--- a/src/HOL/Library/Univ_Poly.thy Sun Aug 25 21:25:17 2013 +0200
+++ b/src/HOL/Library/Univ_Poly.thy Sun Aug 25 23:20:25 2013 +0200
@@ -10,7 +10,8 @@
text{* Application of polynomial as a function. *}
-primrec (in semiring_0) poly :: "'a list => 'a => 'a" where
+primrec (in semiring_0) poly :: "'a list \<Rightarrow> 'a \<Rightarrow> 'a"
+where
poly_Nil: "poly [] x = 0"
| poly_Cons: "poly (h#t) x = h + x * poly t x"
@@ -22,173 +23,171 @@
primrec (in semiring_0) padd :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list" (infixl "+++" 65)
where
padd_Nil: "[] +++ l2 = l2"
-| padd_Cons: "(h#t) +++ l2 = (if l2 = [] then h#t
- else (h + hd l2)#(t +++ tl l2))"
+| padd_Cons: "(h#t) +++ l2 = (if l2 = [] then h#t else (h + hd l2)#(t +++ tl l2))"
text{*Multiplication by a constant*}
primrec (in semiring_0) cmult :: "'a \<Rightarrow> 'a list \<Rightarrow> 'a list" (infixl "%*" 70) where
- cmult_Nil: "c %* [] = []"
-| cmult_Cons: "c %* (h#t) = (c * h)#(c %* t)"
+ cmult_Nil: "c %* [] = []"
+| cmult_Cons: "c %* (h#t) = (c * h)#(c %* t)"
text{*Multiplication by a polynomial*}
primrec (in semiring_0) pmult :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list" (infixl "***" 70)
where
- pmult_Nil: "[] *** l2 = []"
-| pmult_Cons: "(h#t) *** l2 = (if t = [] then h %* l2
+ pmult_Nil: "[] *** l2 = []"
+| pmult_Cons: "(h#t) *** l2 = (if t = [] then h %* l2
else (h %* l2) +++ ((0) # (t *** l2)))"
text{*Repeated multiplication by a polynomial*}
primrec (in semiring_0) mulexp :: "nat \<Rightarrow> 'a list \<Rightarrow> 'a list \<Rightarrow> 'a list" where
- mulexp_zero: "mulexp 0 p q = q"
-| mulexp_Suc: "mulexp (Suc n) p q = p *** mulexp n p q"
+ mulexp_zero: "mulexp 0 p q = q"
+| mulexp_Suc: "mulexp (Suc n) p q = p *** mulexp n p q"
text{*Exponential*}
primrec (in semiring_1) pexp :: "'a list \<Rightarrow> nat \<Rightarrow> 'a list" (infixl "%^" 80) where
- pexp_0: "p %^ 0 = [1]"
-| pexp_Suc: "p %^ (Suc n) = p *** (p %^ n)"
+ pexp_0: "p %^ 0 = [1]"
+| pexp_Suc: "p %^ (Suc n) = p *** (p %^ n)"
text{*Quotient related value of dividing a polynomial by x + a*}
(* Useful for divisor properties in inductive proofs *)
-primrec (in field) "pquot" :: "'a list \<Rightarrow> 'a \<Rightarrow> 'a list" where
- pquot_Nil: "pquot [] a= []"
-| pquot_Cons: "pquot (h#t) a = (if t = [] then [h]
- else (inverse(a) * (h - hd( pquot t a)))#(pquot t a))"
+primrec (in field) "pquot" :: "'a list \<Rightarrow> 'a \<Rightarrow> 'a list"
+where
+ pquot_Nil: "pquot [] a= []"
+| pquot_Cons: "pquot (h#t) a =
+ (if t = [] then [h] else (inverse(a) * (h - hd( pquot t a)))#(pquot t a))"
text{*normalization of polynomials (remove extra 0 coeff)*}
primrec (in semiring_0) pnormalize :: "'a list \<Rightarrow> 'a list" where
pnormalize_Nil: "pnormalize [] = []"
-| pnormalize_Cons: "pnormalize (h#p) = (if ( (pnormalize p) = [])
- then (if (h = 0) then [] else [h])
- else (h#(pnormalize p)))"
+| pnormalize_Cons: "pnormalize (h#p) =
+ (if pnormalize p = [] then (if h = 0 then [] else [h]) else h # pnormalize p)"
definition (in semiring_0) "pnormal p = ((pnormalize p = p) \<and> p \<noteq> [])"
definition (in semiring_0) "nonconstant p = (pnormal p \<and> (\<forall>x. p \<noteq> [x]))"
text{*Other definitions*}
-definition (in ring_1)
- poly_minus :: "'a list => 'a list" ("-- _" [80] 80) where
- "-- p = (- 1) %* p"
+definition (in ring_1) poly_minus :: "'a list \<Rightarrow> 'a list" ("-- _" [80] 80)
+ where "-- p = (- 1) %* p"
-definition (in semiring_0)
- divides :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool" (infixl "divides" 70) where
- "p1 divides p2 = (\<exists>q. poly p2 = poly(p1 *** q))"
+definition (in semiring_0) divides :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool" (infixl "divides" 70)
+ where "p1 divides p2 = (\<exists>q. poly p2 = poly(p1 *** q))"
--{*order of a polynomial*}
-definition (in ring_1) order :: "'a => 'a list => nat" where
- "order a p = (SOME n. ([-a, 1] %^ n) divides p &
- ~ (([-a, 1] %^ (Suc n)) divides p))"
+definition (in ring_1) order :: "'a \<Rightarrow> 'a list \<Rightarrow> nat" where
+ "order a p = (SOME n. ([-a, 1] %^ n) divides p \<and> ~ (([-a, 1] %^ (Suc n)) divides p))"
--{*degree of a polynomial*}
-definition (in semiring_0) degree :: "'a list => nat" where
- "degree p = length (pnormalize p) - 1"
+definition (in semiring_0) degree :: "'a list \<Rightarrow> nat"
+ where "degree p = length (pnormalize p) - 1"
--{*squarefree polynomials --- NB with respect to real roots only.*}
-definition (in ring_1)
- rsquarefree :: "'a list => bool" where
- "rsquarefree p = (poly p \<noteq> poly [] &
- (\<forall>a. (order a p = 0) | (order a p = 1)))"
+definition (in ring_1) rsquarefree :: "'a list \<Rightarrow> bool"
+ where "rsquarefree p \<longleftrightarrow> poly p \<noteq> poly [] \<and> (\<forall>a. order a p = 0 \<or> order a p = 1)"
context semiring_0
begin
lemma padd_Nil2[simp]: "p +++ [] = p"
-by (induct p) auto
+ by (induct p) auto
lemma padd_Cons_Cons: "(h1 # p1) +++ (h2 # p2) = (h1 + h2) # (p1 +++ p2)"
-by auto
+ by auto
lemma pminus_Nil: "-- [] = []"
-by (simp add: poly_minus_def)
+ by (simp add: poly_minus_def)
lemma pmult_singleton: "[h1] *** p1 = h1 %* p1" by simp
+
end
lemma (in semiring_1) poly_ident_mult[simp]: "1 %* t = t" by (induct t) auto
lemma (in semiring_0) poly_simple_add_Cons[simp]: "[a] +++ ((0)#t) = (a#t)"
-by simp
+ by simp
text{*Handy general properties*}
lemma (in comm_semiring_0) padd_commut: "b +++ a = a +++ b"
-proof(induct b arbitrary: a)
- case Nil thus ?case by auto
+proof (induct b arbitrary: a)
+ case Nil
+ thus ?case by auto
next
- case (Cons b bs a) thus ?case by (cases a) (simp_all add: add_commute)
+ case (Cons b bs a)
+ thus ?case by (cases a) (simp_all add: add_commute)
qed
lemma (in comm_semiring_0) padd_assoc: "\<forall>b c. (a +++ b) +++ c = a +++ (b +++ c)"
-apply (induct a)
-apply (simp, clarify)
-apply (case_tac b, simp_all add: add_ac)
-done
+ apply (induct a)
+ apply (simp, clarify)
+ apply (case_tac b, simp_all add: add_ac)
+ done
lemma (in semiring_0) poly_cmult_distr: "a %* ( p +++ q) = (a %* p +++ a %* q)"
-apply (induct p arbitrary: q, simp)
-apply (case_tac q, simp_all add: distrib_left)
-done
+ apply (induct p arbitrary: q)
+ apply simp
+ apply (case_tac q, simp_all add: distrib_left)
+ done
lemma (in ring_1) pmult_by_x[simp]: "[0, 1] *** t = ((0)#t)"
-apply (induct "t", simp)
-apply (auto simp add: padd_commut)
-apply (case_tac t, auto)
-done
+ apply (induct t)
+ apply simp
+ apply (auto simp add: padd_commut)
+ apply (case_tac t, auto)
+ done
text{*properties of evaluation of polynomials.*}
lemma (in semiring_0) poly_add: "poly (p1 +++ p2) x = poly p1 x + poly p2 x"
proof(induct p1 arbitrary: p2)
- case Nil thus ?case by simp
+ case Nil
+ thus ?case by simp
next
- case (Cons a as p2) thus ?case
+ case (Cons a as p2)
+ thus ?case
by (cases p2) (simp_all add: add_ac distrib_left)
qed
lemma (in comm_semiring_0) poly_cmult: "poly (c %* p) x = c * poly p x"
-apply (induct "p")
-apply (case_tac [2] "x=zero")
-apply (auto simp add: distrib_left mult_ac)
-done
+ apply (induct p)
+ apply (case_tac [2] "x = zero")
+ apply (auto simp add: distrib_left mult_ac)
+ done
lemma (in comm_semiring_0) poly_cmult_map: "poly (map (op * c) p) x = c*poly p x"
- by (induct p, auto simp add: distrib_left mult_ac)
+ by (induct p) (auto simp add: distrib_left mult_ac)
lemma (in comm_ring_1) poly_minus: "poly (-- p) x = - (poly p x)"
-apply (simp add: poly_minus_def)
-apply (auto simp add: poly_cmult)
-done
+ apply (simp add: poly_minus_def)
+ apply (auto simp add: poly_cmult)
+ done
lemma (in comm_semiring_0) poly_mult: "poly (p1 *** p2) x = poly p1 x * poly p2 x"
-proof(induct p1 arbitrary: p2)
- case Nil thus ?case by simp
+proof (induct p1 arbitrary: p2)
+ case Nil
+ thus ?case by simp
next
case (Cons a as p2)
- thus ?case by (cases as,
- simp_all add: poly_cmult poly_add distrib_right distrib_left mult_ac)
+ thus ?case by (cases as)
+ (simp_all add: poly_cmult poly_add distrib_right distrib_left mult_ac)
qed
class idom_char_0 = idom + ring_char_0
lemma (in comm_ring_1) poly_exp: "poly (p %^ n) x = (poly p x) ^ n"
-apply (induct "n")
-apply (auto simp add: poly_cmult poly_mult)
-done
+ by (induct n) (auto simp add: poly_cmult poly_mult)
text{*More Polynomial Evaluation Lemmas*}
-lemma (in semiring_0) poly_add_rzero[simp]: "poly (a +++ []) x = poly a x"
-by simp
+lemma (in semiring_0) poly_add_rzero[simp]: "poly (a +++ []) x = poly a x"
+ by simp
lemma (in comm_semiring_0) poly_mult_assoc: "poly ((a *** b) *** c) x = poly (a *** (b *** c)) x"
by (simp add: poly_mult mult_assoc)
lemma (in semiring_0) poly_mult_Nil2[simp]: "poly (p *** []) x = 0"
-by (induct "p", auto)
+ by (induct p) auto
lemma (in comm_semiring_1) poly_exp_add: "poly (p %^ (n + d)) x = poly( p %^ n *** p %^ d) x"
-apply (induct "n")
-apply (auto simp add: poly_mult mult_assoc)
-done
+ by (induct n) (auto simp add: poly_mult mult_assoc)
subsection{*Key Property: if @{term "f(a) = 0"} then @{term "(x - a)"} divides
@{term "p(x)"} *}
@@ -196,11 +195,11 @@
lemma (in comm_ring_1) lemma_poly_linear_rem: "\<forall>h. \<exists>q r. h#t = [r] +++ [-a, 1] *** q"
proof(induct t)
case Nil
- {fix h have "[h] = [h] +++ [- a, 1] *** []" by simp}
+ { fix h have "[h] = [h] +++ [- a, 1] *** []" by simp }
thus ?case by blast
next
case (Cons x xs)
- {fix h
+ { fix h
from Cons.hyps[rule_format, of x]
obtain q r where qr: "x#xs = [r] +++ [- a, 1] *** q" by blast
have "h#x#xs = [a*r + h] +++ [-a, 1] *** (r#q)"
@@ -210,12 +209,12 @@
qed
lemma (in comm_ring_1) poly_linear_rem: "\<exists>q r. h#t = [r] +++ [-a, 1] *** q"
-by (cut_tac t = t and a = a in lemma_poly_linear_rem, auto)
+ using lemma_poly_linear_rem [where t = t and a = a] by auto
lemma (in comm_ring_1) poly_linear_divides: "(poly p a = 0) = ((p = []) | (\<exists>q. p = [-a, 1] *** q))"
-proof-
- {assume p: "p = []" hence ?thesis by simp}
+proof -
+ { assume p: "p = []" hence ?thesis by simp }
moreover
{
fix x xs assume p: "p = x#xs"
@@ -224,59 +223,68 @@
hence "poly p a = 0" by (simp add: poly_add poly_cmult)
}
moreover
- {assume p0: "poly p a = 0"
+ { assume p0: "poly p a = 0"
from poly_linear_rem[of x xs a] obtain q r
where qr: "x#xs = [r] +++ [- a, 1] *** q" by blast
have "r = 0" using p0 by (simp only: p qr poly_mult poly_add) simp
- 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}
- ultimately have ?thesis using p by blast}
- ultimately show ?thesis by (cases p, auto)
+ 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
+ }
+ ultimately have ?thesis using p by blast
+ }
+ ultimately show ?thesis by (cases p) auto
qed
lemma (in semiring_0) lemma_poly_length_mult[simp]: "\<forall>h k a. length (k %* p +++ (h # (a %* p))) = Suc (length p)"
-by (induct "p", auto)
+ by (induct p) auto
lemma (in semiring_0) lemma_poly_length_mult2[simp]: "\<forall>h k. length (k %* p +++ (h # p)) = Suc (length p)"
-by (induct "p", auto)
+ by (induct p) auto
lemma (in ring_1) poly_length_mult[simp]: "length([-a,1] *** q) = Suc (length q)"
-by auto
+ by auto
subsection{*Polynomial length*}
lemma (in semiring_0) poly_cmult_length[simp]: "length (a %* p) = length p"
-by (induct "p", auto)
+ by (induct p) auto
lemma (in semiring_0) poly_add_length: "length (p1 +++ p2) = max (length p1) (length p2)"
-apply (induct p1 arbitrary: p2, simp_all)
-apply arith
-done
+ by (induct p1 arbitrary: p2) (simp_all, arith)
lemma (in semiring_0) poly_root_mult_length[simp]: "length([a,b] *** p) = Suc (length p)"
-by (simp add: poly_add_length)
+ by (simp add: poly_add_length)
lemma (in idom) poly_mult_not_eq_poly_Nil[simp]:
- "poly (p *** q) x \<noteq> poly [] x \<longleftrightarrow> poly p x \<noteq> poly [] x \<and> poly q x \<noteq> poly [] x"
-by (auto simp add: poly_mult)
+ "poly (p *** q) x \<noteq> poly [] x \<longleftrightarrow> poly p x \<noteq> poly [] x \<and> poly q x \<noteq> poly [] x"
+ by (auto simp add: poly_mult)
lemma (in idom) poly_mult_eq_zero_disj: "poly (p *** q) x = 0 \<longleftrightarrow> poly p x = 0 \<or> poly q x = 0"
-by (auto simp add: poly_mult)
+ by (auto simp add: poly_mult)
text{*Normalisation Properties*}
lemma (in semiring_0) poly_normalized_nil: "(pnormalize p = []) --> (poly p x = 0)"
-by (induct "p", auto)
+ by (induct p) auto
text{*A nontrivial polynomial of degree n has no more than n roots*}
lemma (in idom) poly_roots_index_lemma:
assumes p: "poly p x \<noteq> poly [] x" and n: "length p = n"
shows "\<exists>i. \<forall>x. poly p x = 0 \<longrightarrow> (\<exists>m\<le>n. x = i m)"
using p n
-proof(induct n arbitrary: p x)
- case 0 thus ?case by simp
+proof (induct n arbitrary: p x)
+ case 0
+ thus ?case by simp
next
case (Suc n p x)
- {assume C: "\<And>i. \<exists>x. poly p x = 0 \<and> (\<forall>m\<le>Suc n. x \<noteq> i m)"
+ {
+ assume C: "\<And>i. \<exists>x. poly p x = 0 \<and> (\<forall>m\<le>Suc n. x \<noteq> i m)"
from Suc.prems have p0: "poly p x \<noteq> 0" "p\<noteq> []" by auto
from p0(1)[unfolded poly_linear_divides[of p x]]
have "\<forall>q. p \<noteq> [- x, 1] *** q" by blast
@@ -293,48 +301,49 @@
by blast
from y have "y = a \<or> poly q y = 0"
by (simp only: q poly_mult_eq_zero_disj poly_add) (simp add: algebra_simps)
- with i[rule_format, of y] y(1) y(2) have False apply auto
- apply (erule_tac x="m" in allE)
+ with i[rule_format, of y] y(1) y(2) have False
apply auto
- done}
+ apply (erule_tac x = "m" in allE)
+ apply auto
+ done
+ }
thus ?case by blast
qed
-lemma (in idom) poly_roots_index_length: "poly p x \<noteq> poly [] x ==>
- \<exists>i. \<forall>x. (poly p x = 0) --> (\<exists>n. n \<le> length p & x = i n)"
-by (blast intro: poly_roots_index_lemma)
+lemma (in idom) poly_roots_index_length:
+ "poly p x \<noteq> poly [] x \<Longrightarrow> \<exists>i. \<forall>x. (poly p x = 0) \<longrightarrow> (\<exists>n. n \<le> length p \<and> x = i n)"
+ by (blast intro: poly_roots_index_lemma)
-lemma (in idom) poly_roots_finite_lemma1: "poly p x \<noteq> poly [] x ==>
- \<exists>N i. \<forall>x. (poly p x = 0) --> (\<exists>n. (n::nat) < N & x = i n)"
-apply (drule poly_roots_index_length, safe)
-apply (rule_tac x = "Suc (length p)" in exI)
-apply (rule_tac x = i in exI)
-apply (simp add: less_Suc_eq_le)
-done
-
+lemma (in idom) poly_roots_finite_lemma1:
+ "poly p x \<noteq> poly [] x \<Longrightarrow> \<exists>N i. \<forall>x. (poly p x = 0) \<longrightarrow> (\<exists>n. (n::nat) < N \<and> x = i n)"
+ apply (drule poly_roots_index_length, safe)
+ apply (rule_tac x = "Suc (length p)" in exI)
+ apply (rule_tac x = i in exI)
+ apply (simp add: less_Suc_eq_le)
+ done
lemma (in idom) idom_finite_lemma:
- assumes P: "\<forall>x. P x --> (\<exists>n. n < length j & x = j!n)"
+ assumes P: "\<forall>x. P x --> (\<exists>n. n < length j \<and> x = j!n)"
shows "finite {x. P x}"
-proof-
+proof -
let ?M = "{x. P x}"
let ?N = "set j"
have "?M \<subseteq> ?N" using P by auto
thus ?thesis using finite_subset by auto
qed
+lemma (in idom) poly_roots_finite_lemma2:
+ "poly p x \<noteq> poly [] x \<Longrightarrow> \<exists>i. \<forall>x. poly p x = 0 \<longrightarrow> x \<in> set i"
+ apply (drule poly_roots_index_length, safe)
+ apply (rule_tac x="map (\<lambda>n. i n) [0 ..< Suc (length p)]" in exI)
+ apply (auto simp add: image_iff)
+ apply (erule_tac x="x" in allE, clarsimp)
+ apply (case_tac "n = length p")
+ apply (auto simp add: order_le_less)
+ done
-lemma (in idom) poly_roots_finite_lemma2: "poly p x \<noteq> poly [] x ==>
- \<exists>i. \<forall>x. (poly p x = 0) --> x \<in> set i"
-apply (drule poly_roots_index_length, safe)
-apply (rule_tac x="map (\<lambda>n. i n) [0 ..< Suc (length p)]" in exI)
-apply (auto simp add: image_iff)
-apply (erule_tac x="x" in allE, clarsimp)
-by (case_tac "n=length p", auto simp add: order_le_less)
-
-lemma (in ring_char_0) UNIV_ring_char_0_infinte:
- "\<not> (finite (UNIV:: 'a set))"
+lemma (in ring_char_0) UNIV_ring_char_0_infinte: "\<not> (finite (UNIV:: 'a set))"
proof
assume F: "finite (UNIV :: 'a set)"
have "finite (UNIV :: nat set)"
@@ -346,8 +355,7 @@
with infinite_UNIV_nat show False ..
qed
-lemma (in idom_char_0) poly_roots_finite: "(poly p \<noteq> poly []) =
- finite {x. poly p x = 0}"
+lemma (in idom_char_0) poly_roots_finite: "poly p \<noteq> poly [] \<longleftrightarrow> finite {x. poly p x = 0}"
proof
assume H: "poly p \<noteq> poly []"
show "finite {x. poly p x = (0::'a)}"
@@ -357,7 +365,7 @@
apply (rule ccontr)
apply (clarify dest!: poly_roots_finite_lemma2)
using finite_subset
- proof-
+ proof -
fix x i
assume F: "\<not> finite {x. poly p x = (0\<Colon>'a)}"
and P: "\<forall>x. poly p x = (0\<Colon>'a) \<longrightarrow> x \<in> set i"
@@ -373,9 +381,10 @@
text{*Entirety and Cancellation for polynomials*}
lemma (in idom_char_0) poly_entire_lemma2:
- assumes p0: "poly p \<noteq> poly []" and q0: "poly q \<noteq> poly []"
+ assumes p0: "poly p \<noteq> poly []"
+ and q0: "poly q \<noteq> poly []"
shows "poly (p***q) \<noteq> poly []"
-proof-
+proof -
let ?S = "\<lambda>p. {x. poly p x = 0}"
have "?S (p *** q) = ?S p \<union> ?S q" by (auto simp add: poly_mult)
with p0 q0 show ?thesis unfolding poly_roots_finite by auto
@@ -383,74 +392,82 @@
lemma (in idom_char_0) poly_entire:
"poly (p *** q) = poly [] \<longleftrightarrow> poly p = poly [] \<or> poly q = poly []"
-using poly_entire_lemma2[of p q]
-by (auto simp add: fun_eq_iff poly_mult)
+ using poly_entire_lemma2[of p q]
+ by (auto simp add: fun_eq_iff poly_mult)
-lemma (in idom_char_0) poly_entire_neg: "(poly (p *** q) \<noteq> poly []) = ((poly p \<noteq> poly []) & (poly q \<noteq> poly []))"
-by (simp add: poly_entire)
+lemma (in idom_char_0) poly_entire_neg:
+ "poly (p *** q) \<noteq> poly [] \<longleftrightarrow> poly p \<noteq> poly [] \<and> poly q \<noteq> poly []"
+ by (simp add: poly_entire)
-lemma fun_eq: " (f = g) = (\<forall>x. f x = g x)"
-by auto
+lemma fun_eq: "f = g \<longleftrightarrow> (\<forall>x. f x = g x)"
+ by auto
-lemma (in comm_ring_1) poly_add_minus_zero_iff: "(poly (p +++ -- q) = poly []) = (poly p = poly q)"
-by (auto simp add: algebra_simps poly_add poly_minus_def fun_eq poly_cmult)
+lemma (in comm_ring_1) poly_add_minus_zero_iff:
+ "poly (p +++ -- q) = poly [] \<longleftrightarrow> poly p = poly q"
+ by (auto simp add: algebra_simps poly_add poly_minus_def fun_eq poly_cmult)
-lemma (in comm_ring_1) poly_add_minus_mult_eq: "poly (p *** q +++ --(p *** r)) = poly (p *** (q +++ -- r))"
-by (auto simp add: poly_add poly_minus_def fun_eq poly_mult poly_cmult distrib_left)
+lemma (in comm_ring_1) poly_add_minus_mult_eq:
+ "poly (p *** q +++ --(p *** r)) = poly (p *** (q +++ -- r))"
+ by (auto simp add: poly_add poly_minus_def fun_eq poly_mult poly_cmult distrib_left)
subclass (in idom_char_0) comm_ring_1 ..
-lemma (in idom_char_0) poly_mult_left_cancel: "(poly (p *** q) = poly (p *** r)) = (poly p = poly [] | poly q = poly r)"
-proof-
- have "poly (p *** q) = poly (p *** r) \<longleftrightarrow> poly (p *** q +++ -- (p *** r)) = poly []" by (simp only: poly_add_minus_zero_iff)
- also have "\<dots> \<longleftrightarrow> poly p = poly [] | poly q = poly r"
+
+lemma (in idom_char_0) poly_mult_left_cancel:
+ "poly (p *** q) = poly (p *** r) \<longleftrightarrow> poly p = poly [] \<or> poly q = poly r"
+proof -
+ have "poly (p *** q) = poly (p *** r) \<longleftrightarrow> poly (p *** q +++ -- (p *** r)) = poly []"
+ by (simp only: poly_add_minus_zero_iff)
+ also have "\<dots> \<longleftrightarrow> poly p = poly [] \<or> poly q = poly r"
by (auto intro: simp add: poly_add_minus_mult_eq poly_entire poly_add_minus_zero_iff)
finally show ?thesis .
qed
lemma (in idom) poly_exp_eq_zero[simp]:
- "(poly (p %^ n) = poly []) = (poly p = poly [] & n \<noteq> 0)"
-apply (simp only: fun_eq add: HOL.all_simps [symmetric])
-apply (rule arg_cong [where f = All])
-apply (rule ext)
-apply (induct n)
-apply (auto simp add: poly_exp poly_mult)
-done
+ "poly (p %^ n) = poly [] \<longleftrightarrow> poly p = poly [] \<and> n \<noteq> 0"
+ apply (simp only: fun_eq add: HOL.all_simps [symmetric])
+ apply (rule arg_cong [where f = All])
+ apply (rule ext)
+ apply (induct n)
+ apply (auto simp add: poly_exp poly_mult)
+ done
lemma (in comm_ring_1) poly_prime_eq_zero[simp]: "poly [a,1] \<noteq> poly []"
-apply (simp add: fun_eq)
-apply (rule_tac x = "minus one a" in exI)
-apply (unfold diff_minus)
-apply (subst add_commute)
-apply (subst add_assoc)
-apply simp
-done
+ apply (simp add: fun_eq)
+ apply (rule_tac x = "minus one a" in exI)
+ apply (unfold diff_minus)
+ apply (subst add_commute)
+ apply (subst add_assoc)
+ apply simp
+ done
-lemma (in idom) poly_exp_prime_eq_zero: "(poly ([a, 1] %^ n) \<noteq> poly [])"
-by auto
+lemma (in idom) poly_exp_prime_eq_zero: "poly ([a, 1] %^ n) \<noteq> poly []"
+ by auto
text{*A more constructive notion of polynomials being trivial*}
-lemma (in idom_char_0) poly_zero_lemma': "poly (h # t) = poly [] ==> h = 0 & poly t = poly []"
-apply(simp add: fun_eq)
-apply (case_tac "h = zero")
-apply (drule_tac [2] x = zero in spec, auto)
-apply (cases "poly t = poly []", simp)
-proof-
+lemma (in idom_char_0) poly_zero_lemma': "poly (h # t) = poly [] \<Longrightarrow> h = 0 \<and> poly t = poly []"
+ apply (simp add: fun_eq)
+ apply (case_tac "h = zero")
+ apply (drule_tac [2] x = zero in spec, auto)
+ apply (cases "poly t = poly []", simp)
+proof -
fix x
- assume H: "\<forall>x. x = (0\<Colon>'a) \<or> poly t x = (0\<Colon>'a)" and pnz: "poly t \<noteq> poly []"
+ assume H: "\<forall>x. x = (0\<Colon>'a) \<or> poly t x = (0\<Colon>'a)"
+ and pnz: "poly t \<noteq> poly []"
let ?S = "{x. poly t x = 0}"
from H have "\<forall>x. x \<noteq>0 \<longrightarrow> poly t x = 0" by blast
hence th: "?S \<supseteq> UNIV - {0}" by auto
from poly_roots_finite pnz have th': "finite ?S" by blast
- from finite_subset[OF th th'] UNIV_ring_char_0_infinte
- show "poly t x = (0\<Colon>'a)" by simp
- qed
+ from finite_subset[OF th th'] UNIV_ring_char_0_infinte show "poly t x = (0\<Colon>'a)"
+ by simp
+qed
lemma (in idom_char_0) poly_zero: "(poly p = poly []) = list_all (%c. c = 0) p"
-apply (induct "p", simp)
-apply (rule iffI)
-apply (drule poly_zero_lemma', auto)
-done
+ apply (induct p)
+ apply simp
+ apply (rule iffI)
+ apply (drule poly_zero_lemma', auto)
+ done
lemma (in idom_char_0) poly_0: "list_all (\<lambda>c. c = 0) p \<Longrightarrow> poly p x = 0"
unfolding poly_zero[symmetric] by simp
@@ -459,115 +476,126 @@
text{*Basics of divisibility.*}
-lemma (in idom) poly_primes: "([a, 1] divides (p *** q)) = ([a, 1] divides p | [a, 1] divides q)"
-apply (auto simp add: divides_def fun_eq poly_mult poly_add poly_cmult distrib_right [symmetric])
-apply (drule_tac x = "uminus a" in spec)
-apply (simp add: poly_linear_divides poly_add poly_cmult distrib_right [symmetric])
-apply (cases "p = []")
-apply (rule exI[where x="[]"])
-apply simp
-apply (cases "q = []")
-apply (erule allE[where x="[]"], simp)
+lemma (in idom) poly_primes:
+ "[a, 1] divides (p *** q) \<longleftrightarrow> [a, 1] divides p \<or> [a, 1] divides q"
+ apply (auto simp add: divides_def fun_eq poly_mult poly_add poly_cmult distrib_right [symmetric])
+ apply (drule_tac x = "uminus a" in spec)
+ apply (simp add: poly_linear_divides poly_add poly_cmult distrib_right [symmetric])
+ apply (cases "p = []")
+ apply (rule exI[where x="[]"])
+ apply simp
+ apply (cases "q = []")
+ apply (erule allE[where x="[]"], simp)
-apply clarsimp
-apply (cases "\<exists>q\<Colon>'a list. p = a %* q +++ ((0\<Colon>'a) # q)")
-apply (clarsimp simp add: poly_add poly_cmult)
-apply (rule_tac x="qa" in exI)
-apply (simp add: distrib_right [symmetric])
-apply clarsimp
+ apply clarsimp
+ apply (cases "\<exists>q\<Colon>'a list. p = a %* q +++ ((0\<Colon>'a) # q)")
+ apply (clarsimp simp add: poly_add poly_cmult)
+ apply (rule_tac x="qa" in exI)
+ apply (simp add: distrib_right [symmetric])
+ apply clarsimp
-apply (auto simp add: poly_linear_divides poly_add poly_cmult distrib_right [symmetric])
-apply (rule_tac x = "pmult qa q" in exI)
-apply (rule_tac [2] x = "pmult p qa" in exI)
-apply (auto simp add: poly_add poly_mult poly_cmult mult_ac)
-done
+ apply (auto simp add: poly_linear_divides poly_add poly_cmult distrib_right [symmetric])
+ apply (rule_tac x = "pmult qa q" in exI)
+ apply (rule_tac [2] x = "pmult p qa" in exI)
+ apply (auto simp add: poly_add poly_mult poly_cmult mult_ac)
+ done
lemma (in comm_semiring_1) poly_divides_refl[simp]: "p divides p"
-apply (simp add: divides_def)
-apply (rule_tac x = "[one]" in exI)
-apply (auto simp add: poly_mult fun_eq)
-done
+ apply (simp add: divides_def)
+ apply (rule_tac x = "[one]" in exI)
+ apply (auto simp add: poly_mult fun_eq)
+ done
-lemma (in comm_semiring_1) poly_divides_trans: "[| p divides q; q divides r |] ==> p divides r"
-apply (simp add: divides_def, safe)
-apply (rule_tac x = "pmult qa qaa" in exI)
-apply (auto simp add: poly_mult fun_eq mult_assoc)
-done
-
+lemma (in comm_semiring_1) poly_divides_trans: "p divides q \<Longrightarrow> q divides r \<Longrightarrow> p divides r"
+ apply (simp add: divides_def, safe)
+ apply (rule_tac x = "pmult qa qaa" in exI)
+ apply (auto simp add: poly_mult fun_eq mult_assoc)
+ done
-lemma (in comm_semiring_1) poly_divides_exp: "m \<le> n ==> (p %^ m) divides (p %^ n)"
-apply (auto simp add: le_iff_add)
-apply (induct_tac k)
-apply (rule_tac [2] poly_divides_trans)
-apply (auto simp add: divides_def)
-apply (rule_tac x = p in exI)
-apply (auto simp add: poly_mult fun_eq mult_ac)
-done
+lemma (in comm_semiring_1) poly_divides_exp: "m \<le> n \<Longrightarrow> (p %^ m) divides (p %^ n)"
+ apply (auto simp add: le_iff_add)
+ apply (induct_tac k)
+ apply (rule_tac [2] poly_divides_trans)
+ apply (auto simp add: divides_def)
+ apply (rule_tac x = p in exI)
+ apply (auto simp add: poly_mult fun_eq mult_ac)
+ done
-lemma (in comm_semiring_1) poly_exp_divides: "[| (p %^ n) divides q; m\<le>n |] ==> (p %^ m) divides q"
-by (blast intro: poly_divides_exp poly_divides_trans)
+lemma (in comm_semiring_1) poly_exp_divides:
+ "(p %^ n) divides q \<Longrightarrow> m \<le> n \<Longrightarrow> (p %^ m) divides q"
+ by (blast intro: poly_divides_exp poly_divides_trans)
lemma (in comm_semiring_0) poly_divides_add:
- "[| p divides q; p divides r |] ==> p divides (q +++ r)"
-apply (simp add: divides_def, auto)
-apply (rule_tac x = "padd qa qaa" in exI)
-apply (auto simp add: poly_add fun_eq poly_mult distrib_left)
-done
+ "p divides q \<Longrightarrow> p divides r \<Longrightarrow> p divides (q +++ r)"
+ apply (simp add: divides_def, auto)
+ apply (rule_tac x = "padd qa qaa" in exI)
+ apply (auto simp add: poly_add fun_eq poly_mult distrib_left)
+ done
lemma (in comm_ring_1) poly_divides_diff:
- "[| p divides q; p divides (q +++ r) |] ==> p divides r"
-apply (simp add: divides_def, auto)
-apply (rule_tac x = "padd qaa (poly_minus qa)" in exI)
-apply (auto simp add: poly_add fun_eq poly_mult poly_minus algebra_simps)
-done
+ "p divides q \<Longrightarrow> p divides (q +++ r) \<Longrightarrow> p divides r"
+ apply (simp add: divides_def, auto)
+ apply (rule_tac x = "padd qaa (poly_minus qa)" in exI)
+ apply (auto simp add: poly_add fun_eq poly_mult poly_minus algebra_simps)
+ done
-lemma (in comm_ring_1) poly_divides_diff2: "[| p divides r; p divides (q +++ r) |] ==> p divides q"
-apply (erule poly_divides_diff)
-apply (auto simp add: poly_add fun_eq poly_mult divides_def add_ac)
-done
+lemma (in comm_ring_1) poly_divides_diff2:
+ "p divides r \<Longrightarrow> p divides (q +++ r) \<Longrightarrow> p divides q"
+ apply (erule poly_divides_diff)
+ apply (auto simp add: poly_add fun_eq poly_mult divides_def add_ac)
+ done
-lemma (in semiring_0) poly_divides_zero: "poly p = poly [] ==> q divides p"
-apply (simp add: divides_def)
-apply (rule exI[where x="[]"])
-apply (auto simp add: fun_eq poly_mult)
-done
+lemma (in semiring_0) poly_divides_zero: "poly p = poly [] \<Longrightarrow> q divides p"
+ apply (simp add: divides_def)
+ apply (rule exI[where x="[]"])
+ apply (auto simp add: fun_eq poly_mult)
+ done
-lemma (in semiring_0) poly_divides_zero2[simp]: "q divides []"
-apply (simp add: divides_def)
-apply (rule_tac x = "[]" in exI)
-apply (auto simp add: fun_eq)
-done
+lemma (in semiring_0) poly_divides_zero2 [simp]: "q divides []"
+ apply (simp add: divides_def)
+ apply (rule_tac x = "[]" in exI)
+ apply (auto simp add: fun_eq)
+ done
text{*At last, we can consider the order of a root.*}
-lemma (in idom_char_0) poly_order_exists_lemma:
- assumes lp: "length p = d" and p: "poly p \<noteq> poly []"
+lemma (in idom_char_0) poly_order_exists_lemma:
+ assumes lp: "length p = d"
+ and p: "poly p \<noteq> poly []"
shows "\<exists>n q. p = mulexp n [-a, 1] q \<and> poly q a \<noteq> 0"
-using lp p
-proof(induct d arbitrary: p)
- case 0 thus ?case by simp
+ using lp p
+proof (induct d arbitrary: p)
+ case 0
+ thus ?case by simp
next
case (Suc n p)
- {assume p0: "poly p a = 0"
+ show ?case
+ proof (cases "poly p a = 0")
+ case True
from Suc.prems have h: "length p = Suc n" "poly p \<noteq> poly []" by auto
hence pN: "p \<noteq> []" by auto
- from p0[unfolded poly_linear_divides] pN obtain q where
- q: "p = [-a, 1] *** q" by blast
- from q h p0 have qh: "length q = n" "poly q \<noteq> poly []"
+ from True[unfolded poly_linear_divides] pN obtain q where q: "p = [-a, 1] *** q"
+ by blast
+ from q h True have qh: "length q = n" "poly q \<noteq> poly []"
apply -
apply simp
apply (simp only: fun_eq)
apply (rule ccontr)
apply (simp add: fun_eq poly_add poly_cmult)
done
- from Suc.hyps[OF qh] obtain m r where
- mr: "q = mulexp m [-a,1] r" "poly r a \<noteq> 0" by blast
+ from Suc.hyps[OF qh] obtain m r where mr: "q = mulexp m [-a,1] r" "poly r a \<noteq> 0"
+ by blast
from mr q have "p = mulexp (Suc m) [-a,1] r \<and> poly r a \<noteq> 0" by simp
- hence ?case by blast}
- moreover
- {assume p0: "poly p a \<noteq> 0"
- hence ?case using Suc.prems apply simp by (rule exI[where x="0::nat"], simp)}
- ultimately show ?case by blast
+ then show ?thesis by blast
+ next
+ case False
+ then show ?thesis
+ using Suc.prems
+ apply simp
+ apply (rule exI[where x="0::nat"])
+ apply simp
+ done
+ qed
qed
@@ -585,263 +613,240 @@
qed
-
(* FIXME: Tidy up *)
-lemma (in semiring_1)
- zero_power_iff: "0 ^ n = (if n = 0 then 1 else 0)"
+lemma (in semiring_1) zero_power_iff: "0 ^ n = (if n = 0 then 1 else 0)"
by (induct n) simp_all
lemma (in idom_char_0) poly_order_exists:
- assumes lp: "length p = d" and p0: "poly p \<noteq> poly []"
- shows "\<exists>n. ([-a, 1] %^ n) divides p & ~(([-a, 1] %^ (Suc n)) divides p)"
-proof-
-let ?poly = poly
-let ?mulexp = mulexp
-let ?pexp = pexp
-from lp p0
-show ?thesis
-apply -
-apply (drule poly_order_exists_lemma [where a=a], assumption, clarify)
-apply (rule_tac x = n in exI, safe)
-apply (unfold divides_def)
-apply (rule_tac x = q in exI)
-apply (induct_tac "n", simp)
-apply (simp (no_asm_simp) add: poly_add poly_cmult poly_mult distrib_left mult_ac)
-apply safe
-apply (subgoal_tac "?poly (?mulexp n [uminus a, one] q) \<noteq> ?poly (pmult (?pexp [uminus a, one] (Suc n)) qa)")
-apply simp
-apply (induct_tac "n")
-apply (simp del: pmult_Cons pexp_Suc)
-apply (erule_tac Q = "?poly q a = zero" in contrapos_np)
-apply (simp add: poly_add poly_cmult)
-apply (rule pexp_Suc [THEN ssubst])
-apply (rule ccontr)
-apply (simp add: poly_mult_left_cancel poly_mult_assoc del: pmult_Cons pexp_Suc)
-done
-qed
-
+ assumes "length p = d" and "poly p \<noteq> poly []"
+ shows "\<exists>n. ([-a, 1] %^ n) divides p \<and> ~(([-a, 1] %^ (Suc n)) divides p)"
+ using assms
+ apply -
+ apply (drule poly_order_exists_lemma [where a=a], assumption, clarify)
+ apply (rule_tac x = n in exI, safe)
+ apply (unfold divides_def)
+ apply (rule_tac x = q in exI)
+ apply (induct_tac n, simp)
+ apply (simp (no_asm_simp) add: poly_add poly_cmult poly_mult distrib_left mult_ac)
+ apply safe
+ apply (subgoal_tac "poly (mulexp n [uminus a, one] q) \<noteq>
+ poly (pmult (pexp [uminus a, one] (Suc n)) qa)")
+ apply simp
+ apply (induct_tac n)
+ apply (simp del: pmult_Cons pexp_Suc)
+ apply (erule_tac Q = "poly q a = zero" in contrapos_np)
+ apply (simp add: poly_add poly_cmult)
+ apply (rule pexp_Suc [THEN ssubst])
+ apply (rule ccontr)
+ apply (simp add: poly_mult_left_cancel poly_mult_assoc del: pmult_Cons pexp_Suc)
+ done
lemma (in semiring_1) poly_one_divides[simp]: "[1] divides p"
-by (simp add: divides_def, auto)
+ by (auto simp add: divides_def)
-lemma (in idom_char_0) poly_order: "poly p \<noteq> poly []
- ==> EX! n. ([-a, 1] %^ n) divides p &
- ~(([-a, 1] %^ (Suc n)) divides p)"
-apply (auto intro: poly_order_exists simp add: less_linear simp del: pmult_Cons pexp_Suc)
-apply (cut_tac x = y and y = n in less_linear)
-apply (drule_tac m = n in poly_exp_divides)
-apply (auto dest: Suc_le_eq [THEN iffD2, THEN [2] poly_exp_divides]
- simp del: pmult_Cons pexp_Suc)
-done
+lemma (in idom_char_0) poly_order:
+ "poly p \<noteq> poly [] \<Longrightarrow> \<exists>!n. ([-a, 1] %^ n) divides p \<and> \<not> (([-a, 1] %^ Suc n) divides p)"
+ apply (auto intro: poly_order_exists simp add: less_linear simp del: pmult_Cons pexp_Suc)
+ apply (cut_tac x = y and y = n in less_linear)
+ apply (drule_tac m = n in poly_exp_divides)
+ apply (auto dest: Suc_le_eq [THEN iffD2, THEN [2] poly_exp_divides]
+ simp del: pmult_Cons pexp_Suc)
+ done
text{*Order*}
-lemma some1_equalityD: "[| n = (@n. P n); EX! n. P n |] ==> P n"
-by (blast intro: someI2)
+lemma some1_equalityD: "n = (SOME n. P n) \<Longrightarrow> \<exists>!n. P n \<Longrightarrow> P n"
+ by (blast intro: someI2)
lemma (in idom_char_0) order:
- "(([-a, 1] %^ n) divides p &
+ "(([-a, 1] %^ n) divides p \<and>
~(([-a, 1] %^ (Suc n)) divides p)) =
- ((n = order a p) & ~(poly p = poly []))"
-apply (unfold order_def)
-apply (rule iffI)
-apply (blast dest: poly_divides_zero intro!: some1_equality [symmetric] poly_order)
-apply (blast intro!: poly_order [THEN [2] some1_equalityD])
-done
+ ((n = order a p) \<and> ~(poly p = poly []))"
+ apply (unfold order_def)
+ apply (rule iffI)
+ apply (blast dest: poly_divides_zero intro!: some1_equality [symmetric] poly_order)
+ apply (blast intro!: poly_order [THEN [2] some1_equalityD])
+ done
-lemma (in idom_char_0) order2: "[| poly p \<noteq> poly [] |]
- ==> ([-a, 1] %^ (order a p)) divides p &
- ~(([-a, 1] %^ (Suc(order a p))) divides p)"
-by (simp add: order del: pexp_Suc)
+lemma (in idom_char_0) order2:
+ "poly p \<noteq> poly [] \<Longrightarrow>
+ ([-a, 1] %^ (order a p)) divides p \<and> \<not> (([-a, 1] %^ (Suc (order a p))) divides p)"
+ by (simp add: order del: pexp_Suc)
-lemma (in idom_char_0) order_unique: "[| poly p \<noteq> poly []; ([-a, 1] %^ n) divides p;
- ~(([-a, 1] %^ (Suc n)) divides p)
- |] ==> (n = order a p)"
-by (insert order [of a n p], auto)
+lemma (in idom_char_0) order_unique:
+ "poly p \<noteq> poly [] \<Longrightarrow> ([-a, 1] %^ n) divides p \<Longrightarrow> ~(([-a, 1] %^ (Suc n)) divides p) \<Longrightarrow>
+ n = order a p"
+ using order [of a n p] by auto
-lemma (in idom_char_0) order_unique_lemma: "(poly p \<noteq> poly [] & ([-a, 1] %^ n) divides p &
- ~(([-a, 1] %^ (Suc n)) divides p))
- ==> (n = order a p)"
-by (blast intro: order_unique)
+lemma (in idom_char_0) order_unique_lemma:
+ "poly p \<noteq> poly [] \<and> ([-a, 1] %^ n) divides p \<and> ~(([-a, 1] %^ (Suc n)) divides p) \<Longrightarrow>
+ n = order a p"
+ by (blast intro: order_unique)
-lemma (in ring_1) order_poly: "poly p = poly q ==> order a p = order a q"
+lemma (in ring_1) order_poly: "poly p = poly q \<Longrightarrow> order a p = order a q"
by (auto simp add: fun_eq divides_def poly_mult order_def)
lemma (in semiring_1) pexp_one[simp]: "p %^ (Suc 0) = p"
by (induct "p") auto
lemma (in comm_ring_1) lemma_order_root:
- " 0 < n & [- a, 1] %^ n divides p & ~ [- a, 1] %^ (Suc n) divides p
- \<Longrightarrow> poly p a = 0"
-apply (induct n arbitrary: a p, blast)
-apply (auto simp add: divides_def poly_mult simp del: pmult_Cons)
-done
+ "0 < n \<and> [- a, 1] %^ n divides p \<and> ~ [- a, 1] %^ (Suc n) divides p \<Longrightarrow> poly p a = 0"
+ by (induct n arbitrary: a p) (auto simp add: divides_def poly_mult simp del: pmult_Cons)
-lemma (in idom_char_0) order_root: "(poly p a = 0) = ((poly p = poly []) | order a p \<noteq> 0)"
-proof-
- let ?poly = poly
- show ?thesis
-apply (case_tac "?poly p = ?poly []", auto)
-apply (simp add: poly_linear_divides del: pmult_Cons, safe)
-apply (drule_tac [!] a = a in order2)
-apply (rule ccontr)
-apply (simp add: divides_def poly_mult fun_eq del: pmult_Cons, blast)
-using neq0_conv
-apply (blast intro: lemma_order_root)
-done
-qed
+lemma (in idom_char_0) order_root:
+ "poly p a = 0 \<longleftrightarrow> poly p = poly [] \<or> order a p \<noteq> 0"
+ apply (cases "poly p = poly []")
+ apply auto
+ apply (simp add: poly_linear_divides del: pmult_Cons, safe)
+ apply (drule_tac [!] a = a in order2)
+ apply (rule ccontr)
+ apply (simp add: divides_def poly_mult fun_eq del: pmult_Cons, blast)
+ using neq0_conv
+ apply (blast intro: lemma_order_root)
+ done
-lemma (in idom_char_0) order_divides: "(([-a, 1] %^ n) divides p) = ((poly p = poly []) | n \<le> order a p)"
-proof-
- let ?poly = poly
- show ?thesis
-apply (case_tac "?poly p = ?poly []", auto)
-apply (simp add: divides_def fun_eq poly_mult)
-apply (rule_tac x = "[]" in exI)
-apply (auto dest!: order2 [where a=a]
- intro: poly_exp_divides simp del: pexp_Suc)
-done
-qed
+lemma (in idom_char_0) order_divides:
+ "([-a, 1] %^ n) divides p \<longleftrightarrow> poly p = poly [] \<or> n \<le> order a p"
+ apply (cases "poly p = poly []")
+ apply auto
+ apply (simp add: divides_def fun_eq poly_mult)
+ apply (rule_tac x = "[]" in exI)
+ apply (auto dest!: order2 [where a=a] intro: poly_exp_divides simp del: pexp_Suc)
+ done
lemma (in idom_char_0) order_decomp:
- "poly p \<noteq> poly []
- ==> \<exists>q. (poly p = poly (([-a, 1] %^ (order a p)) *** q)) &
- ~([-a, 1] divides q)"
-apply (unfold divides_def)
-apply (drule order2 [where a = a])
-apply (simp add: divides_def del: pexp_Suc pmult_Cons, safe)
-apply (rule_tac x = q in exI, safe)
-apply (drule_tac x = qa in spec)
-apply (auto simp add: poly_mult fun_eq poly_exp mult_ac simp del: pmult_Cons)
-done
+ "poly p \<noteq> poly [] \<Longrightarrow> \<exists>q. poly p = poly (([-a, 1] %^ (order a p)) *** q) \<and> ~([-a, 1] divides q)"
+ apply (unfold divides_def)
+ apply (drule order2 [where a = a])
+ apply (simp add: divides_def del: pexp_Suc pmult_Cons, safe)
+ apply (rule_tac x = q in exI, safe)
+ apply (drule_tac x = qa in spec)
+ apply (auto simp add: poly_mult fun_eq poly_exp mult_ac simp del: pmult_Cons)
+ done
text{*Important composition properties of orders.*}
-lemma order_mult: "poly (p *** q) \<noteq> poly []
- ==> order a (p *** q) = order a p + order (a::'a::{idom_char_0}) q"
-apply (cut_tac a = a and p = "p *** q" and n = "order a p + order a q" in order)
-apply (auto simp add: poly_entire simp del: pmult_Cons)
-apply (drule_tac a = a in order2)+
-apply safe
-apply (simp add: divides_def fun_eq poly_exp_add poly_mult del: pmult_Cons, safe)
-apply (rule_tac x = "qa *** qaa" in exI)
-apply (simp add: poly_mult mult_ac del: pmult_Cons)
-apply (drule_tac a = a in order_decomp)+
-apply safe
-apply (subgoal_tac "[-a,1] divides (qa *** qaa) ")
-apply (simp add: poly_primes del: pmult_Cons)
-apply (auto simp add: divides_def simp del: pmult_Cons)
-apply (rule_tac x = qb in exI)
-apply (subgoal_tac "poly ([-a, 1] %^ (order a p) *** (qa *** qaa)) = poly ([-a, 1] %^ (order a p) *** ([-a, 1] *** qb))")
-apply (drule poly_mult_left_cancel [THEN iffD1], force)
-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))) ")
-apply (drule poly_mult_left_cancel [THEN iffD1], force)
-apply (simp add: fun_eq poly_exp_add poly_mult mult_ac del: pmult_Cons)
-done
+lemma order_mult:
+ "poly (p *** q) \<noteq> poly [] \<Longrightarrow>
+ order a (p *** q) = order a p + order (a::'a::{idom_char_0}) q"
+ apply (cut_tac a = a and p = "p *** q" and n = "order a p + order a q" in order)
+ apply (auto simp add: poly_entire simp del: pmult_Cons)
+ apply (drule_tac a = a in order2)+
+ apply safe
+ apply (simp add: divides_def fun_eq poly_exp_add poly_mult del: pmult_Cons, safe)
+ apply (rule_tac x = "qa *** qaa" in exI)
+ apply (simp add: poly_mult mult_ac del: pmult_Cons)
+ apply (drule_tac a = a in order_decomp)+
+ apply safe
+ apply (subgoal_tac "[-a,1] divides (qa *** qaa) ")
+ apply (simp add: poly_primes del: pmult_Cons)
+ apply (auto simp add: divides_def simp del: pmult_Cons)
+ apply (rule_tac x = qb in exI)
+ apply (subgoal_tac "poly ([-a, 1] %^ (order a p) *** (qa *** qaa)) = poly ([-a, 1] %^ (order a p) *** ([-a, 1] *** qb))")
+ apply (drule poly_mult_left_cancel [THEN iffD1], force)
+ 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))) ")
+ apply (drule poly_mult_left_cancel [THEN iffD1], force)
+ apply (simp add: fun_eq poly_exp_add poly_mult mult_ac del: pmult_Cons)
+ done
lemma (in idom_char_0) order_mult:
- assumes pq0: "poly (p *** q) \<noteq> poly []"
+ assumes "poly (p *** q) \<noteq> poly []"
shows "order a (p *** q) = order a p + order a q"
-proof-
- let ?order = order
- let ?divides = "op divides"
- let ?poly = poly
-from pq0
-show ?thesis
-apply (cut_tac a = a and p = "pmult p q" and n = "?order a p + ?order a q" in order)
-apply (auto simp add: poly_entire simp del: pmult_Cons)
-apply (drule_tac a = a in order2)+
-apply safe
-apply (simp add: divides_def fun_eq poly_exp_add poly_mult del: pmult_Cons, safe)
-apply (rule_tac x = "pmult qa qaa" in exI)
-apply (simp add: poly_mult mult_ac del: pmult_Cons)
-apply (drule_tac a = a in order_decomp)+
-apply safe
-apply (subgoal_tac "?divides [uminus a,one ] (pmult qa qaa) ")
-apply (simp add: poly_primes del: pmult_Cons)
-apply (auto simp add: divides_def simp del: pmult_Cons)
-apply (rule_tac x = qb in exI)
-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))")
-apply (drule poly_mult_left_cancel [THEN iffD1], force)
-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))) ")
-apply (drule poly_mult_left_cancel [THEN iffD1], force)
-apply (simp add: fun_eq poly_exp_add poly_mult mult_ac del: pmult_Cons)
-done
-qed
+ using assms
+ apply (cut_tac a = a and p = "pmult p q" and n = "order a p + order a q" in order)
+ apply (auto simp add: poly_entire simp del: pmult_Cons)
+ apply (drule_tac a = a in order2)+
+ apply safe
+ apply (simp add: divides_def fun_eq poly_exp_add poly_mult del: pmult_Cons, safe)
+ apply (rule_tac x = "pmult qa qaa" in exI)
+ apply (simp add: poly_mult mult_ac del: pmult_Cons)
+ apply (drule_tac a = a in order_decomp)+
+ apply safe
+ apply (subgoal_tac "[uminus a, one] divides pmult qa qaa")
+ apply (simp add: poly_primes del: pmult_Cons)
+ apply (auto simp add: divides_def simp del: pmult_Cons)
+ apply (rule_tac x = qb in exI)
+ 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))")
+ apply (drule poly_mult_left_cancel [THEN iffD1], force)
+ 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)))")
+ apply (drule poly_mult_left_cancel [THEN iffD1], force)
+ apply (simp add: fun_eq poly_exp_add poly_mult mult_ac del: pmult_Cons)
+ done
-lemma (in idom_char_0) order_root2: "poly p \<noteq> poly [] ==> (poly p a = 0) = (order a p \<noteq> 0)"
-by (rule order_root [THEN ssubst], auto)
+lemma (in idom_char_0) order_root2: "poly p \<noteq> poly [] \<Longrightarrow> poly p a = 0 \<longleftrightarrow> order a p \<noteq> 0"
+ by (rule order_root [THEN ssubst]) auto
lemma (in semiring_1) pmult_one[simp]: "[1] *** p = p" by auto
lemma (in semiring_0) poly_Nil_zero: "poly [] = poly [0]"
-by (simp add: fun_eq)
+ by (simp add: fun_eq)
lemma (in idom_char_0) rsquarefree_decomp:
- "[| rsquarefree p; poly p a = 0 |]
- ==> \<exists>q. (poly p = poly ([-a, 1] *** q)) & poly q a \<noteq> 0"
-apply (simp add: rsquarefree_def, safe)
-apply (frule_tac a = a in order_decomp)
-apply (drule_tac x = a in spec)
-apply (drule_tac a = a in order_root2 [symmetric])
-apply (auto simp del: pmult_Cons)
-apply (rule_tac x = q in exI, safe)
-apply (simp add: poly_mult fun_eq)
-apply (drule_tac p1 = q in poly_linear_divides [THEN iffD1])
-apply (simp add: divides_def del: pmult_Cons, safe)
-apply (drule_tac x = "[]" in spec)
-apply (auto simp add: fun_eq)
-done
+ "rsquarefree p \<Longrightarrow> poly p a = 0 \<Longrightarrow>
+ \<exists>q. poly p = poly ([-a, 1] *** q) \<and> poly q a \<noteq> 0"
+ apply (simp add: rsquarefree_def, safe)
+ apply (frule_tac a = a in order_decomp)
+ apply (drule_tac x = a in spec)
+ apply (drule_tac a = a in order_root2 [symmetric])
+ apply (auto simp del: pmult_Cons)
+ apply (rule_tac x = q in exI, safe)
+ apply (simp add: poly_mult fun_eq)
+ apply (drule_tac p1 = q in poly_linear_divides [THEN iffD1])
+ apply (simp add: divides_def del: pmult_Cons, safe)
+ apply (drule_tac x = "[]" in spec)
+ apply (auto simp add: fun_eq)
+ done
text{*Normalization of a polynomial.*}
lemma (in semiring_0) poly_normalize[simp]: "poly (pnormalize p) = poly p"
-apply (induct "p")
-apply (auto simp add: fun_eq)
-done
+ by (induct p) (auto simp add: fun_eq)
text{*The degree of a polynomial.*}
-lemma (in semiring_0) lemma_degree_zero:
- "list_all (%c. c = 0) p \<longleftrightarrow> pnormalize p = []"
-by (induct "p", auto)
+lemma (in semiring_0) lemma_degree_zero: "list_all (%c. c = 0) p \<longleftrightarrow> pnormalize p = []"
+ by (induct p) auto
lemma (in idom_char_0) degree_zero:
- assumes pN: "poly p = poly []" shows"degree p = 0"
-proof-
- let ?pn = pnormalize
- from pN
- show ?thesis
- apply (simp add: degree_def)
- apply (case_tac "?pn p = []")
- apply (auto simp add: poly_zero lemma_degree_zero )
- done
-qed
+ assumes "poly p = poly []"
+ shows "degree p = 0"
+ using assms
+ by (cases "pnormalize p = []") (auto simp add: degree_def poly_zero lemma_degree_zero)
lemma (in semiring_0) pnormalize_sing: "(pnormalize [x] = [x]) \<longleftrightarrow> x \<noteq> 0"
-by simp
-lemma (in semiring_0) pnormalize_pair: "y \<noteq> 0 \<longleftrightarrow> (pnormalize [x, y] = [x, y])" by simp
+ by simp
+
+lemma (in semiring_0) pnormalize_pair: "y \<noteq> 0 \<longleftrightarrow> (pnormalize [x, y] = [x, y])"
+ by simp
+
lemma (in semiring_0) pnormal_cons: "pnormal p \<Longrightarrow> pnormal (c#p)"
unfolding pnormal_def by simp
+
lemma (in semiring_0) pnormal_tail: "p\<noteq>[] \<Longrightarrow> pnormal (c#p) \<Longrightarrow> pnormal p"
unfolding pnormal_def by(auto split: split_if_asm)
-lemma (in semiring_0) pnormal_last_nonzero: "pnormal p ==> last p \<noteq> 0"
-by(induct p) (simp_all add: pnormal_def split: split_if_asm)
+lemma (in semiring_0) pnormal_last_nonzero: "pnormal p \<Longrightarrow> last p \<noteq> 0"
+ by (induct p) (simp_all add: pnormal_def split: split_if_asm)
lemma (in semiring_0) pnormal_length: "pnormal p \<Longrightarrow> 0 < length p"
unfolding pnormal_def length_greater_0_conv by blast
-lemma (in semiring_0) pnormal_last_length: "\<lbrakk>0 < length p ; last p \<noteq> 0\<rbrakk> \<Longrightarrow> pnormal p"
-by (induct p) (auto simp: pnormal_def split: split_if_asm)
+lemma (in semiring_0) pnormal_last_length: "0 < length p \<Longrightarrow> last p \<noteq> 0 \<Longrightarrow> pnormal p"
+ by (induct p) (auto simp: pnormal_def split: split_if_asm)
-lemma (in semiring_0) pnormal_id: "pnormal p \<longleftrightarrow> (0 < length p \<and> last p \<noteq> 0)"
+lemma (in semiring_0) pnormal_id: "pnormal p \<longleftrightarrow> 0 < length p \<and> last p \<noteq> 0"
using pnormal_last_length pnormal_length pnormal_last_nonzero by blast
-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")
+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")
proof
assume eq: ?lhs
hence "\<And>x. poly ((c#cs) +++ -- (d#ds)) x = 0"
@@ -851,18 +856,20 @@
unfolding poly_zero by (simp add: poly_minus_def algebra_simps)
hence "c = d \<and> (\<forall>x. poly (cs +++ -- ds) x = 0)"
unfolding poly_zero[symmetric] by simp
- thus ?rhs by (simp add: poly_minus poly_add algebra_simps fun_eq_iff)
+ then show ?rhs by (simp add: poly_minus poly_add algebra_simps fun_eq_iff)
next
- assume ?rhs then show ?lhs by(simp add:fun_eq_iff)
+ assume ?rhs
+ then show ?lhs by(simp add:fun_eq_iff)
qed
lemma (in idom_char_0) pnormalize_unique: "poly p = poly q \<Longrightarrow> pnormalize p = pnormalize q"
-proof(induct q arbitrary: p)
- case Nil thus ?case by (simp only: poly_zero lemma_degree_zero) simp
+proof (induct q arbitrary: p)
+ case Nil
+ thus ?case by (simp only: poly_zero lemma_degree_zero) simp
next
case (Cons c cs p)
thus ?case
- proof(induct p)
+ proof (induct p)
case Nil
hence "poly [] = poly (c#cs)" by blast
then have "poly (c#cs) = poly [] " by simp
@@ -880,44 +887,51 @@
qed
qed
-lemma (in idom_char_0) degree_unique: assumes pq: "poly p = poly q"
+lemma (in idom_char_0) degree_unique:
+ assumes pq: "poly p = poly q"
shows "degree p = degree q"
-using pnormalize_unique[OF pq] unfolding degree_def by simp
+ using pnormalize_unique[OF pq] unfolding degree_def by simp
-lemma (in semiring_0) pnormalize_length: "length (pnormalize p) \<le> length p" by (induct p, auto)
+lemma (in semiring_0) pnormalize_length:
+ "length (pnormalize p) \<le> length p" by (induct p) auto
lemma (in semiring_0) last_linear_mul_lemma:
- "last ((a %* p) +++ (x#(b %* p))) = (if p=[] then x else b*last p)"
+ "last ((a %* p) +++ (x#(b %* p))) = (if p = [] then x else b * last p)"
+ apply (induct p arbitrary: a x b)
+ apply auto
+ apply (subgoal_tac "padd (cmult aa p) (times b a # cmult b p) \<noteq> []")
+ apply simp
+ apply (induct_tac p)
+ apply auto
+ done
-apply (induct p arbitrary: a x b, auto)
-apply (subgoal_tac "padd (cmult aa p) (times b a # cmult b p) \<noteq> []", simp)
-apply (induct_tac p, auto)
-done
-
-lemma (in semiring_1) last_linear_mul: assumes p:"p\<noteq>[]" shows "last ([a,1] *** p) = last p"
-proof-
- from p obtain c cs where cs: "p = c#cs" by (cases p, auto)
- from cs have eq:"[a,1] *** p = (a %* (c#cs)) +++ (0#(1 %* (c#cs)))"
+lemma (in semiring_1) last_linear_mul:
+ assumes p: "p \<noteq> []"
+ shows "last ([a,1] *** p) = last p"
+proof -
+ from p obtain c cs where cs: "p = c#cs" by (cases p) auto
+ from cs have eq: "[a,1] *** p = (a %* (c#cs)) +++ (0#(1 %* (c#cs)))"
by (simp add: poly_cmult_distr)
show ?thesis using cs
unfolding eq last_linear_mul_lemma by simp
qed
lemma (in semiring_0) pnormalize_eq: "last p \<noteq> 0 \<Longrightarrow> pnormalize p = p"
-by (induct p) (auto split: split_if_asm)
+ by (induct p) (auto split: split_if_asm)
lemma (in semiring_0) last_pnormalize: "pnormalize p \<noteq> [] \<Longrightarrow> last (pnormalize p) \<noteq> 0"
- by (induct p, auto)
+ by (induct p) auto
lemma (in semiring_0) pnormal_degree: "last p \<noteq> 0 \<Longrightarrow> degree p = length p - 1"
using pnormalize_eq[of p] unfolding degree_def by simp
-lemma (in semiring_0) poly_Nil_ext: "poly [] = (\<lambda>x. 0)" by (rule ext) simp
+lemma (in semiring_0) poly_Nil_ext: "poly [] = (\<lambda>x. 0)"
+ by (rule ext) simp
lemma (in idom_char_0) linear_mul_degree:
assumes p: "poly p \<noteq> poly []"
shows "degree ([a,1] *** p) = degree p + 1"
-proof-
+proof -
from p have pnz: "pnormalize p \<noteq> []"
unfolding poly_zero lemma_degree_zero .
@@ -926,7 +940,6 @@
from last_pnormalize[OF pnz] last_linear_mul[OF pnz, of a]
pnormal_degree[OF l0] pnormal_degree[OF last_pnormalize[OF pnz]] pnz
-
have th: "degree ([a,1] *** pnormalize p) = degree (pnormalize p) + 1"
by simp
@@ -938,64 +951,81 @@
lemma (in idom_char_0) linear_pow_mul_degree:
"degree([a,1] %^n *** p) = (if poly p = poly [] then 0 else degree p + n)"
-proof(induct n arbitrary: a p)
+proof (induct n arbitrary: a p)
case (0 a p)
- {assume p: "poly p = poly []"
- hence ?case using degree_unique[OF p] by (simp add: degree_def)}
- moreover
- {assume p: "poly p \<noteq> poly []" hence ?case by (auto simp add: poly_Nil_ext) }
- ultimately show ?case by blast
+ show ?case
+ proof (cases "poly p = poly []")
+ case True
+ then show ?thesis
+ using degree_unique[OF True] by (simp add: degree_def)
+ next
+ case False
+ then show ?thesis by (auto simp add: poly_Nil_ext)
+ qed
next
case (Suc n a p)
have eq: "poly ([a,1] %^(Suc n) *** p) = poly ([a,1] %^ n *** ([a,1] *** p))"
- apply (rule ext, simp add: poly_mult poly_add poly_cmult)
- by (simp add: mult_ac add_ac distrib_left)
+ apply (rule ext)
+ apply (simp add: poly_mult poly_add poly_cmult)
+ apply (simp add: mult_ac add_ac distrib_left)
+ done
note deq = degree_unique[OF eq]
- {assume p: "poly p = poly []"
+ show ?case
+ proof (cases "poly p = poly []")
+ case True
with eq have eq': "poly ([a,1] %^(Suc n) *** p) = poly []"
- by - (rule ext,simp add: poly_mult poly_cmult poly_add)
- from degree_unique[OF eq'] p have ?case by (simp add: degree_def)}
- moreover
- {assume p: "poly p \<noteq> poly []"
- from p have ap: "poly ([a,1] *** p) \<noteq> poly []"
+ apply -
+ apply (rule ext)
+ apply (simp add: poly_mult poly_cmult poly_add)
+ done
+ from degree_unique[OF eq'] True show ?thesis
+ by (simp add: degree_def)
+ next
+ case False
+ then have ap: "poly ([a,1] *** p) \<noteq> poly []"
using poly_mult_not_eq_poly_Nil unfolding poly_entire by auto
have eq: "poly ([a,1] %^(Suc n) *** p) = poly ([a,1]%^n *** ([a,1] *** p))"
- by (rule ext, simp add: poly_mult poly_add poly_exp poly_cmult algebra_simps)
- from ap have ap': "(poly ([a,1] *** p) = poly []) = False" by blast
- have th0: "degree ([a,1]%^n *** ([a,1] *** p)) = degree ([a,1] *** p) + n"
- apply (simp only: Suc.hyps[of a "pmult [a,one] p"] ap')
- by simp
-
- from degree_unique[OF eq] ap p th0 linear_mul_degree[OF p, of a]
- have ?case by (auto simp del: poly.simps)}
- ultimately show ?case by blast
+ by (rule ext, simp add: poly_mult poly_add poly_exp poly_cmult algebra_simps)
+ from ap have ap': "(poly ([a,1] *** p) = poly []) = False"
+ by blast
+ have th0: "degree ([a,1]%^n *** ([a,1] *** p)) = degree ([a,1] *** p) + n"
+ apply (simp only: Suc.hyps[of a "pmult [a,one] p"] ap')
+ apply simp
+ done
+ from degree_unique[OF eq] ap False th0 linear_mul_degree[OF False, of a]
+ show ?thesis by (auto simp del: poly.simps)
+ qed
qed
lemma (in idom_char_0) order_degree:
assumes p0: "poly p \<noteq> poly []"
shows "order a p \<le> degree p"
-proof-
+proof -
from order2[OF p0, unfolded divides_def]
obtain q where q: "poly p = poly ([- a, 1]%^ (order a p) *** q)" by blast
- {assume "poly q = poly []"
- with q p0 have False by (simp add: poly_mult poly_entire)}
- with degree_unique[OF q, unfolded linear_pow_mul_degree]
- show ?thesis by auto
+ {
+ assume "poly q = poly []"
+ with q p0 have False by (simp add: poly_mult poly_entire)
+ }
+ with degree_unique[OF q, unfolded linear_pow_mul_degree] show ?thesis
+ by auto
qed
text{*Tidier versions of finiteness of roots.*}
-lemma (in idom_char_0) poly_roots_finite_set: "poly p \<noteq> poly [] ==> finite {x. poly p x = 0}"
-unfolding poly_roots_finite .
+lemma (in idom_char_0) poly_roots_finite_set:
+ "poly p \<noteq> poly [] \<Longrightarrow> finite {x. poly p x = 0}"
+ unfolding poly_roots_finite .
text{*bound for polynomial.*}
-lemma poly_mono: "abs(x) \<le> k ==> abs(poly p (x::'a::{linordered_idom})) \<le> poly (map abs p) k"
-apply (induct "p", auto)
-apply (rule_tac y = "abs a + abs (x * poly p x)" in order_trans)
-apply (rule abs_triangle_ineq)
-apply (auto intro!: mult_mono simp add: abs_mult)
-done
+lemma poly_mono: "abs(x) \<le> k \<Longrightarrow> abs(poly p (x::'a::{linordered_idom})) \<le> poly (map abs p) k"
+ apply (induct p)
+ apply auto
+ apply (rule_tac y = "abs a + abs (x * poly p x)" in order_trans)
+ apply (rule abs_triangle_ineq)
+ apply (auto intro!: mult_mono simp add: abs_mult)
+ done
lemma (in semiring_0) poly_Sing: "poly [c] x = c" by simp