 |
apply (induct_tac "n") |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
613 |
apply (simp del: pmult_Cons pexp_Suc) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
614 |
apply (erule_tac Q = "?poly q a = zero" in contrapos_np) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
615 |
apply (simp add: poly_add poly_cmult minus_mult_left[symmetric]) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
616 |
apply (rule pexp_Suc [THEN ssubst]) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
617 |
apply (rule ccontr) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
618 |
apply (simp add: poly_mult_left_cancel poly_mult_assoc del: pmult_Cons pexp_Suc) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
619 |
done |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
620 |
qed |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
621 |
|
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
622 |
|
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
623 |
lemma (in semiring_1) poly_one_divides[simp]: "[1] divides p" |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
624 |
by (simp add: divides_def, auto) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
625 |
|
31021 | 626 |
lemma (in idom_char_0) poly_order: "poly p \<noteq> poly [] |
26124
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
627 |
==> EX! n. ([-a, 1] %^ n) divides p & |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
628 |
~(([-a, 1] %^ (Suc n)) divides p)" |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
629 |
apply (auto intro: poly_order_exists simp add: less_linear simp del: pmult_Cons pexp_Suc) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
630 |
apply (cut_tac x = y and y = n in less_linear) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
631 |
apply (drule_tac m = n in poly_exp_divides) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
632 |
apply (auto dest: Suc_le_eq [THEN iffD2, THEN [2] poly_exp_divides] |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
633 |
simp del: pmult_Cons pexp_Suc) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
634 |
done |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
635 |
|
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
636 |
text{*Order*} |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
637 |
|
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
638 |
lemma some1_equalityD: "[| n = (@n. P n); EX! n. P n |] ==> P n" |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
639 |
by (blast intro: someI2) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
640 |
|
31021 | 641 |
lemma (in idom_char_0) order: |
26124
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
642 |
"(([-a, 1] %^ n) divides p & |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
643 |
~(([-a, 1] %^ (Suc n)) divides p)) = |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
644 |
((n = order a p) & ~(poly p = poly []))" |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
645 |
apply (unfold order_def) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
646 |
apply (rule iffI) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
647 |
apply (blast dest: poly_divides_zero intro!: some1_equality [symmetric] poly_order) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
648 |
apply (blast intro!: poly_order [THEN [2] some1_equalityD]) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
649 |
done |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
650 |
|
31021 | 651 |
lemma (in idom_char_0) order2: "[| poly p \<noteq> poly [] |] |
26124
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
652 |
==> ([-a, 1] %^ (order a p)) divides p & |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
653 |
~(([-a, 1] %^ (Suc(order a p))) divides p)" |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
654 |
by (simp add: order del: pexp_Suc) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
655 |
|
31021 | 656 |
lemma (in idom_char_0) order_unique: "[| poly p \<noteq> poly []; ([-a, 1] %^ n) divides p; |
26124
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
657 |
~(([-a, 1] %^ (Suc n)) divides p) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
658 |
|] ==> (n = order a p)" |
30488 | 659 |
by (insert order [of a n p], auto) |
26124
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
660 |
|
31021 | 661 |
lemma (in idom_char_0) order_unique_lemma: "(poly p \<noteq> poly [] & ([-a, 1] %^ n) divides p & |
26124
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
662 |
~(([-a, 1] %^ (Suc n)) divides p)) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
663 |
==> (n = order a p)" |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
664 |
by (blast intro: order_unique) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
665 |
|
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
666 |
lemma (in ring_1) order_poly: "poly p = poly q ==> order a p = order a q" |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
667 |
by (auto simp add: fun_eq divides_def poly_mult order_def) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
668 |
|
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
669 |
lemma (in semiring_1) pexp_one[simp]: "p %^ (Suc 0) = p" |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
670 |
apply (induct "p") |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
671 |
apply (auto simp add: numeral_1_eq_1) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
672 |
done |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
673 |
|
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
674 |
lemma (in comm_ring_1) lemma_order_root: |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
675 |
" 0 < n & [- a, 1] %^ n divides p & ~ [- a, 1] %^ (Suc n) divides p |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
676 |
\<Longrightarrow> poly p a = 0" |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
677 |
apply (induct n arbitrary: a p, blast) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
678 |
apply (auto simp add: divides_def poly_mult simp del: pmult_Cons) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
679 |
done |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
680 |
|
31021 | 681 |
lemma (in idom_char_0) order_root: "(poly p a = 0) = ((poly p = poly []) | order a p \<noteq> 0)" |
26124
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
682 |
proof- |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
683 |
let ?poly = poly |
30488 | 684 |
show ?thesis |
26124
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
685 |
apply (case_tac "?poly p = ?poly []", auto) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
686 |
apply (simp add: poly_linear_divides del: pmult_Cons, safe) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
687 |
apply (drule_tac [!] a = a in order2) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
688 |
apply (rule ccontr) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
689 |
apply (simp add: divides_def poly_mult fun_eq del: pmult_Cons, blast) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
690 |
using neq0_conv |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
691 |
apply (blast intro: lemma_order_root) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
692 |
done |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
693 |
qed |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
694 |
|
31021 | 695 |
lemma (in idom_char_0) order_divides: "(([-a, 1] %^ n) divides p) = ((poly p = poly []) | n \<le> order a p)" |
26124
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
696 |
proof- |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
697 |
let ?poly = poly |
30488 | 698 |
show ?thesis |
26124
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
699 |
apply (case_tac "?poly p = ?poly []", auto) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
700 |
apply (simp add: divides_def fun_eq poly_mult) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
701 |
apply (rule_tac x = "[]" in exI) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
702 |
apply (auto dest!: order2 [where a=a] |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32456
diff
changeset
|
703 |
intro: poly_exp_divides simp del: pexp_Suc) |
26124
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
704 |
done |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
705 |
qed |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
706 |
|
31021 | 707 |
lemma (in idom_char_0) order_decomp: |
26124
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
708 |
"poly p \<noteq> poly [] |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
709 |
==> \<exists>q. (poly p = poly (([-a, 1] %^ (order a p)) *** q)) & |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
710 |
~([-a, 1] divides q)" |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
711 |
apply (unfold divides_def) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
712 |
apply (drule order2 [where a = a]) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
713 |
apply (simp add: divides_def del: pexp_Suc pmult_Cons, safe) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
714 |
apply (rule_tac x = q in exI, safe) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
715 |
apply (drule_tac x = qa in spec) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
716 |
apply (auto simp add: poly_mult fun_eq poly_exp mult_ac simp del: pmult_Cons) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
717 |
done |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
718 |
|
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
719 |
text{*Important composition properties of orders.*} |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
720 |
lemma order_mult: "poly (p *** q) \<noteq> poly [] |
31021 | 721 |
==> order a (p *** q) = order a p + order (a::'a::{idom_char_0}) q" |
26124
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
722 |
apply (cut_tac a = a and p = "p *** q" and n = "order a p + order a q" in order) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
723 |
apply (auto simp add: poly_entire simp del: pmult_Cons) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
724 |
apply (drule_tac a = a in order2)+ |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
725 |
apply safe |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
726 |
apply (simp add: divides_def fun_eq poly_exp_add poly_mult del: pmult_Cons, safe) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
727 |
apply (rule_tac x = "qa *** qaa" in exI) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
728 |
apply (simp add: poly_mult mult_ac del: pmult_Cons) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
729 |
apply (drule_tac a = a in order_decomp)+ |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
730 |
apply safe |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
731 |
apply (subgoal_tac "[-a,1] divides (qa *** qaa) ") |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
732 |
apply (simp add: poly_primes del: pmult_Cons) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
733 |
apply (auto simp add: divides_def simp del: pmult_Cons) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
734 |
apply (rule_tac x = qb in exI) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
735 |
apply (subgoal_tac "poly ([-a, 1] %^ (order a p) *** (qa *** qaa)) = poly ([-a, 1] %^ (order a p) *** ([-a, 1] *** qb))") |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
736 |
apply (drule poly_mult_left_cancel [THEN iffD1], force) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
737 |
apply (subgoal_tac "poly ([-a, 1] %^ (order a q) *** ([-a, 1] %^ (order a p) *** (qa *** qaa))) = poly ([-a, 1] %^ (order a q) *** ([-a, 1] %^ (order a p) *** ([-a, 1] *** qb))) ") |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
738 |
apply (drule poly_mult_left_cancel [THEN iffD1], force) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
739 |
apply (simp add: fun_eq poly_exp_add poly_mult mult_ac del: pmult_Cons) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
740 |
done |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
741 |
|
31021 | 742 |
lemma (in idom_char_0) order_mult: |
26124
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
743 |
assumes pq0: "poly (p *** q) \<noteq> poly []" |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
744 |
shows "order a (p *** q) = order a p + order a q" |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
745 |
proof- |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
746 |
let ?order = order |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
747 |
let ?divides = "op divides" |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
748 |
let ?poly = poly |
30488 | 749 |
from pq0 |
26124
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
750 |
show ?thesis |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
751 |
apply (cut_tac a = a and p = "pmult p q" and n = "?order a p + ?order a q" in order) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
752 |
apply (auto simp add: poly_entire simp del: pmult_Cons) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
753 |
apply (drule_tac a = a in order2)+ |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
754 |
apply safe |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
755 |
apply (simp add: divides_def fun_eq poly_exp_add poly_mult del: pmult_Cons, safe) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
756 |
apply (rule_tac x = "pmult qa qaa" in exI) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
757 |
apply (simp add: poly_mult mult_ac del: pmult_Cons) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
758 |
apply (drule_tac a = a in order_decomp)+ |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
759 |
apply safe |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
760 |
apply (subgoal_tac "?divides [uminus a,one ] (pmult qa qaa) ") |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
761 |
apply (simp add: poly_primes del: pmult_Cons) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
762 |
apply (auto simp add: divides_def simp del: pmult_Cons) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
763 |
apply (rule_tac x = qb in exI) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
764 |
apply (subgoal_tac "?poly (pmult (pexp [uminus a, one] (?order a p)) (pmult qa qaa)) = ?poly (pmult (pexp [uminus a, one] (?order a p)) (pmult [uminus a, one] qb))") |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
765 |
apply (drule poly_mult_left_cancel [THEN iffD1], force) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
766 |
apply (subgoal_tac "?poly (pmult (pexp [uminus a, one ] (order a q)) (pmult (pexp [uminus a, one] (order a p)) (pmult qa qaa))) = ?poly (pmult (pexp [uminus a, one] (order a q)) (pmult (pexp [uminus a, one] (order a p)) (pmult [uminus a, one] qb))) ") |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
767 |
apply (drule poly_mult_left_cancel [THEN iffD1], force) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
768 |
apply (simp add: fun_eq poly_exp_add poly_mult mult_ac del: pmult_Cons) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
769 |
done |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
770 |
qed |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
771 |
|
31021 | 772 |
lemma (in idom_char_0) order_root2: "poly p \<noteq> poly [] ==> (poly p a = 0) = (order a p \<noteq> 0)" |
26124
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
773 |
by (rule order_root [THEN ssubst], auto) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
774 |
|
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
775 |
lemma (in semiring_1) pmult_one[simp]: "[1] *** p = p" by auto |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
776 |
|
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
777 |
lemma (in semiring_0) poly_Nil_zero: "poly [] = poly [0]" |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
778 |
by (simp add: fun_eq) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
779 |
|
31021 | 780 |
lemma (in idom_char_0) rsquarefree_decomp: |
26124
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
781 |
"[| rsquarefree p; poly p a = 0 |] |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
782 |
==> \<exists>q. (poly p = poly ([-a, 1] *** q)) & poly q a \<noteq> 0" |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
783 |
apply (simp add: rsquarefree_def, safe) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
784 |
apply (frule_tac a = a in order_decomp) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
785 |
apply (drule_tac x = a in spec) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
786 |
apply (drule_tac a = a in order_root2 [symmetric]) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
787 |
apply (auto simp del: pmult_Cons) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
788 |
apply (rule_tac x = q in exI, safe) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
789 |
apply (simp add: poly_mult fun_eq) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
790 |
apply (drule_tac p1 = q in poly_linear_divides [THEN iffD1]) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
791 |
apply (simp add: divides_def del: pmult_Cons, safe) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
792 |
apply (drule_tac x = "[]" in spec) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
793 |
apply (auto simp add: fun_eq) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
794 |
done |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
795 |
|
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
796 |
|
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
797 |
text{*Normalization of a polynomial.*} |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
798 |
|
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
799 |
lemma (in semiring_0) poly_normalize[simp]: "poly (pnormalize p) = poly p" |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
800 |
apply (induct "p") |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
801 |
apply (auto simp add: fun_eq) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
802 |
done |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
803 |
|
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
804 |
text{*The degree of a polynomial.*} |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
805 |
|
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
806 |
lemma (in semiring_0) lemma_degree_zero: |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
807 |
"list_all (%c. c = 0) p \<longleftrightarrow> pnormalize p = []" |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
808 |
by (induct "p", auto) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
809 |
|
30488 | 810 |
lemma (in idom_char_0) degree_zero: |
26124
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
811 |
assumes pN: "poly p = poly []" shows"degree p = 0" |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
812 |
proof- |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
813 |
let ?pn = pnormalize |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
814 |
from pN |
30488 | 815 |
show ?thesis |
26124
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
816 |
apply (simp add: degree_def) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
817 |
apply (case_tac "?pn p = []") |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
818 |
apply (auto simp add: poly_zero lemma_degree_zero ) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
819 |
done |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
820 |
qed |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
821 |
|
32456
341c83339aeb
tuned the simp rules for Int involving insert and intervals.
nipkow
parents:
31021
diff
changeset
|
822 |
lemma (in semiring_0) pnormalize_sing: "(pnormalize [x] = [x]) \<longleftrightarrow> x \<noteq> 0" |
341c83339aeb
tuned the simp rules for Int involving insert and intervals.
nipkow
parents:
31021
diff
changeset
|
823 |
by simp |
26124
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
824 |
lemma (in semiring_0) pnormalize_pair: "y \<noteq> 0 \<longleftrightarrow> (pnormalize [x, y] = [x, y])" by simp |
30488 | 825 |
lemma (in semiring_0) pnormal_cons: "pnormal p \<Longrightarrow> pnormal (c#p)" |
26124
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
826 |
unfolding pnormal_def by simp |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
827 |
lemma (in semiring_0) pnormal_tail: "p\<noteq>[] \<Longrightarrow> pnormal (c#p) \<Longrightarrow> pnormal p" |
32456
341c83339aeb
tuned the simp rules for Int involving insert and intervals.
nipkow
parents:
31021
diff
changeset
|
828 |
unfolding pnormal_def by(auto split: split_if_asm) |
26124
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
829 |
|
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
830 |
|
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
831 |
lemma (in semiring_0) pnormal_last_nonzero: "pnormal p ==> last p \<noteq> 0" |
32456
341c83339aeb
tuned the simp rules for Int involving insert and intervals.
nipkow
parents:
31021
diff
changeset
|
832 |
by(induct p) (simp_all add: pnormal_def split: split_if_asm) |
26124
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
833 |
|
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
834 |
lemma (in semiring_0) pnormal_length: "pnormal p \<Longrightarrow> 0 < length p" |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
835 |
unfolding pnormal_def length_greater_0_conv by blast |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
836 |
|
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
837 |
lemma (in semiring_0) pnormal_last_length: "\<lbrakk>0 < length p ; last p \<noteq> 0\<rbrakk> \<Longrightarrow> pnormal p" |
32456
341c83339aeb
tuned the simp rules for Int involving insert and intervals.
nipkow
parents:
31021
diff
changeset
|
838 |
by (induct p) (auto simp: pnormal_def split: split_if_asm) |
341c83339aeb
tuned the simp rules for Int involving insert and intervals.
nipkow
parents:
31021
diff
changeset
|
839 |
|
26124
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
840 |
|
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
841 |
lemma (in semiring_0) pnormal_id: "pnormal p \<longleftrightarrow> (0 < length p \<and> last p \<noteq> 0)" |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
842 |
using pnormal_last_length pnormal_length pnormal_last_nonzero by blast |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
843 |
|
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
844 |
lemma (in idom_char_0) poly_Cons_eq: "poly (c#cs) = poly (d#ds) \<longleftrightarrow> c=d \<and> poly cs = poly ds" (is "?lhs \<longleftrightarrow> ?rhs") |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
845 |
proof |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
846 |
assume eq: ?lhs |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
847 |
hence "\<And>x. poly ((c#cs) +++ -- (d#ds)) x = 0" |
29667 | 848 |
by (simp only: poly_minus poly_add algebra_simps) simp |
39302
d7728f65b353
renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents:
39198
diff
changeset
|
849 |
hence "poly ((c#cs) +++ -- (d#ds)) = poly []" by(simp add: fun_eq_iff) |
26124
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
850 |
hence "c = d \<and> list_all (\<lambda>x. x=0) ((cs +++ -- ds))" |
29667 | 851 |
unfolding poly_zero by (simp add: poly_minus_def algebra_simps) |
26124
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
852 |
hence "c = d \<and> (\<forall>x. poly (cs +++ -- ds) x = 0)" |
30488 | 853 |
unfolding poly_zero[symmetric] by simp |
39302
d7728f65b353
renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents:
39198
diff
changeset
|
854 |
thus ?rhs by (simp add: poly_minus poly_add algebra_simps fun_eq_iff) |
26124
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
855 |
next |
39302
d7728f65b353
renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents:
39198
diff
changeset
|
856 |
assume ?rhs then show ?lhs by(simp add:fun_eq_iff) |
26124
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
857 |
qed |
30488 | 858 |
|
26124
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
859 |
lemma (in idom_char_0) pnormalize_unique: "poly p = poly q \<Longrightarrow> pnormalize p = pnormalize q" |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
860 |
proof(induct q arbitrary: p) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
861 |
case Nil thus ?case by (simp only: poly_zero lemma_degree_zero) simp |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
862 |
next |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
863 |
case (Cons c cs p) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
864 |
thus ?case |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
865 |
proof(induct p) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
866 |
case Nil |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
867 |
hence "poly [] = poly (c#cs)" by blast |
30488 | 868 |
then have "poly (c#cs) = poly [] " by simp |
26124
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
869 |
thus ?case by (simp only: poly_zero lemma_degree_zero) simp |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
870 |
next |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
871 |
case (Cons d ds) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
872 |
hence eq: "poly (d # ds) = poly (c # cs)" by blast |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
873 |
hence eq': "\<And>x. poly (d # ds) x = poly (c # cs) x" by simp |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
874 |
hence "poly (d # ds) 0 = poly (c # cs) 0" by blast |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
875 |
hence dc: "d = c" by auto |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
876 |
with eq have "poly ds = poly cs" |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
877 |
unfolding poly_Cons_eq by simp |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
878 |
with Cons.prems have "pnormalize ds = pnormalize cs" by blast |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
879 |
with dc show ?case by simp |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
880 |
qed |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
881 |
qed |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
882 |
|
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
883 |
lemma (in idom_char_0) degree_unique: assumes pq: "poly p = poly q" |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
884 |
shows "degree p = degree q" |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
885 |
using pnormalize_unique[OF pq] unfolding degree_def by simp |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
886 |
|
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
887 |
lemma (in semiring_0) pnormalize_length: "length (pnormalize p) \<le> length p" by (induct p, auto) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
888 |
|
30488 | 889 |
lemma (in semiring_0) last_linear_mul_lemma: |
26124
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
890 |
"last ((a %* p) +++ (x#(b %* p))) = (if p=[] then x else b*last p)" |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
891 |
|
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
892 |
apply (induct p arbitrary: a x b, auto) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
893 |
apply (subgoal_tac "padd (cmult aa p) (times b a # cmult b p) \<noteq> []", simp) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
894 |
apply (induct_tac p, auto) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
895 |
done |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
896 |
|
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
897 |
lemma (in semiring_1) last_linear_mul: assumes p:"p\<noteq>[]" shows "last ([a,1] *** p) = last p" |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
898 |
proof- |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
899 |
from p obtain c cs where cs: "p = c#cs" by (cases p, auto) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
900 |
from cs have eq:"[a,1] *** p = (a %* (c#cs)) +++ (0#(1 %* (c#cs)))" |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
901 |
by (simp add: poly_cmult_distr) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
902 |
show ?thesis using cs |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
903 |
unfolding eq last_linear_mul_lemma by simp |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
904 |
qed |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
905 |
|
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
906 |
lemma (in semiring_0) pnormalize_eq: "last p \<noteq> 0 \<Longrightarrow> pnormalize p = p" |
32456
341c83339aeb
tuned the simp rules for Int involving insert and intervals.
nipkow
parents:
31021
diff
changeset
|
907 |
by (induct p) (auto split: split_if_asm) |
26124
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
908 |
|
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
909 |
lemma (in semiring_0) last_pnormalize: "pnormalize p \<noteq> [] \<Longrightarrow> last (pnormalize p) \<noteq> 0" |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
910 |
by (induct p, auto) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
911 |
|
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
912 |
lemma (in semiring_0) pnormal_degree: "last p \<noteq> 0 \<Longrightarrow> degree p = length p - 1" |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
913 |
using pnormalize_eq[of p] unfolding degree_def by simp |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
914 |
|
26313 | 915 |
lemma (in semiring_0) poly_Nil_ext: "poly [] = (\<lambda>x. 0)" by (rule ext) simp |
26124
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
916 |
|
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
917 |
lemma (in idom_char_0) linear_mul_degree: assumes p: "poly p \<noteq> poly []" |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
918 |
shows "degree ([a,1] *** p) = degree p + 1" |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
919 |
proof- |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
920 |
from p have pnz: "pnormalize p \<noteq> []" |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
921 |
unfolding poly_zero lemma_degree_zero . |
30488 | 922 |
|
26124
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
923 |
from last_linear_mul[OF pnz, of a] last_pnormalize[OF pnz] |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
924 |
have l0: "last ([a, 1] *** pnormalize p) \<noteq> 0" by simp |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
925 |
from last_pnormalize[OF pnz] last_linear_mul[OF pnz, of a] |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
926 |
pnormal_degree[OF l0] pnormal_degree[OF last_pnormalize[OF pnz]] pnz |
30488 | 927 |
|
26124
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
928 |
|
30488 | 929 |
have th: "degree ([a,1] *** pnormalize p) = degree (pnormalize p) + 1" |
26124
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
930 |
by (auto simp add: poly_length_mult) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
931 |
|
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
932 |
have eqs: "poly ([a,1] *** pnormalize p) = poly ([a,1] *** p)" |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
933 |
by (rule ext) (simp add: poly_mult poly_add poly_cmult) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
934 |
from degree_unique[OF eqs] th |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
935 |
show ?thesis by (simp add: degree_unique[OF poly_normalize]) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
936 |
qed |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
937 |
|
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
938 |
lemma (in idom_char_0) linear_pow_mul_degree: |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
939 |
"degree([a,1] %^n *** p) = (if poly p = poly [] then 0 else degree p + n)" |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
940 |
proof(induct n arbitrary: a p) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
941 |
case (0 a p) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
942 |
{assume p: "poly p = poly []" |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
943 |
hence ?case using degree_unique[OF p] by (simp add: degree_def)} |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
944 |
moreover |
26313 | 945 |
{assume p: "poly p \<noteq> poly []" hence ?case by (auto simp add: poly_Nil_ext) } |
26124
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
946 |
ultimately show ?case by blast |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
947 |
next |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
948 |
case (Suc n a p) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
949 |
have eq: "poly ([a,1] %^(Suc n) *** p) = poly ([a,1] %^ n *** ([a,1] *** p))" |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
950 |
apply (rule ext, simp add: poly_mult poly_add poly_cmult) |
49962
a8cc904a6820
Renamed {left,right}_distrib to distrib_{right,left}.
webertj
parents:
49751
diff
changeset
|
951 |
by (simp add: mult_ac add_ac distrib_left) |
26124
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
952 |
note deq = degree_unique[OF eq] |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
953 |
{assume p: "poly p = poly []" |
30488 | 954 |
with eq have eq': "poly ([a,1] %^(Suc n) *** p) = poly []" |
26124
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
955 |
by - (rule ext,simp add: poly_mult poly_cmult poly_add) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
956 |
from degree_unique[OF eq'] p have ?case by (simp add: degree_def)} |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
957 |
moreover |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
958 |
{assume p: "poly p \<noteq> poly []" |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
959 |
from p have ap: "poly ([a,1] *** p) \<noteq> poly []" |
30488 | 960 |
using poly_mult_not_eq_poly_Nil unfolding poly_entire by auto |
26124
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
961 |
have eq: "poly ([a,1] %^(Suc n) *** p) = poly ([a,1]%^n *** ([a,1] *** p))" |
29667 | 962 |
by (rule ext, simp add: poly_mult poly_add poly_exp poly_cmult algebra_simps) |
26124
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
963 |
from ap have ap': "(poly ([a,1] *** p) = poly []) = False" by blast |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
964 |
have th0: "degree ([a,1]%^n *** ([a,1] *** p)) = degree ([a,1] *** p) + n" |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
965 |
apply (simp only: Suc.hyps[of a "pmult [a,one] p"] ap') |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
966 |
by simp |
30488 | 967 |
|
26124
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
968 |
from degree_unique[OF eq] ap p th0 linear_mul_degree[OF p, of a] |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
969 |
have ?case by (auto simp del: poly.simps)} |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
970 |
ultimately show ?case by blast |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
971 |
qed |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
972 |
|
31021 | 973 |
lemma (in idom_char_0) order_degree: |
26124
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
974 |
assumes p0: "poly p \<noteq> poly []" |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
975 |
shows "order a p \<le> degree p" |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
976 |
proof- |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
977 |
from order2[OF p0, unfolded divides_def] |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
978 |
obtain q where q: "poly p = poly ([- a, 1]%^ (order a p) *** q)" by blast |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
979 |
{assume "poly q = poly []" |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
980 |
with q p0 have False by (simp add: poly_mult poly_entire)} |
30488 | 981 |
with degree_unique[OF q, unfolded linear_pow_mul_degree] |
26124
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
982 |
show ?thesis by auto |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
983 |
qed |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
984 |
|
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
985 |
text{*Tidier versions of finiteness of roots.*} |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
986 |
|
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
987 |
lemma (in idom_char_0) poly_roots_finite_set: "poly p \<noteq> poly [] ==> finite {x. poly p x = 0}" |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
988 |
unfolding poly_roots_finite . |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
989 |
|
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
990 |
text{*bound for polynomial.*} |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
991 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
32960
diff
changeset
|
992 |
lemma poly_mono: "abs(x) \<le> k ==> abs(poly p (x::'a::{linordered_idom})) \<le> poly (map abs p) k" |
26124
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
993 |
apply (induct "p", auto) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
994 |
apply (rule_tac y = "abs a + abs (x * poly p x)" in order_trans) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
995 |
apply (rule abs_triangle_ineq) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
996 |
apply (auto intro!: mult_mono simp add: abs_mult) |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
997 |
done |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
998 |
|
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
999 |
lemma (in semiring_0) poly_Sing: "poly [c] x = c" by simp |
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
1000 |
|
2514f0ade8bc
A library for univariate polynomials -- generalizes old Hyperreal/Poly.thy from reals to locales
chaieb
parents:
diff
changeset
|
1001 |
end |