(* Title: HOL/Computational_Algebra/Fundamental_Theorem_Algebra.thy Author: Amine Chaieb, TU Muenchen *) section ‹Fundamental Theorem of Algebra› theory Fundamental_Theorem_Algebra imports Polynomial Complex_Main begin subsection ‹More lemmas about module of complex numbers› text ‹The triangle inequality for cmod› lemma complex_mod_triangle_sub: "cmod w ≤ cmod (w + z) + norm z" by (metis add_diff_cancel norm_triangle_ineq4) subsection ‹Basic lemmas about polynomials› lemma poly_bound_exists: fixes p :: "'a::{comm_semiring_0,real_normed_div_algebra} poly" shows "∃m. m > 0 ∧ (∀z. norm z ≤ r ⟶ norm (poly p z) ≤ m)" proof (induct p) case 0 then show ?case by (rule exI[where x=1]) simp next case (pCons c cs) from pCons.hyps obtain m where m: "∀z. norm z ≤ r ⟶ norm (poly cs z) ≤ m" by blast let ?k = " 1 + norm c + ¦r * m¦" have kp: "?k > 0" using abs_ge_zero[of "r*m"] norm_ge_zero[of c] by arith have "norm (poly (pCons c cs) z) ≤ ?k" if H: "norm z ≤ r" for z proof - from m H have th: "norm (poly cs z) ≤ m" by blast from H have rp: "r ≥ 0" using norm_ge_zero[of z] by arith have "norm (poly (pCons c cs) z) ≤ norm c + norm (z * poly cs z)" using norm_triangle_ineq[of c "z* poly cs z"] by simp also have "… ≤ ?k" using mult_mono[OF H th rp norm_ge_zero[of "poly cs z"]] by (simp add: norm_mult) finally show ?thesis . qed with kp show ?case by blast qed text ‹Offsetting the variable in a polynomial gives another of same degree› definition offset_poly :: "'a::comm_semiring_0 poly ⇒ 'a ⇒ 'a poly" where "offset_poly p h = fold_coeffs (λa q. smult h q + pCons a q) p 0" lemma offset_poly_0: "offset_poly 0 h = 0" by (simp add: offset_poly_def) lemma offset_poly_pCons: "offset_poly (pCons a p) h = smult h (offset_poly p h) + pCons a (offset_poly p h)" by (cases "p = 0 ∧ a = 0") (auto simp add: offset_poly_def) lemma offset_poly_single [simp]: "offset_poly [:a:] h = [:a:]" by (simp add: offset_poly_pCons offset_poly_0) lemma poly_offset_poly: "poly (offset_poly p h) x = poly p (h + x)" by (induct p) (auto simp add: offset_poly_0 offset_poly_pCons algebra_simps) lemma offset_poly_eq_0_lemma: "smult c p + pCons a p = 0 ⟹ p = 0" by (induct p arbitrary: a) (simp, force) lemma offset_poly_eq_0_iff [simp]: "offset_poly p h = 0 ⟷ p = 0" proof show "offset_poly p h = 0 ⟹ p = 0" proof(induction p) case 0 then show ?case by blast next case (pCons a p) then show ?case by (metis offset_poly_eq_0_lemma offset_poly_pCons offset_poly_single) qed qed (simp add: offset_poly_0) lemma degree_offset_poly [simp]: "degree (offset_poly p h) = degree p" proof(induction p) case 0 then show ?case by (simp add: offset_poly_0) next case (pCons a p) have "p ≠ 0 ⟹ degree (offset_poly (pCons a p) h) = Suc (degree p)" by (metis degree_add_eq_right degree_pCons_eq degree_smult_le le_imp_less_Suc offset_poly_eq_0_iff offset_poly_pCons pCons.IH) then show ?case by simp qed definition "psize p = (if p = 0 then 0 else Suc (degree p))" lemma psize_eq_0_iff [simp]: "psize p = 0 ⟷ p = 0" unfolding psize_def by simp lemma poly_offset: fixes p :: "'a::comm_ring_1 poly" shows "∃q. psize q = psize p ∧ (∀x. poly q x = poly p (a + x))" by (metis degree_offset_poly offset_poly_eq_0_iff poly_offset_poly psize_def) text ‹An alternative useful formulation of completeness of the reals› lemma real_sup_exists: assumes ex: "∃x. P x" and bz: "∃z. ∀x. P x ⟶ x < z" shows "∃s::real. ∀y. (∃x. P x ∧ y < x) ⟷ y < s" proof from bz have "bdd_above (Collect P)" by (force intro: less_imp_le) then show "∀y. (∃x. P x ∧ y < x) ⟷ y < Sup (Collect P)" using ex bz by (subst less_cSup_iff) auto qed subsection ‹Fundamental theorem of algebra› lemma unimodular_reduce_norm: assumes md: "cmod z = 1" shows "cmod (z + 1) < 1 ∨ cmod (z - 1) < 1 ∨ cmod (z + 𝗂) < 1 ∨ cmod (z - 𝗂) < 1" proof - obtain x y where z: "z = Complex x y " by (cases z) auto from md z have xy: "x⇧^{2}+ y⇧^{2}= 1" by (simp add: cmod_def) have False if "cmod (z + 1) ≥ 1" "cmod (z - 1) ≥ 1" "cmod (z + 𝗂) ≥ 1" "cmod (z - 𝗂) ≥ 1" proof - from that z xy have *: "2 * x ≤ 1" "2 * x ≥ -1" "2 * y ≤ 1" "2 * y ≥ -1" by (simp_all add: cmod_def power2_eq_square algebra_simps) then have "¦2 * x¦ ≤ 1" "¦2 * y¦ ≤ 1" by simp_all then have "¦2 * x¦⇧^{2}≤ 1⇧^{2}" "¦2 * y¦⇧^{2}≤ 1⇧^{2}" by (metis abs_square_le_1 one_power2 power2_abs)+ with xy * show ?thesis by (smt (verit, best) four_x_squared square_le_1) qed then show ?thesis by force qed text ‹Hence we can always reduce modulus of ‹1 + b z^n› if nonzero› lemma reduce_poly_simple: assumes b: "b ≠ 0" and n: "n ≠ 0" shows "∃z. cmod (1 + b * z^n) < 1" using n proof (induct n rule: nat_less_induct) fix n assume IH: "∀m<n. m ≠ 0 ⟶ (∃z. cmod (1 + b * z ^ m) < 1)" assume n: "n ≠ 0" let ?P = "λz n. cmod (1 + b * z ^ n) < 1" show "∃z. ?P z n" proof cases assume "even n" then obtain m where m: "n = 2 * m" and "m ≠ 0" "m < n" using n by auto with IH obtain z where z: "?P z m" by blast from z have "?P (csqrt z) n" by (simp add: m power_mult) then show ?thesis .. next assume "odd n" then have "∃m. n = Suc (2 * m)" by presburger+ then obtain m where m: "n = Suc (2 * m)" by blast have 0: "cmod (complex_of_real (cmod b) / b) = 1" using b by (simp add: norm_divide) have "∃v. cmod (complex_of_real (cmod b) / b + v^n) < 1" proof (cases "cmod (complex_of_real (cmod b) / b + 1) < 1") case True then show ?thesis by (metis power_one) next case F1: False show ?thesis proof (cases "cmod (complex_of_real (cmod b) / b - 1) < 1") case True with ‹odd n› show ?thesis by (metis add_uminus_conv_diff neg_one_odd_power) next case F2: False show ?thesis proof (cases "cmod (complex_of_real (cmod b) / b + 𝗂) < 1") case T1: True show ?thesis proof (cases "even m") case True with T1 show ?thesis by (rule_tac x="𝗂" in exI) (simp add: m power_mult) next case False with T1 show ?thesis by (rule_tac x="- 𝗂" in exI) (simp add: m power_mult) qed next case False then have lt1: "cmod (of_real (cmod b) / b - 𝗂) < 1" using "0" F1 F2 unimodular_reduce_norm by blast show ?thesis proof (cases "even m") case True with m lt1 show ?thesis by (rule_tac x="- 𝗂" in exI) (simp add: power_mult) next case False with m lt1 show ?thesis by (rule_tac x="𝗂" in exI) (simp add: power_mult) qed qed qed qed then obtain v where v: "cmod (complex_of_real (cmod b) / b + v^n) < 1" by blast let ?w = "v / complex_of_real (root n (cmod b))" from odd_real_root_pow[OF ‹odd n›, of "cmod b"] have 1: "?w ^ n = v^n / complex_of_real (cmod b)" by (simp add: power_divide of_real_power[symmetric]) have 2:"cmod (complex_of_real (cmod b) / b) = 1" using b by (simp add: norm_divide) then have 3: "cmod (complex_of_real (cmod b) / b) ≥ 0" by simp have 4: "cmod (complex_of_real (cmod b) / b) * cmod (1 + b * (v ^ n / complex_of_real (cmod b))) < cmod (complex_of_real (cmod b) / b) * 1" apply (simp only: norm_mult[symmetric] distrib_left) using b v apply (simp add: 2) done show ?thesis by (metis 1 mult_left_less_imp_less[OF 4 3]) qed qed text ‹Bolzano-Weierstrass type property for closed disc in complex plane.› lemma metric_bound_lemma: "cmod (x - y) ≤ ¦Re x - Re y¦ + ¦Im x - Im y¦" using real_sqrt_sum_squares_triangle_ineq[of "Re x - Re y" 0 0 "Im x - Im y"] unfolding cmod_def by simp lemma Bolzano_Weierstrass_complex_disc: assumes r: "∀n. cmod (s n) ≤ r" shows "∃f z. strict_mono (f :: nat ⇒ nat) ∧ (∀e >0. ∃N. ∀n ≥ N. cmod (s (f n) - z) < e)" proof - from seq_monosub[of "Re ∘ s"] obtain f where f: "strict_mono f" "monoseq (λn. Re (s (f n)))" unfolding o_def by blast from seq_monosub[of "Im ∘ s ∘ f"] obtain g where g: "strict_mono g" "monoseq (λn. Im (s (f (g n))))" unfolding o_def by blast let ?h = "f ∘ g" have "r ≥ 0" by (meson norm_ge_zero order_trans r) have "∀n. r + 1 ≥ ¦Re (s n)¦" by (smt (verit, ccfv_threshold) abs_Re_le_cmod r) then have conv1: "convergent (λn. Re (s (f n)))" by (metis Bseq_monoseq_convergent f(2) BseqI' real_norm_def) have "∀n. r + 1 ≥ ¦Im (s n)¦" by (smt (verit) abs_Im_le_cmod r) then have conv2: "convergent (λn. Im (s (f (g n))))" by (metis Bseq_monoseq_convergent g(2) BseqI' real_norm_def) obtain x where x: "∀r>0. ∃n0. ∀n≥n0. ¦Re (s (f n)) - x¦ < r" using conv1[unfolded convergent_def] LIMSEQ_iff real_norm_def by metis obtain y where y: "∀r>0. ∃n0. ∀n≥n0. ¦Im (s (f (g n))) - y¦ < r" using conv2[unfolded convergent_def] LIMSEQ_iff real_norm_def by metis let ?w = "Complex x y" from f(1) g(1) have hs: "strict_mono ?h" unfolding strict_mono_def by auto have "∃N. ∀n≥N. cmod (s (?h n) - ?w) < e" if "e > 0" for e proof - from that have e2: "e/2 > 0" by simp from x y e2 obtain N1 N2 where N1: "∀n≥N1. ¦Re (s (f n)) - x¦ < e / 2" and N2: "∀n≥N2. ¦Im (s (f (g n))) - y¦ < e / 2" by blast have "cmod (s (?h n) - ?w) < e" if "n ≥ N1 + N2" for n proof - from that have nN1: "g n ≥ N1" and nN2: "n ≥ N2" using seq_suble[OF g(1), of n] by arith+ show ?thesis using metric_bound_lemma[of "s (f (g n))" ?w] N1 N2 nN1 nN2 by fastforce qed then show ?thesis by blast qed with hs show ?thesis by blast qed text ‹Polynomial is continuous.› lemma poly_cont: fixes p :: "'a::{comm_semiring_0,real_normed_div_algebra} poly" assumes ep: "e > 0" shows "∃d >0. ∀w. 0 < norm (w - z) ∧ norm (w - z) < d ⟶ norm (poly p w - poly p z) < e" proof - obtain q where "degree q = degree p" and q: "⋀w. poly p w = poly q (w - z)" by (metis add.commute degree_offset_poly diff_add_cancel poly_offset_poly) show ?thesis unfolding q proof (induct q) case 0 then show ?case using ep by auto next case (pCons c cs) obtain m where m: "m > 0" "norm z ≤ 1 ⟹ norm (poly cs z) ≤ m" for z using poly_bound_exists[of 1 "cs"] by blast with ep have em0: "e/m > 0" by (simp add: field_simps) obtain d where d: "d > 0" "d < 1" "d < e / m" by (meson em0 field_lbound_gt_zero zero_less_one) then have "⋀w. norm (w - z) < d ⟹ norm (w - z) * norm (poly cs (w - z)) < e" by (smt (verit, del_insts) m mult_left_mono norm_ge_zero pos_less_divide_eq) with d show ?case by (force simp add: norm_mult) qed qed text ‹Hence a polynomial attains minimum on a closed disc in the complex plane.› lemma poly_minimum_modulus_disc: "∃z. ∀w. cmod w ≤ r ⟶ cmod (poly p z) ≤ cmod (poly p w)" proof - show ?thesis proof (cases "r ≥ 0") case False then show ?thesis by (metis norm_ge_zero order.trans) next case True then have mth1: "∃x z. cmod z ≤ r ∧ cmod (poly p z) = - x" by (metis add.inverse_inverse norm_zero) obtain s where s: "∀y. (∃x. (∃z. cmod z ≤ r ∧ cmod (poly p z) = - x) ∧ y < x) ⟷ y < s" by (smt (verit, del_insts) real_sup_exists[OF mth1] norm_zero zero_less_norm_iff) let ?m = "- s" have s1: "(∃z. cmod z ≤ r ∧ - (- cmod (poly p z)) < y) ⟷ ?m < y" for y by (metis add.inverse_inverse minus_less_iff s) then have s1m: "⋀z. cmod z ≤ r ⟹ cmod (poly p z) ≥ ?m" by force have "∃z. cmod z ≤ r ∧ cmod (poly p z) < - s + 1 / real (Suc n)" for n using s1[of "?m + 1/real (Suc n)"] by simp then obtain g where g: "∀n. cmod (g n) ≤ r" "∀n. cmod (poly p (g n)) <?m + 1 /real(Suc n)" by metis from Bolzano_Weierstrass_complex_disc[OF g(1)] obtain f::"nat ⇒ nat" and z where fz: "strict_mono f" "∀e>0. ∃N. ∀n≥N. cmod (g (f n) - z) < e" by blast { fix w assume wr: "cmod w ≤ r" let ?e = "¦cmod (poly p z) - ?m¦" { assume e: "?e > 0" then have e2: "?e/2 > 0" by simp with poly_cont obtain d where "d > 0" and d: "⋀w. 0<cmod (w - z)∧ cmod(w - z) < d ⟶ cmod(poly p w - poly p z) < ?e/2" by blast have 1: "cmod(poly p w - poly p z) < ?e / 2" if w: "cmod (w - z) < d" for w using d[of w] w e by (cases "w = z") simp_all from fz(2) ‹d > 0› obtain N1 where N1: "∀n≥N1. cmod (g (f n) - z) < d" by blast from reals_Archimedean2 obtain N2 :: nat where N2: "2/?e < real N2" by blast have 2: "cmod (poly p (g (f (N1 + N2))) - poly p z) < ?e/2" using N1 1 by auto have 0: "a < e2 ⟹ ¦b - m¦ < e2 ⟹ 2 * e2 ≤ ¦b - m¦ + a ⟹ False" for a b e2 m :: real by arith from seq_suble[OF fz(1), of "N1 + N2"] have 00: "?m + 1 / real (Suc (f (N1 + N2))) ≤ ?m + 1 / real (Suc (N1 + N2))" by (simp add: frac_le) from N2 e2 less_imp_inverse_less[of "2/?e" "real (Suc (N1 + N2))"] have "?e/2 > 1/ real (Suc (N1 + N2))" by (simp add: inverse_eq_divide) with order_less_le_trans[OF _ 00] have 1: "¦cmod (poly p (g (f (N1 + N2)))) - ?m¦ < ?e/2" using g s1 by (smt (verit)) with 0[OF 2] have False by (smt (verit) field_sum_of_halves norm_triangle_ineq3) } then have "?e = 0" by auto with s1m[OF wr] have "cmod (poly p z) ≤ cmod (poly p w)" by simp } then show ?thesis by blast qed qed text ‹Nonzero polynomial in z goes to infinity as z does.› lemma poly_infinity: fixes p:: "'a::{comm_semiring_0,real_normed_div_algebra} poly" assumes ex: "p ≠ 0" shows "∃r. ∀z. r ≤ norm z ⟶ d ≤ norm (poly (pCons a p) z)" using ex proof (induct p arbitrary: a d) case 0 then show ?case by simp next case (pCons c cs a d) show ?case proof (cases "cs = 0") case False with pCons.hyps obtain r where r: "∀z. r ≤ norm z ⟶ d + norm a ≤ norm (poly (pCons c cs) z)" by blast let ?r = "1 + ¦r¦" have "d ≤ norm (poly (pCons a (pCons c cs)) z)" if "1 + ¦r¦ ≤ norm z" for z proof - have "d ≤ norm(z * poly (pCons c cs) z) - norm a" by (smt (verit, best) norm_ge_zero mult_less_cancel_right2 norm_mult r that) with norm_diff_ineq add.commute show ?thesis by (metis order.trans poly_pCons) qed then show ?thesis by blast next case True have "d ≤ norm (poly (pCons a (pCons c cs)) z)" if "(¦d¦ + norm a) / norm c ≤ norm z" for z :: 'a proof - have "¦d¦ + norm a ≤ norm (z * c)" by (metis that True norm_mult pCons.hyps(1) pos_divide_le_eq zero_less_norm_iff) also have "… ≤ norm (a + z * c) + norm a" by (simp add: add.commute norm_add_leD) finally show ?thesis using True by auto qed then show ?thesis by blast qed qed text ‹Hence polynomial's modulus attains its minimum somewhere.› lemma poly_minimum_modulus: "∃z.∀w. cmod (poly p z) ≤ cmod (poly p w)" proof (induct p) case 0 then show ?case by simp next case (pCons c cs) show ?case proof (cases "cs = 0") case False from poly_infinity[OF False, of "cmod (poly (pCons c cs) 0)" c] obtain r where r: "cmod (poly (pCons c cs) 0) ≤ cmod (poly (pCons c cs) z)" if "r ≤ cmod z" for z by blast from poly_minimum_modulus_disc[of "¦r¦" "pCons c cs"] show ?thesis by (smt (verit, del_insts) order.trans linorder_linear r) qed (use pCons.hyps in auto) qed text ‹Constant function (non-syntactic characterization).› definition "constant f ⟷ (∀x y. f x = f y)" lemma nonconstant_length: "¬ constant (poly p) ⟹ psize p ≥ 2" by (induct p) (auto simp: constant_def psize_def) lemma poly_replicate_append: "poly (monom 1 n * p) (x::'a::comm_ring_1) = x^n * poly p x" by (simp add: poly_monom) text ‹Decomposition of polynomial, skipping zero coefficients after the first.› lemma poly_decompose_lemma: assumes nz: "¬ (∀z. z ≠ 0 ⟶ poly p z = (0::'a::idom))" shows "∃k a q. a ≠ 0 ∧ Suc (psize q + k) = psize p ∧ (∀z. poly p z = z^k * poly (pCons a q) z)" unfolding psize_def using nz proof (induct p) case 0 then show ?case by simp next case (pCons c cs) show ?case proof (cases "c = 0") case True from pCons.hyps pCons.prems True show ?thesis apply auto apply (rule_tac x="k+1" in exI) apply (rule_tac x="a" in exI) apply clarsimp apply (rule_tac x="q" in exI) apply auto done qed force qed lemma poly_decompose: fixes p :: "'a::idom poly" assumes nc: "¬ constant (poly p)" shows "∃k a q. a ≠ 0 ∧ k ≠ 0 ∧ psize q + k + 1 = psize p ∧ (∀z. poly p z = poly p 0 + z^k * poly (pCons a q) z)" using nc proof (induct p) case 0 then show ?case by (simp add: constant_def) next case (pCons c cs) have "¬ (∀z. z ≠ 0 ⟶ poly cs z = 0)" by (smt (verit) constant_def mult_eq_0_iff pCons.prems poly_pCons) from poly_decompose_lemma[OF this] obtain k a q where *: "a ≠ 0 ∧ Suc (psize q + k) = psize cs ∧ (∀z. poly cs z = z ^ k * poly (pCons a q) z)" by blast then have "psize q + k + 2 = psize (pCons c cs)" by (auto simp add: psize_def split: if_splits) then show ?case using "*" by force qed text ‹Fundamental theorem of algebra› theorem fundamental_theorem_of_algebra: assumes nc: "¬ constant (poly p)" shows "∃z::complex. poly p z = 0" using nc proof (induct "psize p" arbitrary: p rule: less_induct) case less let ?p = "poly p" let ?ths = "∃z. ?p z = 0" from nonconstant_length[OF less(2)] have n2: "psize p ≥ 2" . from poly_minimum_modulus obtain c where c: "∀w. cmod (?p c) ≤ cmod (?p w)" by blast show ?ths proof (cases "?p c = 0") case True then show ?thesis by blast next case False obtain q where q: "psize q = psize p" "∀x. poly q x = ?p (c + x)" using poly_offset[of p c] by blast then have qnc: "¬ constant (poly q)" by (metis (no_types, opaque_lifting) add.commute constant_def diff_add_cancel less.prems) from q(2) have pqc0: "?p c = poly q 0" by simp from c pqc0 have cq0: "∀w. cmod (poly q 0) ≤ cmod (?p w)" by simp let ?a0 = "poly q 0" from False pqc0 have a00: "?a0 ≠ 0" by simp from a00 have qr: "∀z. poly q z = poly (smult (inverse ?a0) q) z * ?a0" by simp let ?r = "smult (inverse ?a0) q" have lgqr: "psize q = psize ?r" by (simp add: a00 psize_def) have rnc: "¬ constant (poly ?r)" using constant_def qnc qr by fastforce have r01: "poly ?r 0 = 1" by (simp add: a00) have mrmq_eq: "cmod (poly ?r w) < 1 ⟷ cmod (poly q w) < cmod ?a0" for w by (smt (verit, del_insts) a00 mult_less_cancel_right2 norm_mult qr zero_less_norm_iff) from poly_decompose[OF rnc] obtain k a s where kas: "a ≠ 0" "k ≠ 0" "psize s + k + 1 = psize ?r" "∀z. poly ?r z = poly ?r 0 + z^k* poly (pCons a s) z" by blast have "∃w. cmod (poly ?r w) < 1" proof (cases "psize p = k + 1") case True with kas q have s0: "s = 0" by (simp add: lgqr) with reduce_poly_simple kas show ?thesis by (metis mult.commute mult.right_neutral poly_1 poly_smult r01 smult_one) next case False note kn = this from kn kas(3) q(1) lgqr have k1n: "k + 1 < psize p" by simp have 01: "¬ constant (poly (pCons 1 (monom a (k - 1))))" unfolding constant_def poly_pCons poly_monom by (metis add_cancel_left_right kas(1) mult.commute mult_cancel_right2 power_one) have 02: "k + 1 = psize (pCons 1 (monom a (k - 1)))" using kas by (simp add: psize_def degree_monom_eq) from less(1) [OF _ 01] k1n 02 obtain w where w: "1 + w^k * a = 0" by (metis kas(2) mult.commute mult.left_commute poly_monom poly_pCons power_eq_if) from poly_bound_exists[of "cmod w" s] obtain m where m: "m > 0" "∀z. cmod z ≤ cmod w ⟶ cmod (poly s z) ≤ m" by blast have "w ≠ 0" using kas(2) w by (auto simp add: power_0_left) from w have wm1: "w^k * a = - 1" by (simp add: add_eq_0_iff) have inv0: "0 < inverse (cmod w ^ (k + 1) * m)" by (simp add: ‹w ≠ 0› m(1)) with field_lbound_gt_zero[OF zero_less_one] obtain t where t: "t > 0" "t < 1" "t < inverse (cmod w ^ (k + 1) * m)" by blast let ?ct = "complex_of_real t" let ?w = "?ct * w" have "1 + ?w^k * (a + ?w * poly s ?w) = 1 + ?ct^k * (w^k * a) + ?w^k * ?w * poly s ?w" using kas(1) by (simp add: algebra_simps power_mult_distrib) also have "… = complex_of_real (1 - t^k) + ?w^k * ?w * poly s ?w" unfolding wm1 by simp finally have "cmod (1 + ?w^k * (a + ?w * poly s ?w)) = cmod (complex_of_real (1 - t^k) + ?w^k * ?w * poly s ?w)" by metis with norm_triangle_ineq[of "complex_of_real (1 - t^k)" "?w^k * ?w * poly s ?w"] have 11: "cmod (1 + ?w^k * (a + ?w * poly s ?w)) ≤ ¦1 - t^k¦ + cmod (?w^k * ?w * poly s ?w)" unfolding norm_of_real by simp have ath: "⋀x t::real. 0 ≤ x ⟹ x < t ⟹ t ≤ 1 ⟹ ¦1 - t¦ + x < 1" by arith have tw: "cmod ?w ≤ cmod w" by (smt (verit) mult_le_cancel_right2 norm_ge_zero norm_mult norm_of_real t) have "t * (cmod w ^ (k + 1) * m) < 1" by (smt (verit, best) inv0 inverse_positive_iff_positive left_inverse mult_strict_right_mono t(3)) with zero_less_power[OF t(1), of k] have 30: "t^k * (t* (cmod w ^ (k + 1) * m)) < t^k" by simp have "cmod (?w^k * ?w * poly s ?w) = t^k * (t* (cmod w ^ (k + 1) * cmod (poly s ?w)))" using ‹w ≠ 0› t(1) by (simp add: algebra_simps norm_power norm_mult) with 30 have 120: "cmod (?w^k * ?w * poly s ?w) < t^k" by (smt (verit, ccfv_SIG) m(2) mult_left_mono norm_ge_zero t(1) tw zero_le_power) from power_strict_mono[OF t(2), of k] t(1) kas(2) have 121: "t^k ≤ 1" by auto from ath[OF norm_ge_zero[of "?w^k * ?w * poly s ?w"] 120 121] show ?thesis by (smt (verit) "11" kas(4) poly_pCons r01) qed with cq0 q(2) show ?thesis by (smt (verit) mrmq_eq) qed qed text ‹Alternative version with a syntactic notion of constant polynomial.› lemma fundamental_theorem_of_algebra_alt: assumes nc: "¬ (∃a l. a ≠ 0 ∧ l = 0 ∧ p = pCons a l)" shows "∃z. poly p z = (0::complex)" proof (rule ccontr) assume N: "∄z. poly p z = 0" then have "¬ constant (poly p)" unfolding constant_def by (metis (no_types, opaque_lifting) nc poly_pcompose pcompose_0' pcompose_const poly_0_coeff_0 poly_all_0_iff_0 poly_diff right_minus_eq) then show False using N fundamental_theorem_of_algebra by blast qed subsection ‹Nullstellensatz, degrees and divisibility of polynomials› lemma nullstellensatz_lemma: fixes p :: "complex poly" assumes "∀x. poly p x = 0 ⟶ poly q x = 0" and "degree p = n" and "n ≠ 0" shows "p dvd (q ^ n)" using assms proof (induct n arbitrary: p q rule: nat_less_induct) fix n :: nat fix p q :: "complex poly" assume IH: "∀m<n. ∀p q. (∀x. poly p x = (0::complex) ⟶ poly q x = 0) ⟶ degree p = m ⟶ m ≠ 0 ⟶ p dvd (q ^ m)" and pq0: "∀x. poly p x = 0 ⟶ poly q x = 0" and dpn: "degree p = n" and n0: "n ≠ 0" from dpn n0 have pne: "p ≠ 0" by auto show "p dvd (q ^ n)" proof (cases "∃a. poly p a = 0") case True then obtain a where a: "poly p a = 0" .. have ?thesis if oa: "order a p ≠ 0" proof - let ?op = "order a p" from pne have ap: "([:- a, 1:] ^ ?op) dvd p" "¬ [:- a, 1:] ^ (Suc ?op) dvd p" using order by blast+ note oop = order_degree[OF pne, unfolded dpn] show ?thesis proof (cases "q = 0") case True with n0 show ?thesis by (simp add: power_0_left) next case False from pq0[rule_format, OF a, unfolded poly_eq_0_iff_dvd] obtain r where r: "q = [:- a, 1:] * r" by (rule dvdE) from ap(1) obtain s where s: "p = [:- a, 1:] ^ ?op * s" by (rule dvdE) have sne: "s ≠ 0" using s pne by auto show ?thesis proof (cases "degree s = 0") case True then obtain k where kpn: "s = [:k:]" by (cases s) (auto split: if_splits) from sne kpn have k: "k ≠ 0" by simp let ?w = "([:1/k:] * ([:-a,1:] ^ (n - ?op))) * (r ^ n)" have "q^n = [:- a, 1:] ^ n * r ^ n" using power_mult_distrib r by blast also have "... = [:- a, 1:] ^ order a p * [:k:] * ([:1 / k:] * [:- a, 1:] ^ (n - order a p) * r ^ n)" using k oop [of a] by (simp flip: power_add) also have "... = p * ?w" by (metis s kpn) finally show ?thesis unfolding dvd_def by blast next case False with sne dpn s oa have dsn: "degree s < n" by (metis add_diff_cancel_right' degree_0 degree_linear_power degree_mult_eq gr0I zero_less_diff) have "poly r x = 0" if h: "poly s x = 0" for x proof - have "x ≠ a" by (metis ap(2) dvd_refl mult_dvd_mono poly_eq_0_iff_dvd power_Suc power_commutes s that) moreover have "poly p x = 0" by (metis (no_types) mult_eq_0_iff poly_mult s that) ultimately show ?thesis using pq0 r by auto qed with False IH dsn obtain u where u: "r ^ (degree s) = s * u" by blast then have u': "⋀x. poly s x * poly u x = poly r x ^ degree s" by (simp only: poly_mult[symmetric] poly_power[symmetric]) have "q^n = [:- a, 1:] ^ n * r ^ n" using power_mult_distrib r by blast also have "... = [:- a, 1:] ^ order a p * (s * u * ([:- a, 1:] ^ (n - order a p) * r ^ (n - degree s)))" by (smt (verit, del_insts) s u mult_ac power_add add_diff_cancel_right' degree_linear_power degree_mult_eq dpn mult_zero_left) also have "... = p * (u * ([:-a,1:] ^ (n - ?op))) * (r ^ (n - degree s))" using s by force finally show ?thesis unfolding dvd_def by auto qed qed qed then show ?thesis using a order_root pne by blast next case False then show ?thesis using dpn n0 fundamental_theorem_of_algebra_alt[of p] by fastforce qed qed lemma nullstellensatz_univariate: "(∀x. poly p x = (0::complex) ⟶ poly q x = 0) ⟷ p dvd (q ^ (degree p)) ∨ (p = 0 ∧ q = 0)" proof - consider "p = 0" | "p ≠ 0" "degree p = 0" | n where "p ≠ 0" "degree p = Suc n" by (cases "degree p") auto then show ?thesis proof cases case p: 1 then have "(∀x. poly p x = (0::complex) ⟶ poly q x = 0) ⟷ q = 0" by (auto simp add: poly_all_0_iff_0) with p show ?thesis by force next case dp: 2 then show ?thesis by (meson dvd_trans is_unit_iff_degree poly_eq_0_iff_dvd unit_imp_dvd) next case dp: 3 have False if "p dvd (q ^ (Suc n))" "poly p x = 0" "poly q x ≠ 0" for x by (metis dvd_trans poly_eq_0_iff_dvd poly_power power_eq_0_iff that) with dp nullstellensatz_lemma[of p q "degree p"] show ?thesis by auto qed qed text ‹Useful lemma› lemma constant_degree: fixes p :: "'a::{idom,ring_char_0} poly" shows "constant (poly p) ⟷ degree p = 0" (is "?lhs = ?rhs") proof show ?rhs if ?lhs proof - from that[unfolded constant_def, rule_format, of _ "0"] have "poly p = poly [:poly p 0:]" by auto then show ?thesis by (metis degree_pCons_0 poly_eq_poly_eq_iff) qed show ?lhs if ?rhs unfolding constant_def by (metis degree_eq_zeroE pcompose_const poly_0 poly_pcompose that) qed lemma complex_poly_decompose: "smult (lead_coeff p) (∏z|poly p z = 0. [:-z, 1:] ^ order z p) = (p :: complex poly)" proof (induction p rule: poly_root_order_induct) case (no_roots p) show ?case proof (cases "degree p = 0") case False hence "¬constant (poly p)" by (subst constant_degree) with fundamental_theorem_of_algebra and no_roots show ?thesis by blast qed (auto elim!: degree_eq_zeroE) next case (root p x n) from root have *: "{z. poly ([:- x, 1:] ^ n * p) z = 0} = insert x {z. poly p z = 0}" by auto have "smult (lead_coeff ([:-x, 1:] ^ n * p)) (∏z|poly ([:-x,1:] ^ n * p) z = 0. [:-z, 1:] ^ order z ([:- x, 1:] ^ n * p)) = [:- x, 1:] ^ order x ([:- x, 1:] ^ n * p) * smult (lead_coeff p) (∏z∈{z. poly p z = 0}. [:- z, 1:] ^ order z ([:- x, 1:] ^ n * p))" by (subst *, subst prod.insert) (insert root, auto intro: poly_roots_finite simp: mult_ac lead_coeff_mult lead_coeff_power) also have "order x ([:- x, 1:] ^ n * p) = n" using root by (subst order_mult) (auto simp: order_power_n_n order_0I) also have "(∏z∈{z. poly p z = 0}. [:- z, 1:] ^ order z ([:- x, 1:] ^ n * p)) = (∏z∈{z. poly p z = 0}. [:- z, 1:] ^ order z p)" proof (intro prod.cong refl, goal_cases) case (1 y) with root have "order y ([:-x,1:] ^ n) = 0" by (intro order_0I) auto thus ?case using root by (subst order_mult) auto qed also note root.IH finally show ?case . qed simp_all instance complex :: alg_closed_field by standard (use fundamental_theorem_of_algebra constant_degree neq0_conv in blast) lemma size_proots_complex: "size (proots (p :: complex poly)) = degree p" proof (cases "p = 0") case [simp]: False show "size (proots p) = degree p" by (subst (1 2) complex_poly_decompose [symmetric]) (simp add: proots_prod proots_power degree_prod_sum_eq degree_power_eq) qed auto lemma complex_poly_decompose_multiset: "smult (lead_coeff p) (∏x∈#proots p. [:-x, 1:]) = (p :: complex poly)" proof (cases "p = 0") case False hence "(∏x∈#proots p. [:-x, 1:]) = (∏x | poly p x = 0. [:-x, 1:] ^ order x p)" by (subst image_prod_mset_multiplicity) simp_all also have "smult (lead_coeff p) … = p" by (rule complex_poly_decompose) finally show ?thesis . qed auto lemma complex_poly_decompose': obtains root where "smult (lead_coeff p) (∏i<degree p. [:-root i, 1:]) = (p :: complex poly)" proof - obtain roots where roots: "mset roots = proots p" using ex_mset by blast have "p = smult (lead_coeff p) (∏x∈#proots p. [:-x, 1:])" by (rule complex_poly_decompose_multiset [symmetric]) also have "(∏x∈#proots p. [:-x, 1:]) = (∏x←roots. [:-x, 1:])" by (subst prod_mset_prod_list [symmetric]) (simp add: roots) also have "… = (∏i<length roots. [:-roots ! i, 1:])" by (subst prod.list_conv_set_nth) (auto simp: atLeast0LessThan) finally have eq: "p = smult (lead_coeff p) (∏i<length roots. [:-roots ! i, 1:])" . also have [simp]: "degree p = length roots" using roots by (subst eq) (auto simp: degree_prod_sum_eq) finally show ?thesis by (intro that[of "λi. roots ! i"]) auto qed lemma complex_poly_decompose_rsquarefree: assumes "rsquarefree p" shows "smult (lead_coeff p) (∏z|poly p z = 0. [:-z, 1:]) = (p :: complex poly)" proof (cases "p = 0") case False have "(∏z|poly p z = 0. [:-z, 1:]) = (∏z|poly p z = 0. [:-z, 1:] ^ order z p)" using assms False by (intro prod.cong) (auto simp: rsquarefree_root_order) also have "smult (lead_coeff p) … = p" by (rule complex_poly_decompose) finally show ?thesis . qed auto text ‹Arithmetic operations on multivariate polynomials.› lemma mpoly_base_conv: fixes x :: "'a::comm_ring_1" shows "0 = poly 0 x" "c = poly [:c:] x" "x = poly [:0,1:] x" by simp_all lemma mpoly_norm_conv: fixes x :: "'a::comm_ring_1" shows "poly [:0:] x = poly 0 x" "poly [:poly 0 y:] x = poly 0 x" by simp_all lemma mpoly_sub_conv: fixes x :: "'a::comm_ring_1" shows "poly p x - poly q x = poly p x + -1 * poly q x" by simp lemma poly_pad_rule: "poly p x = 0 ⟹ poly (pCons 0 p) x = 0" by simp lemma poly_cancel_eq_conv: fixes x :: "'a::field" shows "x = 0 ⟹ a ≠ 0 ⟹ y = 0 ⟷ a * y - b * x = 0" by auto lemma poly_divides_pad_rule: fixes p:: "('a::comm_ring_1) poly" assumes pq: "p dvd q" shows "p dvd (pCons 0 q)" by (metis add_0 dvd_def mult_pCons_right pq smult_0_left) lemma poly_divides_conv0: fixes p:: "'a::field poly" assumes lgpq: "degree q < degree p" and lq: "p ≠ 0" shows "p dvd q ⟷ q = 0" using lgpq mod_poly_less by fastforce lemma poly_divides_conv1: fixes p :: "'a::field poly" assumes a0: "a ≠ 0" and pp': "p dvd p'" and qrp': "smult a q - p' = r" shows "p dvd q ⟷ p dvd r" by (metis a0 diff_add_cancel dvd_add_left_iff dvd_smult_iff pp' qrp') lemma basic_cqe_conv1: "(∃x. poly p x = 0 ∧ poly 0 x ≠ 0) ⟷ False" "(∃x. poly 0 x ≠ 0) ⟷ False" "(∃x. poly [:c:] x ≠ 0) ⟷ c ≠ 0" "(∃x. poly 0 x = 0) ⟷ True" "(∃x. poly [:c:] x = 0) ⟷ c = 0" by simp_all lemma basic_cqe_conv2: assumes l: "p ≠ 0" shows "∃x. poly (pCons a (pCons b p)) x = (0::complex)" by (meson fundamental_theorem_of_algebra_alt l pCons_eq_0_iff pCons_eq_iff) lemma basic_cqe_conv_2b: "(∃x. poly p x ≠ (0::complex)) ⟷ p ≠ 0" by (metis poly_all_0_iff_0) lemma basic_cqe_conv3: fixes p q :: "complex poly" assumes l: "p ≠ 0" shows "(∃x. poly (pCons a p) x = 0 ∧ poly q x ≠ 0) ⟷ ¬ (pCons a p) dvd (q ^ psize p)" by (metis degree_pCons_eq_if l nullstellensatz_univariate pCons_eq_0_iff psize_def) lemma basic_cqe_conv4: fixes p q :: "complex poly" assumes h: "⋀x. poly (q ^ n) x = poly r x" shows "p dvd (q ^ n) ⟷ p dvd r" by (metis (no_types) basic_cqe_conv_2b h poly_diff right_minus_eq) lemma poly_const_conv: fixes x :: "'a::comm_ring_1" shows "poly [:c:] x = y ⟷ c = y" by simp end