# Theory Univ_Poly

theory Univ_Poly
imports Main
`(*  Title:      HOL/Library/Univ_Poly.thy    Author:     Amine Chaieb*)header {* Univariate Polynomials *}theory Univ_Polyimports Mainbegintext{* Application of polynomial as a function. *}primrec (in semiring_0) poly :: "'a list => 'a => 'a"where  poly_Nil:  "poly [] x = 0"| poly_Cons: "poly (h#t) x = h + x * poly t x"subsection{*Arithmetic Operations on Polynomials*}text{*addition*}primrec (in semiring_0) padd :: "'a list => 'a list => '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))"text{*Multiplication by a constant*}primrec (in semiring_0) cmult :: "'a => 'a list => 'a list"  (infixl "%*" 70) where  cmult_Nil:  "c %* [] = []"| cmult_Cons: "c %* (h#t) = (c * h)#(c %* t)"text{*Multiplication by a polynomial*}primrec (in semiring_0) pmult :: "'a list => 'a list => 'a list"  (infixl "***" 70)where  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 => 'a list => 'a  list => 'a list" where  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 => nat => 'a list"  (infixl "%^" 80) where  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 => 'a => '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 => '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)"definition (in semiring_0) "pnormal p = ((pnormalize p = p) ∧ p ≠ [])"definition (in semiring_0) "nonconstant p = (pnormal p ∧ (∀x. p ≠ [x]))"text{*Other definitions*}definition (in ring_1) poly_minus :: "'a list => 'a list" ("-- _" [80] 80)  where "-- p = (- 1) %* p"definition (in semiring_0) divides :: "'a list => 'a list => bool"  (infixl "divides" 70)  where "p1 divides p2 = (∃q. poly p2 = poly(p1 *** q))"lemma (in semiring_0) dividesI:  "poly p2 = poly (p1 *** q) ==> p1 divides p2"  by (auto simp add: divides_def)lemma (in semiring_0) dividesE:  assumes "p1 divides p2"  obtains q where "poly p2 = poly (p1 *** q)"  using assms by (auto simp add: divides_def)    --{*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))"     --{*degree of a polynomial*}definition (in semiring_0) degree :: "'a list => 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 ≠ poly [] ∧ (∀a. order a p = 0 ∨ order a p = 1)"context semiring_0beginlemma padd_Nil2[simp]: "p +++ [] = p"  by (induct p) autolemma padd_Cons_Cons: "(h1 # p1) +++ (h2 # p2) = (h1 + h2) # (p1 +++ p2)"  by autolemma pminus_Nil: "-- [] = []"  by (simp add: poly_minus_def)lemma pmult_singleton: "[h1] *** p1 = h1 %* p1" by simpendlemma (in semiring_1) poly_ident_mult[simp]: "1 %* t = t" by (induct t) autolemma (in semiring_0) poly_simple_add_Cons[simp]: "[a] +++ ((0)#t) = (a#t)"  by simptext{*Handy general properties*}lemma (in comm_semiring_0) padd_commut: "b +++ a = a +++ b"proof (induct b arbitrary: a)  case Nil  thus ?case by autonext  case (Cons b bs a)  thus ?case by (cases a) (simp_all add: add_commute)qedlemma (in comm_semiring_0) padd_assoc: "∀b c. (a +++ b) +++ c = a +++ (b +++ c)"  apply (induct a)  apply (simp, clarify)  apply (case_tac b, simp_all add: add_ac)  donelemma (in semiring_0) poly_cmult_distr: "a %* ( p +++ q) = (a %* p +++ a %* q)"  apply (induct p arbitrary: q)  apply simp  apply (case_tac q, simp_all add: distrib_left)  donelemma (in ring_1) pmult_by_x[simp]: "[0, 1] *** t = ((0)#t)"  apply (induct t)  apply simp  apply (auto simp add: padd_commut)  apply (case_tac t, auto)  donetext{*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 simpnext  case (Cons a as p2)  thus ?case    by (cases p2) (simp_all  add: add_ac distrib_left)qedlemma (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)  donelemma (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)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)  donelemma (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 simpnext  case (Cons a as p2)  thus ?case by (cases as)    (simp_all add: poly_cmult poly_add distrib_right distrib_left mult_ac)qedclass idom_char_0 = idom + ring_char_0lemma (in comm_ring_1) poly_exp: "poly (p %^ n) x = (poly p x) ^ n"  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 simplemma (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) autolemma (in comm_semiring_1) poly_exp_add: "poly (p %^ (n + d)) x = poly( p %^ n *** p %^ d) x"  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)"} *}lemma (in comm_ring_1) lemma_poly_linear_rem: "∀h. ∃q r. h#t = [r] +++ [-a, 1] *** q"proof(induct t)  case Nil  { fix h have "[h] = [h] +++ [- a, 1] *** []" by simp }  thus ?case by blastnext  case (Cons  x xs)  { 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)"      using qr by (cases q) (simp_all add: algebra_simps)    hence "∃q r. h#x#xs = [r] +++ [-a, 1] *** q" by blast}  thus ?case by blastqedlemma (in comm_ring_1) poly_linear_rem: "∃q r. h#t = [r] +++ [-a, 1] *** q"  using lemma_poly_linear_rem [where t = t and a = a] by autolemma (in comm_ring_1) poly_linear_divides: "(poly p a = 0) = ((p = []) | (∃q. p = [-a, 1] *** q))"proof -  { assume p: "p = []" hence ?thesis by simp }  moreover  {    fix x xs assume p: "p = x#xs"    {      fix q assume "p = [-a, 1] *** q"      hence "poly p a = 0" by (simp add: poly_add poly_cmult)    }    moreover    { 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 "∃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) autoqedlemma (in semiring_0) lemma_poly_length_mult[simp]: "∀h k a. length (k %* p +++  (h # (a %* p))) = Suc (length p)"  by (induct p) autolemma (in semiring_0) lemma_poly_length_mult2[simp]: "∀h k. length (k %* p +++  (h # p)) = Suc (length p)"  by (induct p) autolemma (in ring_1) poly_length_mult[simp]: "length([-a,1] *** q) = Suc (length q)"  by autosubsection{*Polynomial length*}lemma (in semiring_0) poly_cmult_length[simp]: "length (a %* p) = length p"  by (induct p) autolemma (in semiring_0) poly_add_length: "length (p1 +++ p2) = max (length p1) (length p2)"  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)lemma (in idom) poly_mult_not_eq_poly_Nil[simp]:  "poly (p *** q) x ≠ poly [] x <-> poly p x ≠ poly [] x ∧ poly q x ≠ poly [] x"  by (auto simp add: poly_mult)lemma (in idom) poly_mult_eq_zero_disj: "poly (p *** q) x = 0 <-> poly p x = 0 ∨ poly q x = 0"  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) autotext{*A nontrivial polynomial of degree n has no more than n roots*}lemma (in idom) poly_roots_index_lemma:   assumes p: "poly p x ≠ poly [] x" and n: "length p = n"  shows "∃i. ∀x. poly p x = 0 --> (∃m≤n. x = i m)"  using p nproof (induct n arbitrary: p x)  case 0  thus ?case by simpnext  case (Suc n p x)  {    assume C: "!!i. ∃x. poly p x = 0 ∧ (∀m≤Suc n. x ≠ i m)"    from Suc.prems have p0: "poly p x ≠ 0" "p≠ []" by auto    from p0(1)[unfolded poly_linear_divides[of p x]]    have "∀q. p ≠ [- x, 1] *** q" by blast    from C obtain a where a: "poly p a = 0" by blast    from a[unfolded poly_linear_divides[of p a]] p0(2)    obtain q where q: "p = [-a, 1] *** q" by blast    have lg: "length q = n" using q Suc.prems(2) by simp    from q p0 have qx: "poly q x ≠ poly [] x"      by (auto simp add: poly_mult poly_add poly_cmult)    from Suc.hyps[OF qx lg] obtain i where      i: "∀x. poly q x = 0 --> (∃m≤n. x = i m)" by blast    let ?i = "λm. if m = Suc n then a else i m"    from C[of ?i] obtain y where y: "poly p y = 0" "∀m≤ Suc n. y ≠ ?i m"      by blast    from y have "y = a ∨ 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)      apply auto      done  }  thus ?case by blastqedlemma (in idom) poly_roots_index_length:  "poly p x ≠ poly [] x ==> ∃i. ∀x. (poly p x = 0) --> (∃n. n ≤ length p ∧ x = i n)"  by (blast intro: poly_roots_index_lemma)lemma (in idom) poly_roots_finite_lemma1:  "poly p x ≠ poly [] x ==> ∃N i. ∀x. (poly p x = 0) --> (∃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)  donelemma (in idom) idom_finite_lemma:  assumes P: "∀x. P x --> (∃n. n < length j ∧ x = j!n)"  shows "finite {x. P x}"proof -  let ?M = "{x. P x}"  let ?N = "set j"  have "?M ⊆ ?N" using P by auto  thus ?thesis using finite_subset by autoqedlemma (in idom) poly_roots_finite_lemma2:  "poly p x ≠ poly [] x ==> ∃i. ∀x. poly p x = 0 --> x ∈ set i"  apply (drule poly_roots_index_length, safe)  apply (rule_tac x="map (λn. 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)  donelemma (in ring_char_0) UNIV_ring_char_0_infinte: "¬ (finite (UNIV:: 'a set))"proof  assume F: "finite (UNIV :: 'a set)"  have "finite (UNIV :: nat set)"  proof (rule finite_imageD)    have "of_nat ` UNIV ⊆ UNIV" by simp    then show "finite (of_nat ` UNIV :: 'a set)" using F by (rule finite_subset)    show "inj (of_nat :: nat => 'a)" by (simp add: inj_on_def)  qed  with infinite_UNIV_nat show False ..qedlemma (in idom_char_0) poly_roots_finite: "poly p ≠ poly [] <-> finite {x. poly p x = 0}"proof  assume H: "poly p ≠ poly []"  show "finite {x. poly p x = (0::'a)}"    using H    apply -    apply (erule contrapos_np, rule ext)    apply (rule ccontr)    apply (clarify dest!: poly_roots_finite_lemma2)    using finite_subset  proof -    fix x i    assume F: "¬ finite {x. poly p x = (0::'a)}"      and P: "∀x. poly p x = (0::'a) --> x ∈ set i"    let ?M= "{x. poly p x = (0::'a)}"    from P have "?M ⊆ set i" by auto    with finite_subset F show False by auto  qednext  assume F: "finite {x. poly p x = (0::'a)}"  show "poly p ≠ poly []" using F UNIV_ring_char_0_infinte by autoqedtext{*Entirety and Cancellation for polynomials*}lemma (in idom_char_0) poly_entire_lemma2:  assumes p0: "poly p ≠ poly []"    and q0: "poly q ≠ poly []"  shows "poly (p***q) ≠ poly []"proof -  let ?S = "λp. {x. poly p x = 0}"  have "?S (p *** q) = ?S p ∪ ?S q" by (auto simp add: poly_mult)  with p0 q0 show ?thesis  unfolding poly_roots_finite by autoqedlemma (in idom_char_0) poly_entire:  "poly (p *** q) = poly [] <-> poly p = poly [] ∨ poly q = poly []"  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) ≠ poly [] <-> poly p ≠ poly [] ∧ poly q ≠ poly []"  by (simp add: poly_entire)lemma fun_eq: "f = g <-> (∀x. f x = g x)"  by autolemma (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_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) <-> poly (p *** q +++ -- (p *** r)) = poly []"    by (simp only: poly_add_minus_zero_iff)  also have "… <-> poly p = poly [] ∨ poly q = poly r"    by (auto intro: simp add: poly_add_minus_mult_eq poly_entire poly_add_minus_zero_iff)  finally show ?thesis .qedlemma (in idom) poly_exp_eq_zero[simp]:  "poly (p %^ n) = poly [] <-> poly p = poly [] ∧ n ≠ 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)  donelemma (in comm_ring_1) poly_prime_eq_zero[simp]: "poly [a,1] ≠ 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  donelemma (in idom) poly_exp_prime_eq_zero: "poly ([a, 1] %^ n) ≠ poly []"  by autotext{*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 -  fix x  assume H: "∀x. x = (0::'a) ∨ poly t x = (0::'a)"    and pnz: "poly t ≠ poly []"  let ?S = "{x. poly t x = 0}"  from H have "∀x. x ≠0 --> poly t x = 0" by blast  hence th: "?S ⊇ 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::'a)"    by simpqedlemma (in idom_char_0) poly_zero: "(poly p = poly []) = list_all (%c. c = 0) p"  apply (induct p)  apply simp  apply (rule iffI)  apply (drule poly_zero_lemma', auto)  donelemma (in idom_char_0) poly_0: "list_all (λc. c = 0) p ==> poly p x = 0"  unfolding poly_zero[symmetric] by simptext{*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)  apply clarsimp  apply (cases "∃q::'a list. p = a %* q +++ ((0::'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)  donelemma (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)  donelemma (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)  donelemma (in comm_semiring_1) poly_divides_exp: "m ≤ 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)  donelemma (in comm_semiring_1) poly_exp_divides:  "(p %^ n) divides q ==> m ≤ n ==> (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)  donelemma (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)  donelemma (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)  donelemma (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)  donelemma (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)  donetext{*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 ≠ poly []"  shows "∃n q. p = mulexp n [-a, 1] q ∧ poly q a ≠ 0"  using lp pproof (induct d arbitrary: p)  case 0  thus ?case by simpnext  case (Suc n p)  show ?case  proof (cases "poly p a = 0")    case True    from Suc.prems have h: "length p = Suc n" "poly p ≠ poly []" by auto    hence pN: "p ≠ []" by auto    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 ≠ 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 ≠ 0"      by blast    from mr q have "p = mulexp (Suc m) [-a,1] r ∧ poly r a ≠ 0" by simp    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  qedqedlemma (in comm_semiring_1) poly_mulexp: "poly (mulexp n p q) x = (poly p x) ^ n * poly q x"  by (induct n) (auto simp add: poly_mult mult_ac)lemma (in comm_semiring_1) divides_left_mult:  assumes d:"(p***q) divides r" shows "p divides r ∧ q divides r"proof-  from d obtain t where r:"poly r = poly (p***q *** t)"    unfolding divides_def by blast  hence "poly r = poly (p *** (q *** t))"    "poly r = poly (q *** (p***t))" by(auto simp add: fun_eq poly_mult mult_ac)  thus ?thesis unfolding divides_def by blastqed(* FIXME: Tidy up *)lemma (in semiring_1) zero_power_iff: "0 ^ n = (if n = 0 then 1 else 0)"  by (induct n) simp_alllemma (in idom_char_0) poly_order_exists:  assumes "length p = d" and "poly p ≠ poly []"  shows "∃n. [- a, 1] %^ n divides p ∧ ¬ [- a, 1] %^ Suc n divides p"proof -  from assms have "∃n q. p = mulexp n [- a, 1] q ∧ poly q a ≠ 0"    by (rule poly_order_exists_lemma)  then obtain n q where p: "p = mulexp n [- a, 1] q" and "poly q a ≠ 0" by blast  have "[- a, 1] %^ n divides mulexp n [- a, 1] q"  proof (rule dividesI)    show "poly (mulexp n [- a, 1] q) = poly ([- a, 1] %^ n *** q)"      by (induct n) (simp_all add: poly_add poly_cmult poly_mult distrib_left mult_ac)  qed  moreover have "¬ [- a, 1] %^ Suc n divides mulexp n [- a, 1] q"  proof    assume "[- a, 1] %^ Suc n divides mulexp n [- a, 1] q"    then obtain m where "poly (mulexp n [- a, 1] q) = poly ([- a, 1] %^ Suc n *** m)"      by (rule dividesE)    moreover have "poly (mulexp n [- a, 1] q) ≠ poly ([- a, 1] %^ Suc n *** m)"    proof (induct n)      case 0 show ?case      proof (rule ccontr)        assume "¬ poly (mulexp 0 [- a, 1] q) ≠ poly ([- a, 1] %^ Suc 0 *** m)"        then have "poly q a = 0"          by (simp add: poly_add poly_cmult)        with `poly q a ≠ 0` show False by simp      qed    next      case (Suc n) show ?case        by (rule pexp_Suc [THEN ssubst], rule ccontr)          (simp add: poly_mult_left_cancel poly_mult_assoc Suc del: pmult_Cons pexp_Suc)    qed    ultimately show False by simp  qed  ultimately show ?thesis by (auto simp add: p)qedlemma (in semiring_1) poly_one_divides[simp]: "[1] divides p"  by (auto simp add: divides_def)lemma (in idom_char_0) poly_order:  "poly p ≠ poly [] ==> ∃!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)  donetext{*Order*}lemma some1_equalityD: "n = (SOME n. P n) ==> ∃!n. P n ==> P n"  by (blast intro: someI2)lemma (in idom_char_0) order:      "(([-a, 1] %^ n) divides p ∧        ~(([-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])  donelemma (in idom_char_0) order2:  "poly p ≠ 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) order_unique:  "poly p ≠ poly [] ==> ([-a, 1] %^ n) divides p ==> ~(([-a, 1] %^ (Suc n)) divides p) ==>    n = order a p"  using order [of a n p] by autolemma (in idom_char_0) order_unique_lemma:  "poly p ≠ poly [] ∧ ([-a, 1] %^ n) divides p ∧ ~(([-a, 1] %^ (Suc n)) divides p) ==>    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"  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") autolemma (in comm_ring_1) lemma_order_root:  "0 < n ∧ [- a, 1] %^ n divides p ∧ ~ [- a, 1] %^ (Suc n) divides p ==> 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 ≠ 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)  donelemma (in idom_char_0) order_divides:  "([-a, 1] %^ n) divides p <-> poly p = poly [] ∨ n ≤ 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)  donelemma (in idom_char_0) order_decomp:  "poly p ≠ poly [] ==> ∃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)  donetext{*Important composition properties of orders.*}lemma order_mult:  "poly (p *** q) ≠ 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)  donelemma (in idom_char_0) order_mult:  assumes "poly (p *** q) ≠ poly []"  shows "order a (p *** q) = order a p + order a q"  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)  donelemma (in idom_char_0) order_root2: "poly p ≠ poly [] ==> poly p a = 0 <-> order a p ≠ 0"  by (rule order_root [THEN ssubst]) autolemma (in semiring_1) pmult_one[simp]: "[1] *** p = p" by autolemma (in semiring_0) poly_Nil_zero: "poly [] = poly [0]"  by (simp add: fun_eq)lemma (in idom_char_0) rsquarefree_decomp:  "rsquarefree p ==> poly p a = 0 ==>    ∃q. poly p = poly ([-a, 1] *** q) ∧ poly q a ≠ 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)  donetext{*Normalization of a polynomial.*}lemma (in semiring_0) poly_normalize[simp]: "poly (pnormalize p) = poly p"  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 <-> pnormalize p = []"  by (induct p) autolemma (in idom_char_0) degree_zero:  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]) <-> x ≠ 0"  by simplemma (in semiring_0) pnormalize_pair: "y ≠ 0 <-> (pnormalize [x, y] = [x, y])"  by simplemma (in semiring_0) pnormal_cons: "pnormal p ==> pnormal (c#p)"  unfolding pnormal_def by simplemma (in semiring_0) pnormal_tail: "p≠[] ==> pnormal (c#p) ==> pnormal p"  unfolding pnormal_def by(auto split: split_if_asm)lemma (in semiring_0) pnormal_last_nonzero: "pnormal p ==> last p ≠ 0"  by (induct p) (simp_all add: pnormal_def split: split_if_asm)lemma (in semiring_0) pnormal_length: "pnormal p ==> 0 < length p"  unfolding pnormal_def length_greater_0_conv by blastlemma (in semiring_0) pnormal_last_length: "0 < length p ==> last p ≠ 0 ==> pnormal p"  by (induct p) (auto simp: pnormal_def  split: split_if_asm)lemma (in semiring_0) pnormal_id: "pnormal p <-> 0 < length p ∧ last p ≠ 0"  using pnormal_last_length pnormal_length pnormal_last_nonzero by blastlemma (in idom_char_0) poly_Cons_eq:  "poly (c # cs) = poly (d # ds) <-> c = d ∧ poly cs = poly ds"  (is "?lhs <-> ?rhs")proof  assume eq: ?lhs  hence "!!x. poly ((c#cs) +++ -- (d#ds)) x = 0"    by (simp only: poly_minus poly_add algebra_simps) simp  hence "poly ((c#cs) +++ -- (d#ds)) = poly []" by(simp add: fun_eq_iff)  hence "c = d ∧ list_all (λx. x=0) ((cs +++ -- ds))"    unfolding poly_zero by (simp add: poly_minus_def algebra_simps)  hence "c = d ∧ (∀x. poly (cs +++ -- ds) x = 0)"    unfolding poly_zero[symmetric] by simp  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)qedlemma (in idom_char_0) pnormalize_unique: "poly p = poly q ==> pnormalize p = pnormalize q"proof (induct q arbitrary: p)  case Nil  thus ?case by (simp only: poly_zero lemma_degree_zero) simpnext  case (Cons c cs p)  thus ?case  proof (induct p)    case Nil    hence "poly [] = poly (c#cs)" by blast    then have "poly (c#cs) = poly [] " by simp    thus ?case by (simp only: poly_zero lemma_degree_zero) simp  next    case (Cons d ds)    hence eq: "poly (d # ds) = poly (c # cs)" by blast    hence eq': "!!x. poly (d # ds) x = poly (c # cs) x" by simp    hence "poly (d # ds) 0 = poly (c # cs) 0" by blast    hence dc: "d = c" by auto    with eq have "poly ds = poly cs"      unfolding  poly_Cons_eq by simp    with Cons.prems have "pnormalize ds = pnormalize cs" by blast    with dc show ?case by simp  qedqedlemma (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 simplemma (in semiring_0) pnormalize_length:  "length (pnormalize p) ≤ length p" by (induct p) autolemma (in semiring_0) last_linear_mul_lemma:  "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) ≠ []")  apply simp  apply (induct_tac p)  apply auto  donelemma (in semiring_1) last_linear_mul:  assumes p: "p ≠ []"  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 simpqedlemma (in semiring_0) pnormalize_eq: "last p ≠ 0 ==> pnormalize p = p"  by (induct p) (auto split: split_if_asm)lemma (in semiring_0) last_pnormalize: "pnormalize p ≠ [] ==> last (pnormalize p) ≠ 0"  by (induct p) autolemma (in semiring_0) pnormal_degree: "last p ≠ 0 ==> degree p = length p - 1"  using pnormalize_eq[of p] unfolding degree_def by simplemma (in semiring_0) poly_Nil_ext: "poly [] = (λx. 0)"  by (rule ext) simplemma (in idom_char_0) linear_mul_degree:  assumes p: "poly p ≠ poly []"  shows "degree ([a,1] *** p) = degree p + 1"proof -  from p have pnz: "pnormalize p ≠ []"    unfolding poly_zero lemma_degree_zero .  from last_linear_mul[OF pnz, of a] last_pnormalize[OF pnz]  have l0: "last ([a, 1] *** pnormalize p) ≠ 0" by simp  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  have eqs: "poly ([a,1] *** pnormalize p) = poly ([a,1] *** p)"    by (rule ext) (simp add: poly_mult poly_add poly_cmult)  from degree_unique[OF eqs] th  show ?thesis by (simp add: degree_unique[OF poly_normalize])qedlemma (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)  case (0 a p)  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)  qednext  case (Suc n a p)  have eq: "poly ([a,1] %^(Suc n) *** p) = poly ([a,1] %^ n *** ([a,1] *** p))"    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]  show ?case  proof (cases "poly p = poly []")    case True    with eq have eq': "poly ([a,1] %^(Suc n) *** p) = 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) ≠ 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')      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)  qedqedlemma (in idom_char_0) order_degree:  assumes p0: "poly p ≠ poly []"  shows "order a p ≤ degree p"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 autoqedtext{*Tidier versions of finiteness of roots.*}lemma (in idom_char_0) poly_roots_finite_set:  "poly p ≠ poly [] ==> finite {x. poly p x = 0}"  unfolding poly_roots_finite .text{*bound for polynomial.*}lemma poly_mono: "abs(x) ≤ k ==> abs(poly p (x::'a::{linordered_idom})) ≤ 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)  donelemma (in semiring_0) poly_Sing: "poly [c] x = c" by simpend`