(* Title: Univ_Poly.thy Author: Amine Chaieb *) header {* Univariate Polynomials *} theory Univ_Poly imports Plain List begin text{* 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 [code del]: "p1 divides p2 = (\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))" --{*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_0 begin lemma padd_Nil2[simp]: "p +++ [] = p" by (induct p) auto lemma padd_Cons_Cons: "(h1 # p1) +++ (h2 # p2) = (h1 + h2) # (p1 +++ p2)" by auto lemma pminus_Nil[simp]: "-- [] = []" 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 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 next case (Cons b bs a) thus ?case by (cases a, simp_all add: add_commute) qed lemma (in comm_semiring_0) padd_assoc: "\b c. (a +++ b) +++ c = a +++ (b +++ c)" apply (induct a arbitrary: b c) 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: right_distrib) done lemma (in ring_1) pmult_by_x[simp]: "[0, 1] *** t = ((0)#t)" apply (induct "t", simp) apply (auto simp add: mult_zero_left poly_ident_mult 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 next case (Cons a as p2) thus ?case by (cases p2, simp_all add: add_ac right_distrib) 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: right_distrib 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: right_distrib 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 minus_mult_left[symmetric]) 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 next case (Cons a as p2) thus ?case by (cases as, simp_all add: poly_cmult poly_add left_distrib right_distrib mult_ac) qed class recpower_semiring = semiring + recpower class recpower_semiring_1 = semiring_1 + recpower class recpower_semiring_0 = semiring_0 + recpower class recpower_ring = ring + recpower class recpower_ring_1 = ring_1 + recpower subclass (in recpower_ring_1) recpower_ring .. class recpower_comm_semiring_1 = recpower + comm_semiring_1 class recpower_comm_ring_1 = recpower + comm_ring_1 subclass (in recpower_comm_ring_1) recpower_comm_semiring_1 .. class recpower_idom = recpower + idom subclass (in recpower_idom) recpower_comm_ring_1 .. class idom_char_0 = idom + ring_char_0 class recpower_idom_char_0 = recpower + idom_char_0 subclass (in recpower_idom_char_0) recpower_idom .. lemma (in recpower_comm_ring_1) poly_exp: "poly (p %^ n) x = (poly p x) ^ n" apply (induct "n") apply (auto simp add: poly_cmult poly_mult power_Suc) done text{*More Polynomial Evaluation Lemmas*} 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) 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 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 blast next 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 diff_def[symmetric] minus_mult_left[symmetric] right_minus) hence "\q r. h#x#xs = [r] +++ [-a, 1] *** q" by blast} thus ?case by blast qed lemma (in comm_ring_1) poly_linear_rem: "\q r. h#t = [r] +++ [-a, 1] *** q" by (cut_tac t = t and a = a in lemma_poly_linear_rem, auto) lemma (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 minus_mult_left[symmetric])} 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, auto) qed lemma (in semiring_0) lemma_poly_length_mult[simp]: "\h k a. length (k %* p +++ (h # (a %* p))) = Suc (length p)" by (induct "p", auto) lemma (in semiring_0) lemma_poly_length_mult2[simp]: "\h k. length (k %* p +++ (h # p)) = Suc (length p)" by (induct "p", auto) lemma (in ring_1) poly_length_mult[simp]: "length([-a,1] *** q) = Suc (length q)" by auto subsection{*Polynomial length*} lemma (in semiring_0) poly_cmult_length[simp]: "length (a %* p) = length p" 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 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", 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 \ poly [] x" and n: "length p = n" shows "\i. \x. poly p x = 0 \ (\m\n. x = i m)" using p n proof(induct n arbitrary: p x) case 0 thus ?case by simp next 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 = "\m. 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 blast qed lemma (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) done lemma (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 auto qed lemma (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 (\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: "\ (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 .. qed lemma (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 qed next assume F: "finite {x. poly p x = (0\'a)}" show "poly p \ poly []" using F UNIV_ring_char_0_infinte by auto qed text{*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 = "\p. {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 auto qed lemma (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: expand_fun_eq 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 (auto intro!: ext) 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 minus_mult_left[symmetric]) 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 right_distrib minus_mult_left[symmetric] minus_mult_right[symmetric]) 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: ext simp add: poly_add_minus_mult_eq poly_entire poly_add_minus_zero_iff) finally show ?thesis . qed lemma (in recpower_idom) poly_exp_eq_zero[simp]: "(poly (p %^ n) = poly []) = (poly p = poly [] & n \ 0)" apply (simp only: fun_eq add: 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 semiring_1) one_neq_zero[simp]: "1 \ 0" using zero_neq_one by blast lemma (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 done lemma (in recpower_idom) poly_exp_prime_eq_zero: "(poly ([a, 1] %^ n) \ 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- 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 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 lemma (in idom_char_0) poly_0: "list_all (\c. c = 0) p \ poly p x = 0" unfolding poly_zero[symmetric] by simp 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 left_distrib [symmetric]) apply (drule_tac x = "uminus a" in spec) apply (simp add: poly_linear_divides poly_add poly_cmult left_distrib [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: left_distrib [symmetric]) apply clarsimp apply (auto simp add: right_minus poly_linear_divides poly_add poly_cmult left_distrib [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 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 recpower_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) done lemma (in recpower_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 right_distrib) 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 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 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_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 \ poly []" shows "\n q. p = mulexp n [-a, 1] q \ poly q a \ 0" 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" from Suc.prems have h: "length p = Suc n" "poly p \ poly []" by auto hence pN: "p \ []" 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 \ poly []" apply - apply simp apply (simp only: fun_eq) apply (rule ccontr) apply (simp add: fun_eq poly_add poly_cmult minus_mult_left[symmetric]) 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 hence ?case by blast} moreover {assume p0: "poly p a \ 0" hence ?case using Suc.prems apply simp by (rule exI[where x="0::nat"], simp)} ultimately show ?case by blast qed lemma (in recpower_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 power_Suc 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 blast qed (* FIXME: Tidy up *) lemma (in recpower_semiring_1) zero_power_iff: "0 ^ n = (if n = 0 then 1 else 0)" by (induct n, simp_all add: power_Suc) lemma (in recpower_idom_char_0) poly_order_exists: assumes lp: "length p = d" and p0: "poly p \ poly []" shows "\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 right_distrib mult_ac) apply safe apply (subgoal_tac "?poly (?mulexp n [uminus a, one] q) \ ?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 minus_mult_left[symmetric]) 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 lemma (in semiring_1) poly_one_divides[simp]: "[1] divides p" by (simp add: divides_def, auto) lemma (in recpower_idom_char_0) poly_order: "poly p \ 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 text{*Order*} lemma some1_equalityD: "[| n = (@n. P n); EX! n. P n |] ==> P n" by (blast intro: someI2) lemma (in recpower_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]) done lemma (in recpower_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 recpower_idom_char_0) order_unique: "[| poly p \ 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 recpower_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" apply (induct "p") apply (auto simp add: numeral_1_eq_1) done lemma (in comm_ring_1) lemma_order_root: " 0 < n & [- a, 1] %^ n divides p & ~ [- a, 1] %^ (Suc n) divides p \ poly p a = 0" apply (induct n arbitrary: a p, blast) apply (auto simp add: divides_def poly_mult simp del: pmult_Cons) done lemma (in recpower_idom_char_0) order_root: "(poly p a = 0) = ((poly p = poly []) | order a p \ 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 recpower_idom_char_0) order_divides: "(([-a, 1] %^ n) divides p) = ((poly p = poly []) | n \ 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 recpower_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) done text{*Important composition properties of orders.*} lemma order_mult: "poly (p *** q) \ poly [] ==> order a (p *** q) = order a p + order (a::'a::{recpower_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 recpower_idom_char_0) order_mult: assumes pq0: "poly (p *** q) \ 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 lemma (in recpower_idom_char_0) order_root2: "poly p \ poly [] ==> (poly p a = 0) = (order a p \ 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) lemma (in recpower_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) 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 text{*The degree of a polynomial.*} lemma (in semiring_0) lemma_degree_zero: "list_all (%c. c = 0) p \ 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 lemma (in semiring_0) pnormalize_sing: "(pnormalize [x] = [x]) \ x \ 0" by simp lemma (in semiring_0) pnormalize_pair: "y \ 0 \ (pnormalize [x, y] = [x, y])" by simp lemma (in semiring_0) pnormal_cons: "pnormal p \ pnormal (c#p)" unfolding pnormal_def by simp lemma (in semiring_0) pnormal_tail: "p\[] \ pnormal (c#p) \ pnormal p" unfolding pnormal_def apply (cases "pnormalize p = []", auto) by (cases "c = 0", auto) lemma (in semiring_0) pnormal_last_nonzero: "pnormal p ==> last p \ 0" proof(induct p) case Nil thus ?case by (simp add: pnormal_def) next case (Cons a as) thus ?case apply (simp add: pnormal_def) apply (cases "pnormalize as = []", simp_all) apply (cases "as = []", simp_all) apply (cases "a=0", simp_all) apply (cases "a=0", simp_all) done qed lemma (in semiring_0) pnormal_length: "pnormal p \ 0 < length p" unfolding pnormal_def length_greater_0_conv by blast lemma (in semiring_0) pnormal_last_length: "\0 < length p ; last p \ 0\ \ pnormal p" apply (induct p, auto) apply (case_tac "p = []", auto) apply (simp add: pnormal_def) by (rule pnormal_cons, auto) lemma (in semiring_0) pnormal_id: "pnormal p \ (0 < length p \ last p \ 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) \ 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:expand_fun_eq) hence "c = d \ list_all (\x. 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 thus ?rhs by (simp add: poly_minus poly_add algebra_simps expand_fun_eq) next assume ?rhs then show ?lhs by(simp add:expand_fun_eq) qed lemma (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) simp next 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 qed qed 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 lemma (in semiring_0) pnormalize_length: "length (pnormalize p) \ 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)" apply (induct p arbitrary: a x b, auto) apply (subgoal_tac "padd (cmult aa p) (times b a # cmult b p) \ []", simp) apply (induct_tac p, auto) done lemma (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 simp qed lemma (in semiring_0) pnormalize_eq: "last p \ 0 \ pnormalize p = p" apply (induct p, auto) apply (case_tac p, auto)+ done lemma (in semiring_0) last_pnormalize: "pnormalize p \ [] \ last (pnormalize p) \ 0" by (induct p, auto) lemma (in semiring_0) pnormal_degree: "last p \ 0 \ degree p = length p - 1" using pnormalize_eq[of p] unfolding degree_def by simp lemma (in semiring_0) poly_Nil_ext: "poly [] = (\x. 0)" by (rule ext) simp lemma (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 (auto simp add: poly_length_mult) 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]) qed 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) 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 \ poly []" hence ?case by (auto simp add: poly_Nil_ext) } ultimately show ?case by blast 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 right_distrib) note deq = degree_unique[OF eq] {assume p: "poly p = poly []" 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 \ poly []" from p 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') 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 qed lemma (in recpower_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 auto qed text{*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::{ordered_idom})) \ 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 (in semiring_0) poly_Sing: "poly [c] x = c" by simp end