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(* Title: HOL/Decision_Procs/Polynomial_List.thy
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Author: Amine Chaieb
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*)
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header{*Univariate Polynomials as Lists *}
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theory Polynomial_List
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imports Main
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begin
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text{* Application of polynomial as a real function. *}
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consts poly :: "'a list => 'a => ('a::{comm_ring})"
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primrec
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poly_Nil: "poly [] x = 0"
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poly_Cons: "poly (h#t) x = h + x * poly t x"
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subsection{*Arithmetic Operations on Polynomials*}
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text{*addition*}
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consts padd :: "['a list, 'a list] => ('a::comm_ring_1) list" (infixl "+++" 65)
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primrec
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padd_Nil: "[] +++ l2 = l2"
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padd_Cons: "(h#t) +++ l2 = (if l2 = [] then h#t
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else (h + hd l2)#(t +++ tl l2))"
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text{*Multiplication by a constant*}
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consts cmult :: "['a :: comm_ring_1, 'a list] => 'a list" (infixl "%*" 70)
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primrec
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cmult_Nil: "c %* [] = []"
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cmult_Cons: "c %* (h#t) = (c * h)#(c %* t)"
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text{*Multiplication by a polynomial*}
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consts pmult :: "['a list, 'a list] => ('a::comm_ring_1) list" (infixl "***" 70)
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primrec
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pmult_Nil: "[] *** l2 = []"
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pmult_Cons: "(h#t) *** l2 = (if t = [] then h %* l2
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else (h %* l2) +++ ((0) # (t *** l2)))"
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text{*Repeated multiplication by a polynomial*}
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consts mulexp :: "[nat, 'a list, 'a list] => ('a ::comm_ring_1) list"
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primrec
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mulexp_zero: "mulexp 0 p q = q"
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mulexp_Suc: "mulexp (Suc n) p q = p *** mulexp n p q"
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text{*Exponential*}
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consts pexp :: "['a list, nat] => ('a::comm_ring_1) list" (infixl "%^" 80)
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primrec
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pexp_0: "p %^ 0 = [1]"
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pexp_Suc: "p %^ (Suc n) = p *** (p %^ n)"
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text{*Quotient related value of dividing a polynomial by x + a*}
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(* Useful for divisor properties in inductive proofs *)
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consts "pquot" :: "['a list, 'a::field] => 'a list"
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primrec
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pquot_Nil: "pquot [] a= []"
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pquot_Cons: "pquot (h#t) a = (if t = [] then [h]
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else (inverse(a) * (h - hd( pquot t a)))#(pquot t a))"
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text{*normalization of polynomials (remove extra 0 coeff)*}
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consts pnormalize :: "('a::comm_ring_1) list => 'a list"
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primrec
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pnormalize_Nil: "pnormalize [] = []"
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pnormalize_Cons: "pnormalize (h#p) = (if ( (pnormalize p) = [])
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then (if (h = 0) then [] else [h])
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else (h#(pnormalize p)))"
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definition "pnormal p = ((pnormalize p = p) \<and> p \<noteq> [])"
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definition "nonconstant p = (pnormal p \<and> (\<forall>x. p \<noteq> [x]))"
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text{*Other definitions*}
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definition
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poly_minus :: "'a list => ('a :: comm_ring_1) list" ("-- _" [80] 80) where
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"-- p = (- 1) %* p"
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definition
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divides :: "[('a::comm_ring_1) list, 'a list] => bool" (infixl "divides" 70) where
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"p1 divides p2 = (\<exists>q. poly p2 = poly(p1 *** q))"
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definition
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order :: "('a::comm_ring_1) => 'a list => nat" where
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--{*order of a polynomial*}
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"order a p = (SOME n. ([-a, 1] %^ n) divides p &
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~ (([-a, 1] %^ (Suc n)) divides p))"
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definition
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degree :: "('a::comm_ring_1) list => nat" where
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--{*degree of a polynomial*}
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"degree p = length (pnormalize p) - 1"
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definition
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rsquarefree :: "('a::comm_ring_1) list => bool" where
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--{*squarefree polynomials --- NB with respect to real roots only.*}
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"rsquarefree p = (poly p \<noteq> poly [] &
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(\<forall>a. (order a p = 0) | (order a p = 1)))"
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lemma padd_Nil2: "p +++ [] = p"
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by (induct p) auto
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declare padd_Nil2 [simp]
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lemma padd_Cons_Cons: "(h1 # p1) +++ (h2 # p2) = (h1 + h2) # (p1 +++ p2)"
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by auto
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lemma pminus_Nil: "-- [] = []"
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by (simp add: poly_minus_def)
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declare pminus_Nil [simp]
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lemma pmult_singleton: "[h1] *** p1 = h1 %* p1"
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by simp
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lemma poly_ident_mult: "1 %* t = t"
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by (induct "t", auto)
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declare poly_ident_mult [simp]
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lemma poly_simple_add_Cons: "[a] +++ ((0)#t) = (a#t)"
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by simp
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declare poly_simple_add_Cons [simp]
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text{*Handy general properties*}
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lemma padd_commut: "b +++ a = a +++ b"
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apply (subgoal_tac "\<forall>a. b +++ a = a +++ b")
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apply (induct_tac [2] "b", auto)
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apply (rule padd_Cons [THEN ssubst])
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apply (case_tac "aa", auto)
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done
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lemma padd_assoc [rule_format]: "\<forall>b c. (a +++ b) +++ c = a +++ (b +++ c)"
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apply (induct "a", simp, clarify)
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apply (case_tac b, simp_all)
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done
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lemma poly_cmult_distr [rule_format]:
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"\<forall>q. a %* ( p +++ q) = (a %* p +++ a %* q)"
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apply (induct "p", simp, clarify)
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apply (case_tac "q")
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apply (simp_all add: right_distrib)
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done
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lemma pmult_by_x[simp]: "[0, 1] *** t = ((0)#t)"
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apply (induct "t", simp)
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by (auto simp add: mult_zero_left poly_ident_mult padd_commut)
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text{*properties of evaluation of polynomials.*}
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lemma poly_add: "poly (p1 +++ p2) x = poly p1 x + poly p2 x"
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apply (subgoal_tac "\<forall>p2. poly (p1 +++ p2) x = poly (p1) x + poly (p2) x")
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apply (induct_tac [2] "p1", auto)
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apply (case_tac "p2")
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apply (auto simp add: right_distrib)
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done
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lemma poly_cmult: "poly (c %* p) x = c * poly p x"
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apply (induct "p")
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apply (case_tac [2] "x=0")
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apply (auto simp add: right_distrib mult_ac)
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done
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lemma poly_minus: "poly (-- p) x = - (poly p x)"
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apply (simp add: poly_minus_def)
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apply (auto simp add: poly_cmult)
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done
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lemma poly_mult: "poly (p1 *** p2) x = poly p1 x * poly p2 x"
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apply (subgoal_tac "\<forall>p2. poly (p1 *** p2) x = poly p1 x * poly p2 x")
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apply (simp (no_asm_simp))
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apply (induct "p1")
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apply (auto simp add: poly_cmult)
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apply (case_tac p1)
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apply (auto simp add: poly_cmult poly_add left_distrib right_distrib mult_ac)
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done
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lemma poly_exp: "poly (p %^ n) (x::'a::comm_ring_1) = (poly p x) ^ n"
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apply (induct "n")
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apply (auto simp add: poly_cmult poly_mult power_Suc)
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done
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text{*More Polynomial Evaluation Lemmas*}
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lemma poly_add_rzero: "poly (a +++ []) x = poly a x"
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by simp
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declare poly_add_rzero [simp]
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lemma poly_mult_assoc: "poly ((a *** b) *** c) x = poly (a *** (b *** c)) x"
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by (simp add: poly_mult mult_assoc)
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lemma poly_mult_Nil2: "poly (p *** []) x = 0"
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by (induct "p", auto)
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declare poly_mult_Nil2 [simp]
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lemma poly_exp_add: "poly (p %^ (n + d)) x = poly( p %^ n *** p %^ d) x"
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apply (induct "n")
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apply (auto simp add: poly_mult mult_assoc)
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done
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subsection{*Key Property: if @{term "f(a) = 0"} then @{term "(x - a)"} divides
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@{term "p(x)"} *}
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lemma lemma_poly_linear_rem: "\<forall>h. \<exists>q r. h#t = [r] +++ [-a, 1] *** q"
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apply (induct "t", safe)
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apply (rule_tac x = "[]" in exI)
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apply (rule_tac x = h in exI, simp)
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apply (drule_tac x = aa in spec, safe)
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apply (rule_tac x = "r#q" in exI)
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apply (rule_tac x = "a*r + h" in exI)
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apply (case_tac "q", auto)
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done
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lemma poly_linear_rem: "\<exists>q r. h#t = [r] +++ [-a, 1] *** q"
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by (cut_tac t = t and a = a in lemma_poly_linear_rem, auto)
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lemma poly_linear_divides: "(poly p a = 0) = ((p = []) | (\<exists>q. p = [-a, 1] *** q))"
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apply (auto simp add: poly_add poly_cmult right_distrib)
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apply (case_tac "p", simp)
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apply (cut_tac h = aa and t = list and a = a in poly_linear_rem, safe)
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apply (case_tac "q", auto)
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apply (drule_tac x = "[]" in spec, simp)
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apply (auto simp add: poly_add poly_cmult add_assoc)
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apply (drule_tac x = "aa#lista" in spec, auto)
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done
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lemma lemma_poly_length_mult: "\<forall>h k a. length (k %* p +++ (h # (a %* p))) = Suc (length p)"
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by (induct "p", auto)
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declare lemma_poly_length_mult [simp]
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lemma lemma_poly_length_mult2: "\<forall>h k. length (k %* p +++ (h # p)) = Suc (length p)"
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by (induct "p", auto)
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declare lemma_poly_length_mult2 [simp]
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lemma poly_length_mult: "length([-a,1] *** q) = Suc (length q)"
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by auto
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declare poly_length_mult [simp]
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subsection{*Polynomial length*}
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lemma poly_cmult_length: "length (a %* p) = length p"
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by (induct "p", auto)
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declare poly_cmult_length [simp]
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lemma poly_add_length [rule_format]:
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"\<forall>p2. length (p1 +++ p2) =
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(if (length p1 < length p2) then length p2 else length p1)"
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apply (induct "p1", simp_all)
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apply arith
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done
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lemma poly_root_mult_length: "length([a,b] *** p) = Suc (length p)"
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by (simp add: poly_cmult_length poly_add_length)
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declare poly_root_mult_length [simp]
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lemma poly_mult_not_eq_poly_Nil: "(poly (p *** q) x \<noteq> poly [] x) =
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(poly p x \<noteq> poly [] x & poly q x \<noteq> poly [] (x::'a::idom))"
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apply (auto simp add: poly_mult)
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done
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declare poly_mult_not_eq_poly_Nil [simp]
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lemma poly_mult_eq_zero_disj: "(poly (p *** q) (x::'a::idom) = 0) = (poly p x = 0 | poly q x = 0)"
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by (auto simp add: poly_mult)
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text{*Normalisation Properties*}
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lemma poly_normalized_nil: "(pnormalize p = []) --> (poly p x = 0)"
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by (induct "p", auto)
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text{*A nontrivial polynomial of degree n has no more than n roots*}
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lemma poly_roots_index_lemma0 [rule_format]:
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"\<forall>p x. poly p x \<noteq> poly [] x & length p = n
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--> (\<exists>i. \<forall>x. (poly p x = (0::'a::idom)) --> (\<exists>m. (m \<le> n & x = i m)))"
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apply (induct "n", safe)
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apply (rule ccontr)
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apply (subgoal_tac "\<exists>a. poly p a = 0", safe)
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apply (drule poly_linear_divides [THEN iffD1], safe)
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apply (drule_tac x = q in spec)
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apply (drule_tac x = x in spec)
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apply (simp del: poly_Nil pmult_Cons)
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apply (erule exE)
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apply (drule_tac x = "%m. if m = Suc n then a else i m" in spec, safe)
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apply (drule poly_mult_eq_zero_disj [THEN iffD1], safe)
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apply (drule_tac x = "Suc (length q)" in spec)
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apply (auto simp add: ring_simps)
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apply (drule_tac x = xa in spec)
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apply (clarsimp simp add: ring_simps)
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apply (drule_tac x = m in spec)
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apply (auto simp add:ring_simps)
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done
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lemmas poly_roots_index_lemma1 = conjI [THEN poly_roots_index_lemma0, standard]
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lemma poly_roots_index_length0: "poly p (x::'a::idom) \<noteq> poly [] x ==>
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\<exists>i. \<forall>x. (poly p x = 0) --> (\<exists>n. n \<le> length p & x = i n)"
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by (blast intro: poly_roots_index_lemma1)
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lemma poly_roots_finite_lemma: "poly p (x::'a::idom) \<noteq> poly [] x ==>
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\<exists>N i. \<forall>x. (poly p x = 0) --> (\<exists>n. (n::nat) < N & x = i n)"
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apply (drule poly_roots_index_length0, safe)
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apply (rule_tac x = "Suc (length p)" in exI)
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apply (rule_tac x = i in exI)
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apply (simp add: less_Suc_eq_le)
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done
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lemma real_finite_lemma:
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assumes P: "\<forall>x. P x --> (\<exists>n. n < length j & x = j!n)"
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shows "finite {(x::'a::idom). P x}"
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proof-
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let ?M = "{x. P x}"
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let ?N = "set j"
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have "?M \<subseteq> ?N" using P by auto
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thus ?thesis using finite_subset by auto
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qed
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lemma poly_roots_index_lemma [rule_format]:
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"\<forall>p x. poly p x \<noteq> poly [] x & length p = n
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--> (\<exists>i. \<forall>x. (poly p x = (0::'a::{idom})) --> x \<in> set i)"
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apply (induct "n", safe)
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apply (rule ccontr)
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apply (subgoal_tac "\<exists>a. poly p a = 0", safe)
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apply (drule poly_linear_divides [THEN iffD1], safe)
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apply (drule_tac x = q in spec)
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apply (drule_tac x = x in spec)
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apply (auto simp del: poly_Nil pmult_Cons)
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apply (drule_tac x = "a#i" in spec)
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apply (auto simp only: poly_mult List.list.size)
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apply (drule_tac x = xa in spec)
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apply (clarsimp simp add: ring_simps)
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done
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lemmas poly_roots_index_lemma2 = conjI [THEN poly_roots_index_lemma, standard]
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lemma poly_roots_index_length: "poly p (x::'a::idom) \<noteq> poly [] x ==>
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\<exists>i. \<forall>x. (poly p x = 0) --> x \<in> set i"
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by (blast intro: poly_roots_index_lemma2)
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lemma poly_roots_finite_lemma': "poly p (x::'a::idom) \<noteq> poly [] x ==>
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\<exists>i. \<forall>x. (poly p x = 0) --> x \<in> set i"
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by (drule poly_roots_index_length, safe)
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343 |
lemma UNIV_nat_infinite: "\<not> finite (UNIV :: nat set)"
|
|
344 |
unfolding finite_conv_nat_seg_image
|
|
345 |
proof(auto simp add: expand_set_eq image_iff)
|
|
346 |
fix n::nat and f:: "nat \<Rightarrow> nat"
|
|
347 |
let ?N = "{i. i < n}"
|
|
348 |
let ?fN = "f ` ?N"
|
|
349 |
let ?y = "Max ?fN + 1"
|
|
350 |
from nat_seg_image_imp_finite[of "?fN" "f" n]
|
|
351 |
have thfN: "finite ?fN" by simp
|
|
352 |
{assume "n =0" hence "\<exists>x. \<forall>xa<n. x \<noteq> f xa" by auto}
|
|
353 |
moreover
|
|
354 |
{assume nz: "n \<noteq> 0"
|
|
355 |
hence thne: "?fN \<noteq> {}" by (auto simp add: neq0_conv)
|
|
356 |
have "\<forall>x\<in> ?fN. Max ?fN \<ge> x" using nz Max_ge_iff[OF thfN thne] by auto
|
|
357 |
hence "\<forall>x\<in> ?fN. ?y > x" by (auto simp add: less_Suc_eq_le)
|
|
358 |
hence "?y \<notin> ?fN" by auto
|
|
359 |
hence "\<exists>x. \<forall>xa<n. x \<noteq> f xa" by auto }
|
|
360 |
ultimately show "\<exists>x. \<forall>xa<n. x \<noteq> f xa" by blast
|
|
361 |
qed
|
|
362 |
|
|
363 |
lemma UNIV_ring_char_0_infinte: "\<not> finite (UNIV:: ('a::ring_char_0) set)"
|
|
364 |
proof
|
|
365 |
assume F: "finite (UNIV :: 'a set)"
|
|
366 |
have th0: "of_nat ` UNIV \<subseteq> (UNIV:: 'a set)" by simp
|
|
367 |
from finite_subset[OF th0 F] have th: "finite (of_nat ` UNIV :: 'a set)" .
|
|
368 |
have th': "inj_on (of_nat::nat \<Rightarrow> 'a) (UNIV)"
|
|
369 |
unfolding inj_on_def by auto
|
|
370 |
from finite_imageD[OF th th'] UNIV_nat_infinite
|
|
371 |
show False by blast
|
|
372 |
qed
|
|
373 |
|
|
374 |
lemma poly_roots_finite: "(poly p \<noteq> poly []) =
|
|
375 |
finite {x. poly p x = (0::'a::{idom, ring_char_0})}"
|
|
376 |
proof
|
|
377 |
assume H: "poly p \<noteq> poly []"
|
|
378 |
show "finite {x. poly p x = (0::'a)}"
|
|
379 |
using H
|
|
380 |
apply -
|
|
381 |
apply (erule contrapos_np, rule ext)
|
|
382 |
apply (rule ccontr)
|
|
383 |
apply (clarify dest!: poly_roots_finite_lemma')
|
|
384 |
using finite_subset
|
|
385 |
proof-
|
|
386 |
fix x i
|
|
387 |
assume F: "\<not> finite {x. poly p x = (0\<Colon>'a)}"
|
|
388 |
and P: "\<forall>x. poly p x = (0\<Colon>'a) \<longrightarrow> x \<in> set i"
|
|
389 |
let ?M= "{x. poly p x = (0\<Colon>'a)}"
|
|
390 |
from P have "?M \<subseteq> set i" by auto
|
|
391 |
with finite_subset F show False by auto
|
|
392 |
qed
|
|
393 |
next
|
|
394 |
assume F: "finite {x. poly p x = (0\<Colon>'a)}"
|
|
395 |
show "poly p \<noteq> poly []" using F UNIV_ring_char_0_infinte by auto
|
|
396 |
qed
|
|
397 |
|
|
398 |
text{*Entirety and Cancellation for polynomials*}
|
|
399 |
|
|
400 |
lemma poly_entire_lemma: "[| poly (p:: ('a::{idom,ring_char_0}) list) \<noteq> poly [] ; poly q \<noteq> poly [] |]
|
|
401 |
==> poly (p *** q) \<noteq> poly []"
|
|
402 |
by (auto simp add: poly_roots_finite poly_mult Collect_disj_eq)
|
|
403 |
|
|
404 |
lemma poly_entire: "(poly (p *** q) = poly ([]::('a::{idom,ring_char_0}) list)) = ((poly p = poly []) | (poly q = poly []))"
|
|
405 |
apply (auto intro: ext dest: fun_cong simp add: poly_entire_lemma poly_mult)
|
|
406 |
apply (blast intro: ccontr dest: poly_entire_lemma poly_mult [THEN subst])
|
|
407 |
done
|
|
408 |
|
|
409 |
lemma poly_entire_neg: "(poly (p *** q) \<noteq> poly ([]::('a::{idom,ring_char_0}) list)) = ((poly p \<noteq> poly []) & (poly q \<noteq> poly []))"
|
|
410 |
by (simp add: poly_entire)
|
|
411 |
|
|
412 |
lemma fun_eq: " (f = g) = (\<forall>x. f x = g x)"
|
|
413 |
by (auto intro!: ext)
|
|
414 |
|
|
415 |
lemma poly_add_minus_zero_iff: "(poly (p +++ -- q) = poly []) = (poly p = poly q)"
|
|
416 |
by (auto simp add: ring_simps poly_add poly_minus_def fun_eq poly_cmult)
|
|
417 |
|
|
418 |
lemma poly_add_minus_mult_eq: "poly (p *** q +++ --(p *** r)) = poly (p *** (q +++ -- r))"
|
|
419 |
by (auto simp add: poly_add poly_minus_def fun_eq poly_mult poly_cmult right_distrib)
|
|
420 |
|
|
421 |
lemma poly_mult_left_cancel: "(poly (p *** q) = poly (p *** r)) = (poly p = poly ([]::('a::{idom, ring_char_0}) list) | poly q = poly r)"
|
|
422 |
apply (rule_tac p1 = "p *** q" in poly_add_minus_zero_iff [THEN subst])
|
|
423 |
apply (auto intro: ext simp add: poly_add_minus_mult_eq poly_entire poly_add_minus_zero_iff)
|
|
424 |
done
|
|
425 |
|
|
426 |
lemma poly_exp_eq_zero:
|
|
427 |
"(poly (p %^ n) = poly ([]::('a::idom) list)) = (poly p = poly [] & n \<noteq> 0)"
|
|
428 |
apply (simp only: fun_eq add: all_simps [symmetric])
|
|
429 |
apply (rule arg_cong [where f = All])
|
|
430 |
apply (rule ext)
|
|
431 |
apply (induct_tac "n")
|
|
432 |
apply (auto simp add: poly_mult)
|
|
433 |
done
|
|
434 |
declare poly_exp_eq_zero [simp]
|
|
435 |
|
|
436 |
lemma poly_prime_eq_zero: "poly [a,(1::'a::comm_ring_1)] \<noteq> poly []"
|
|
437 |
apply (simp add: fun_eq)
|
|
438 |
apply (rule_tac x = "1 - a" in exI, simp)
|
|
439 |
done
|
|
440 |
declare poly_prime_eq_zero [simp]
|
|
441 |
|
|
442 |
lemma poly_exp_prime_eq_zero: "(poly ([a, (1::'a::idom)] %^ n) \<noteq> poly [])"
|
|
443 |
by auto
|
|
444 |
declare poly_exp_prime_eq_zero [simp]
|
|
445 |
|
|
446 |
text{*A more constructive notion of polynomials being trivial*}
|
|
447 |
|
|
448 |
lemma poly_zero_lemma': "poly (h # t) = poly [] ==> h = (0::'a::{idom,ring_char_0}) & poly t = poly []"
|
|
449 |
apply(simp add: fun_eq)
|
|
450 |
apply (case_tac "h = 0")
|
|
451 |
apply (drule_tac [2] x = 0 in spec, auto)
|
|
452 |
apply (case_tac "poly t = poly []", simp)
|
|
453 |
proof-
|
|
454 |
fix x
|
|
455 |
assume H: "\<forall>x. x = (0\<Colon>'a) \<or> poly t x = (0\<Colon>'a)" and pnz: "poly t \<noteq> poly []"
|
|
456 |
let ?S = "{x. poly t x = 0}"
|
|
457 |
from H have "\<forall>x. x \<noteq>0 \<longrightarrow> poly t x = 0" by blast
|
|
458 |
hence th: "?S \<supseteq> UNIV - {0}" by auto
|
|
459 |
from poly_roots_finite pnz have th': "finite ?S" by blast
|
|
460 |
from finite_subset[OF th th'] UNIV_ring_char_0_infinte[where ?'a = 'a]
|
|
461 |
show "poly t x = (0\<Colon>'a)" by simp
|
|
462 |
qed
|
|
463 |
|
|
464 |
lemma poly_zero: "(poly p = poly []) = list_all (%c. c = (0::'a::{idom,ring_char_0})) p"
|
|
465 |
apply (induct "p", simp)
|
|
466 |
apply (rule iffI)
|
|
467 |
apply (drule poly_zero_lemma', auto)
|
|
468 |
done
|
|
469 |
|
|
470 |
|
|
471 |
|
|
472 |
text{*Basics of divisibility.*}
|
|
473 |
|
|
474 |
lemma poly_primes: "([a, (1::'a::idom)] divides (p *** q)) = ([a, 1] divides p | [a, 1] divides q)"
|
|
475 |
apply (auto simp add: divides_def fun_eq poly_mult poly_add poly_cmult left_distrib [symmetric])
|
|
476 |
apply (drule_tac x = "-a" in spec)
|
|
477 |
apply (auto simp add: poly_linear_divides poly_add poly_cmult left_distrib [symmetric])
|
|
478 |
apply (rule_tac x = "qa *** q" in exI)
|
|
479 |
apply (rule_tac [2] x = "p *** qa" in exI)
|
|
480 |
apply (auto simp add: poly_add poly_mult poly_cmult mult_ac)
|
|
481 |
done
|
|
482 |
|
|
483 |
lemma poly_divides_refl: "p divides p"
|
|
484 |
apply (simp add: divides_def)
|
|
485 |
apply (rule_tac x = "[1]" in exI)
|
|
486 |
apply (auto simp add: poly_mult fun_eq)
|
|
487 |
done
|
|
488 |
declare poly_divides_refl [simp]
|
|
489 |
|
|
490 |
lemma poly_divides_trans: "[| p divides q; q divides r |] ==> p divides r"
|
|
491 |
apply (simp add: divides_def, safe)
|
|
492 |
apply (rule_tac x = "qa *** qaa" in exI)
|
|
493 |
apply (auto simp add: poly_mult fun_eq mult_assoc)
|
|
494 |
done
|
|
495 |
|
|
496 |
lemma poly_divides_exp: "m \<le> n ==> (p %^ m) divides (p %^ n)"
|
|
497 |
apply (auto simp add: le_iff_add)
|
|
498 |
apply (induct_tac k)
|
|
499 |
apply (rule_tac [2] poly_divides_trans)
|
|
500 |
apply (auto simp add: divides_def)
|
|
501 |
apply (rule_tac x = p in exI)
|
|
502 |
apply (auto simp add: poly_mult fun_eq mult_ac)
|
|
503 |
done
|
|
504 |
|
|
505 |
lemma poly_exp_divides: "[| (p %^ n) divides q; m\<le>n |] ==> (p %^ m) divides q"
|
|
506 |
by (blast intro: poly_divides_exp poly_divides_trans)
|
|
507 |
|
|
508 |
lemma poly_divides_add:
|
|
509 |
"[| p divides q; p divides r |] ==> p divides (q +++ r)"
|
|
510 |
apply (simp add: divides_def, auto)
|
|
511 |
apply (rule_tac x = "qa +++ qaa" in exI)
|
|
512 |
apply (auto simp add: poly_add fun_eq poly_mult right_distrib)
|
|
513 |
done
|
|
514 |
|
|
515 |
lemma poly_divides_diff:
|
|
516 |
"[| p divides q; p divides (q +++ r) |] ==> p divides r"
|
|
517 |
apply (simp add: divides_def, auto)
|
|
518 |
apply (rule_tac x = "qaa +++ -- qa" in exI)
|
|
519 |
apply (auto simp add: poly_add fun_eq poly_mult poly_minus right_diff_distrib algebra_simps)
|
|
520 |
done
|
|
521 |
|
|
522 |
lemma poly_divides_diff2: "[| p divides r; p divides (q +++ r) |] ==> p divides q"
|
|
523 |
apply (erule poly_divides_diff)
|
|
524 |
apply (auto simp add: poly_add fun_eq poly_mult divides_def add_ac)
|
|
525 |
done
|
|
526 |
|
|
527 |
lemma poly_divides_zero: "poly p = poly [] ==> q divides p"
|
|
528 |
apply (simp add: divides_def)
|
|
529 |
apply (rule exI[where x="[]"])
|
|
530 |
apply (auto simp add: fun_eq poly_mult)
|
|
531 |
done
|
|
532 |
|
|
533 |
lemma poly_divides_zero2: "q divides []"
|
|
534 |
apply (simp add: divides_def)
|
|
535 |
apply (rule_tac x = "[]" in exI)
|
|
536 |
apply (auto simp add: fun_eq)
|
|
537 |
done
|
|
538 |
declare poly_divides_zero2 [simp]
|
|
539 |
|
|
540 |
text{*At last, we can consider the order of a root.*}
|
|
541 |
|
|
542 |
|
|
543 |
lemma poly_order_exists_lemma [rule_format]:
|
|
544 |
"\<forall>p. length p = d --> poly p \<noteq> poly []
|
|
545 |
--> (\<exists>n q. p = mulexp n [-a, (1::'a::{idom,ring_char_0})] q & poly q a \<noteq> 0)"
|
|
546 |
apply (induct "d")
|
|
547 |
apply (simp add: fun_eq, safe)
|
|
548 |
apply (case_tac "poly p a = 0")
|
|
549 |
apply (drule_tac poly_linear_divides [THEN iffD1], safe)
|
|
550 |
apply (drule_tac x = q in spec)
|
|
551 |
apply (drule_tac poly_entire_neg [THEN iffD1], safe, force)
|
|
552 |
apply (rule_tac x = "Suc n" in exI)
|
|
553 |
apply (rule_tac x = qa in exI)
|
|
554 |
apply (simp del: pmult_Cons)
|
|
555 |
apply (rule_tac x = 0 in exI, force)
|
|
556 |
done
|
|
557 |
|
|
558 |
(* FIXME: Tidy up *)
|
|
559 |
lemma poly_order_exists:
|
|
560 |
"[| length p = d; poly p \<noteq> poly [] |]
|
|
561 |
==> \<exists>n. ([-a, 1] %^ n) divides p &
|
|
562 |
~(([-a, (1::'a::{idom,ring_char_0})] %^ (Suc n)) divides p)"
|
|
563 |
apply (drule poly_order_exists_lemma [where a=a], assumption, clarify)
|
|
564 |
apply (rule_tac x = n in exI, safe)
|
|
565 |
apply (unfold divides_def)
|
|
566 |
apply (rule_tac x = q in exI)
|
|
567 |
apply (induct_tac "n", simp)
|
|
568 |
apply (simp (no_asm_simp) add: poly_add poly_cmult poly_mult right_distrib mult_ac)
|
|
569 |
apply safe
|
|
570 |
apply (subgoal_tac "poly (mulexp n [- a, 1] q) \<noteq> poly ([- a, 1] %^ Suc n *** qa)")
|
|
571 |
apply simp
|
|
572 |
apply (induct_tac "n")
|
|
573 |
apply (simp del: pmult_Cons pexp_Suc)
|
|
574 |
apply (erule_tac Q = "poly q a = 0" in contrapos_np)
|
|
575 |
apply (simp add: poly_add poly_cmult)
|
|
576 |
apply (rule pexp_Suc [THEN ssubst])
|
|
577 |
apply (rule ccontr)
|
|
578 |
apply (simp add: poly_mult_left_cancel poly_mult_assoc del: pmult_Cons pexp_Suc)
|
|
579 |
done
|
|
580 |
|
|
581 |
lemma poly_one_divides: "[1] divides p"
|
|
582 |
by (simp add: divides_def, auto)
|
|
583 |
declare poly_one_divides [simp]
|
|
584 |
|
|
585 |
lemma poly_order: "poly p \<noteq> poly []
|
|
586 |
==> EX! n. ([-a, (1::'a::{idom,ring_char_0})] %^ n) divides p &
|
|
587 |
~(([-a, 1] %^ (Suc n)) divides p)"
|
|
588 |
apply (auto intro: poly_order_exists simp add: less_linear simp del: pmult_Cons pexp_Suc)
|
|
589 |
apply (cut_tac x = y and y = n in less_linear)
|
|
590 |
apply (drule_tac m = n in poly_exp_divides)
|
|
591 |
apply (auto dest: Suc_le_eq [THEN iffD2, THEN [2] poly_exp_divides]
|
|
592 |
simp del: pmult_Cons pexp_Suc)
|
|
593 |
done
|
|
594 |
|
|
595 |
text{*Order*}
|
|
596 |
|
|
597 |
lemma some1_equalityD: "[| n = (@n. P n); EX! n. P n |] ==> P n"
|
|
598 |
by (blast intro: someI2)
|
|
599 |
|
|
600 |
lemma order:
|
|
601 |
"(([-a, (1::'a::{idom,ring_char_0})] %^ n) divides p &
|
|
602 |
~(([-a, 1] %^ (Suc n)) divides p)) =
|
|
603 |
((n = order a p) & ~(poly p = poly []))"
|
|
604 |
apply (unfold order_def)
|
|
605 |
apply (rule iffI)
|
|
606 |
apply (blast dest: poly_divides_zero intro!: some1_equality [symmetric] poly_order)
|
|
607 |
apply (blast intro!: poly_order [THEN [2] some1_equalityD])
|
|
608 |
done
|
|
609 |
|
|
610 |
lemma order2: "[| poly p \<noteq> poly [] |]
|
|
611 |
==> ([-a, (1::'a::{idom,ring_char_0})] %^ (order a p)) divides p &
|
|
612 |
~(([-a, 1] %^ (Suc(order a p))) divides p)"
|
|
613 |
by (simp add: order del: pexp_Suc)
|
|
614 |
|
|
615 |
lemma order_unique: "[| poly p \<noteq> poly []; ([-a, 1] %^ n) divides p;
|
|
616 |
~(([-a, (1::'a::{idom,ring_char_0})] %^ (Suc n)) divides p)
|
|
617 |
|] ==> (n = order a p)"
|
|
618 |
by (insert order [of a n p], auto)
|
|
619 |
|
|
620 |
lemma order_unique_lemma: "(poly p \<noteq> poly [] & ([-a, 1] %^ n) divides p &
|
|
621 |
~(([-a, (1::'a::{idom,ring_char_0})] %^ (Suc n)) divides p))
|
|
622 |
==> (n = order a p)"
|
|
623 |
by (blast intro: order_unique)
|
|
624 |
|
|
625 |
lemma order_poly: "poly p = poly q ==> order a p = order a q"
|
|
626 |
by (auto simp add: fun_eq divides_def poly_mult order_def)
|
|
627 |
|
|
628 |
lemma pexp_one: "p %^ (Suc 0) = p"
|
|
629 |
apply (induct "p")
|
|
630 |
apply (auto simp add: numeral_1_eq_1)
|
|
631 |
done
|
|
632 |
declare pexp_one [simp]
|
|
633 |
|
|
634 |
lemma lemma_order_root [rule_format]:
|
|
635 |
"\<forall>p a. 0 < n & [- a, 1] %^ n divides p & ~ [- a, 1] %^ (Suc n) divides p
|
|
636 |
--> poly p a = 0"
|
|
637 |
apply (induct "n", blast)
|
|
638 |
apply (auto simp add: divides_def poly_mult simp del: pmult_Cons)
|
|
639 |
done
|
|
640 |
|
|
641 |
lemma order_root: "(poly p a = (0::'a::{idom,ring_char_0})) = ((poly p = poly []) | order a p \<noteq> 0)"
|
|
642 |
apply (case_tac "poly p = poly []", auto)
|
|
643 |
apply (simp add: poly_linear_divides del: pmult_Cons, safe)
|
|
644 |
apply (drule_tac [!] a = a in order2)
|
|
645 |
apply (rule ccontr)
|
|
646 |
apply (simp add: divides_def poly_mult fun_eq del: pmult_Cons, blast)
|
|
647 |
using neq0_conv
|
|
648 |
apply (blast intro: lemma_order_root)
|
|
649 |
done
|
|
650 |
|
|
651 |
lemma order_divides: "(([-a, 1::'a::{idom,ring_char_0}] %^ n) divides p) = ((poly p = poly []) | n \<le> order a p)"
|
|
652 |
apply (case_tac "poly p = poly []", auto)
|
|
653 |
apply (simp add: divides_def fun_eq poly_mult)
|
|
654 |
apply (rule_tac x = "[]" in exI)
|
|
655 |
apply (auto dest!: order2 [where a=a]
|
|
656 |
intro: poly_exp_divides simp del: pexp_Suc)
|
|
657 |
done
|
|
658 |
|
|
659 |
lemma order_decomp:
|
|
660 |
"poly p \<noteq> poly []
|
|
661 |
==> \<exists>q. (poly p = poly (([-a, 1] %^ (order a p)) *** q)) &
|
|
662 |
~([-a, 1::'a::{idom,ring_char_0}] divides q)"
|
|
663 |
apply (unfold divides_def)
|
|
664 |
apply (drule order2 [where a = a])
|
|
665 |
apply (simp add: divides_def del: pexp_Suc pmult_Cons, safe)
|
|
666 |
apply (rule_tac x = q in exI, safe)
|
|
667 |
apply (drule_tac x = qa in spec)
|
|
668 |
apply (auto simp add: poly_mult fun_eq poly_exp mult_ac simp del: pmult_Cons)
|
|
669 |
done
|
|
670 |
|
|
671 |
text{*Important composition properties of orders.*}
|
|
672 |
|
|
673 |
lemma order_mult: "poly (p *** q) \<noteq> poly []
|
|
674 |
==> order a (p *** q) = order a p + order (a::'a::{idom,ring_char_0}) q"
|
|
675 |
apply (cut_tac a = a and p = "p***q" and n = "order a p + order a q" in order)
|
|
676 |
apply (auto simp add: poly_entire simp del: pmult_Cons)
|
|
677 |
apply (drule_tac a = a in order2)+
|
|
678 |
apply safe
|
|
679 |
apply (simp add: divides_def fun_eq poly_exp_add poly_mult del: pmult_Cons, safe)
|
|
680 |
apply (rule_tac x = "qa *** qaa" in exI)
|
|
681 |
apply (simp add: poly_mult mult_ac del: pmult_Cons)
|
|
682 |
apply (drule_tac a = a in order_decomp)+
|
|
683 |
apply safe
|
|
684 |
apply (subgoal_tac "[-a,1] divides (qa *** qaa) ")
|
|
685 |
apply (simp add: poly_primes del: pmult_Cons)
|
|
686 |
apply (auto simp add: divides_def simp del: pmult_Cons)
|
|
687 |
apply (rule_tac x = qb in exI)
|
|
688 |
apply (subgoal_tac "poly ([-a, 1] %^ (order a p) *** (qa *** qaa)) = poly ([-a, 1] %^ (order a p) *** ([-a, 1] *** qb))")
|
|
689 |
apply (drule poly_mult_left_cancel [THEN iffD1], force)
|
|
690 |
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))) ")
|
|
691 |
apply (drule poly_mult_left_cancel [THEN iffD1], force)
|
|
692 |
apply (simp add: fun_eq poly_exp_add poly_mult mult_ac del: pmult_Cons)
|
|
693 |
done
|
|
694 |
|
|
695 |
|
|
696 |
|
|
697 |
lemma order_root2: "poly p \<noteq> poly [] ==> (poly p a = 0) = (order (a::'a::{idom,ring_char_0}) p \<noteq> 0)"
|
|
698 |
by (rule order_root [THEN ssubst], auto)
|
|
699 |
|
|
700 |
|
|
701 |
lemma pmult_one: "[1] *** p = p"
|
|
702 |
by auto
|
|
703 |
declare pmult_one [simp]
|
|
704 |
|
|
705 |
lemma poly_Nil_zero: "poly [] = poly [0]"
|
|
706 |
by (simp add: fun_eq)
|
|
707 |
|
|
708 |
lemma rsquarefree_decomp:
|
|
709 |
"[| rsquarefree p; poly p a = (0::'a::{idom,ring_char_0}) |]
|
|
710 |
==> \<exists>q. (poly p = poly ([-a, 1] *** q)) & poly q a \<noteq> 0"
|
|
711 |
apply (simp add: rsquarefree_def, safe)
|
|
712 |
apply (frule_tac a = a in order_decomp)
|
|
713 |
apply (drule_tac x = a in spec)
|
|
714 |
apply (drule_tac a = a in order_root2 [symmetric])
|
|
715 |
apply (auto simp del: pmult_Cons)
|
|
716 |
apply (rule_tac x = q in exI, safe)
|
|
717 |
apply (simp add: poly_mult fun_eq)
|
|
718 |
apply (drule_tac p1 = q in poly_linear_divides [THEN iffD1])
|
|
719 |
apply (simp add: divides_def del: pmult_Cons, safe)
|
|
720 |
apply (drule_tac x = "[]" in spec)
|
|
721 |
apply (auto simp add: fun_eq)
|
|
722 |
done
|
|
723 |
|
|
724 |
|
|
725 |
text{*Normalization of a polynomial.*}
|
|
726 |
|
|
727 |
lemma poly_normalize: "poly (pnormalize p) = poly p"
|
|
728 |
apply (induct "p")
|
|
729 |
apply (auto simp add: fun_eq)
|
|
730 |
done
|
|
731 |
declare poly_normalize [simp]
|
|
732 |
|
|
733 |
|
|
734 |
text{*The degree of a polynomial.*}
|
|
735 |
|
|
736 |
lemma lemma_degree_zero:
|
|
737 |
"list_all (%c. c = 0) p \<longleftrightarrow> pnormalize p = []"
|
|
738 |
by (induct "p", auto)
|
|
739 |
|
|
740 |
lemma degree_zero: "(poly p = poly ([]:: (('a::{idom,ring_char_0}) list))) \<Longrightarrow> (degree p = 0)"
|
|
741 |
apply (simp add: degree_def)
|
|
742 |
apply (case_tac "pnormalize p = []")
|
|
743 |
apply (auto simp add: poly_zero lemma_degree_zero )
|
|
744 |
done
|
|
745 |
|
|
746 |
lemma pnormalize_sing: "(pnormalize [x] = [x]) \<longleftrightarrow> x \<noteq> 0" by simp
|
|
747 |
lemma pnormalize_pair: "y \<noteq> 0 \<longleftrightarrow> (pnormalize [x, y] = [x, y])" by simp
|
|
748 |
lemma pnormal_cons: "pnormal p \<Longrightarrow> pnormal (c#p)"
|
|
749 |
unfolding pnormal_def by simp
|
|
750 |
lemma pnormal_tail: "p\<noteq>[] \<Longrightarrow> pnormal (c#p) \<Longrightarrow> pnormal p"
|
|
751 |
unfolding pnormal_def
|
|
752 |
apply (cases "pnormalize p = []", auto)
|
|
753 |
by (cases "c = 0", auto)
|
|
754 |
lemma pnormal_last_nonzero: "pnormal p ==> last p \<noteq> 0"
|
|
755 |
apply (induct p, auto simp add: pnormal_def)
|
|
756 |
apply (case_tac "pnormalize p = []", auto)
|
|
757 |
by (case_tac "a=0", auto)
|
|
758 |
lemma pnormal_length: "pnormal p \<Longrightarrow> 0 < length p"
|
|
759 |
unfolding pnormal_def length_greater_0_conv by blast
|
|
760 |
lemma pnormal_last_length: "\<lbrakk>0 < length p ; last p \<noteq> 0\<rbrakk> \<Longrightarrow> pnormal p"
|
|
761 |
apply (induct p, auto)
|
|
762 |
apply (case_tac "p = []", auto)
|
|
763 |
apply (simp add: pnormal_def)
|
|
764 |
by (rule pnormal_cons, auto)
|
|
765 |
lemma pnormal_id: "pnormal p \<longleftrightarrow> (0 < length p \<and> last p \<noteq> 0)"
|
|
766 |
using pnormal_last_length pnormal_length pnormal_last_nonzero by blast
|
|
767 |
|
|
768 |
text{*Tidier versions of finiteness of roots.*}
|
|
769 |
|
|
770 |
lemma poly_roots_finite_set: "poly p \<noteq> poly [] ==> finite {x::'a::{idom,ring_char_0}. poly p x = 0}"
|
|
771 |
unfolding poly_roots_finite .
|
|
772 |
|
|
773 |
text{*bound for polynomial.*}
|
|
774 |
|
|
775 |
lemma poly_mono: "abs(x) \<le> k ==> abs(poly p (x::'a::{ordered_idom})) \<le> poly (map abs p) k"
|
|
776 |
apply (induct "p", auto)
|
|
777 |
apply (rule_tac y = "abs a + abs (x * poly p x)" in order_trans)
|
|
778 |
apply (rule abs_triangle_ineq)
|
|
779 |
apply (auto intro!: mult_mono simp add: abs_mult)
|
|
780 |
done
|
|
781 |
|
|
782 |
lemma poly_Sing: "poly [c] x = c" by simp
|
|
783 |
end
|