(* 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" using complex_mod_triangle_ineq2[of "w + z" "-z"] by auto 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 "… ≤ norm c + r * m" using mult_mono[OF H th rp norm_ge_zero[of "poly cs z"]] by (simp add: norm_mult) also have "… ≤ ?k" by simp 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: "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)" apply (induct p) apply (simp add: offset_poly_0) apply (simp add: offset_poly_pCons algebra_simps) done 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: "offset_poly p h = 0 ⟷ p = 0" apply (safe intro!: offset_poly_0) apply (induct p) apply simp apply (simp add: offset_poly_pCons) apply (frule offset_poly_eq_0_lemma, simp) done lemma degree_offset_poly: "degree (offset_poly p h) = degree p" apply (induct p) apply (simp add: offset_poly_0) apply (case_tac "p = 0") apply (simp add: offset_poly_0 offset_poly_pCons) apply (simp add: offset_poly_pCons) apply (subst degree_add_eq_right) apply (rule le_less_trans [OF degree_smult_le]) apply (simp add: offset_poly_eq_0_iff) apply (simp add: offset_poly_eq_0_iff) done 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))" proof (intro exI conjI) show "psize (offset_poly p a) = psize p" unfolding psize_def by (simp add: offset_poly_eq_0_iff degree_offset_poly) show "∀x. poly (offset_poly p a) x = poly p (a + x)" by (simp add: poly_offset_poly) qed 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 - (rule power_mono, simp, simp)+ then have th0: "4 * x⇧^{2}≤ 1" "4 * y⇧^{2}≤ 1" by (simp_all add: power_mult_distrib) from add_mono[OF th0] xy show ?thesis by simp qed then show ?thesis unfolding linorder_not_le[symmetric] by blast 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 have "∃m. n = 2 * m" by presburger then obtain m where m: "n = 2 * m" by blast from n m have "m ≠ 0" "m < n" by presburger+ with IH[rule_format, of m] 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 th0: "cmod (complex_of_real (cmod b) / b) = 1" using b by (simp add: norm_divide) from unimodular_reduce_norm[OF th0] ‹odd n› have "∃v. cmod (complex_of_real (cmod b) / b + v^n) < 1" apply (cases "cmod (complex_of_real (cmod b) / b + 1) < 1") apply (rule_tac x="1" in exI) apply simp apply (cases "cmod (complex_of_real (cmod b) / b - 1) < 1") apply (rule_tac x="-1" in exI) apply simp apply (cases "cmod (complex_of_real (cmod b) / b + 𝗂) < 1") apply (cases "even m") apply (rule_tac x="𝗂" in exI) apply (simp add: m power_mult) apply (rule_tac x="- 𝗂" in exI) apply (simp add: m power_mult) apply (cases "even m") apply (rule_tac x="- 𝗂" in exI) apply (simp add: m power_mult) apply (auto simp add: m power_mult) apply (rule_tac x="𝗂" in exI) apply (auto simp add: m power_mult) done 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 th1: "?w ^ n = v^n / complex_of_real (cmod b)" by (simp add: power_divide of_real_power[symmetric]) have th2:"cmod (complex_of_real (cmod b) / b) = 1" using b by (simp add: norm_divide) then have th3: "cmod (complex_of_real (cmod b) / b) ≥ 0" by simp have th4: "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: th2) done from mult_left_less_imp_less[OF th4 th3] have "?P ?w n" unfolding th1 . then show ?thesis .. 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" from r[rule_format, of 0] have rp: "r ≥ 0" using norm_ge_zero[of "s 0"] by arith have th: "∀n. r + 1 ≥ ¦Re (s n)¦" proof fix n from abs_Re_le_cmod[of "s n"] r[rule_format, of n] show "¦Re (s n)¦ ≤ r + 1" by arith qed have conv1: "convergent (λn. Re (s (f n)))" apply (rule Bseq_monoseq_convergent) apply (simp add: Bseq_def) apply (metis gt_ex le_less_linear less_trans order.trans th) apply (rule f(2)) done have th: "∀n. r + 1 ≥ ¦Im (s n)¦" proof fix n from abs_Im_le_cmod[of "s n"] r[rule_format, of n] show "¦Im (s n)¦ ≤ r + 1" by arith qed have conv2: "convergent (λn. Im (s (f (g n))))" apply (rule Bseq_monoseq_convergent) apply (simp add: Bseq_def) apply (metis gt_ex le_less_linear less_trans order.trans th) apply (rule g(2)) done from conv1[unfolded convergent_def] obtain x where "LIMSEQ (λn. Re (s (f n))) x" by blast then have x: "∀r>0. ∃n0. ∀n≥n0. ¦Re (s (f n)) - x¦ < r" unfolding LIMSEQ_iff real_norm_def . from conv2[unfolded convergent_def] obtain y where "LIMSEQ (λn. Im (s (f (g n)))) y" by blast then have y: "∀r>0. ∃n0. ∀n≥n0. ¦Im (s (f (g n))) - y¦ < r" unfolding LIMSEQ_iff real_norm_def . 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[rule_format, OF e2] y[rule_format, OF 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+ from add_strict_mono[OF N1[rule_format, OF nN1] N2[rule_format, OF nN2]] show ?thesis using metric_bound_lemma[of "s (f (g n))" ?w] by simp 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 q: "degree q = degree p" "poly q x = poly p (z + x)" for x proof show "degree (offset_poly p z) = degree p" by (rule degree_offset_poly) show "⋀x. poly (offset_poly p z) x = poly p (z + x)" by (rule poly_offset_poly) qed have th: "⋀w. poly q (w - z) = poly p w" using q(2)[of "w - z" for w] by simp show ?thesis unfolding th[symmetric] proof (induct q) case 0 then show ?case using ep by auto next case (pCons c cs) from poly_bound_exists[of 1 "cs"] obtain m where m: "m > 0" "norm z ≤ 1 ⟹ norm (poly cs z) ≤ m" for z by blast from ep m(1) have em0: "e/m > 0" by (simp add: field_simps) have one0: "1 > (0::real)" by arith from field_lbound_gt_zero[OF one0 em0] obtain d where d: "d > 0" "d < 1" "d < e / m" by blast from d(1,3) m(1) have dm: "d * m > 0" "d * m < e" by (simp_all add: field_simps) show ?case proof (rule ex_forward[OF field_lbound_gt_zero[OF one0 em0]], clarsimp simp add: norm_mult) fix d w assume H: "d > 0" "d < 1" "d < e/m" "w ≠ z" "norm (w - z) < d" then have d1: "norm (w-z) ≤ 1" "d ≥ 0" by simp_all from H(3) m(1) have dme: "d*m < e" by (simp add: field_simps) from H have th: "norm (w - z) ≤ d" by simp from mult_mono[OF th m(2)[OF d1(1)] d1(2) norm_ge_zero] dme show "norm (w - z) * norm (poly cs (w - z)) < e" by simp qed 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 "cmod 0 ≤ r ∧ cmod (poly p 0) = - (- cmod (poly p 0))" by simp then have mth1: "∃x z. cmod z ≤ r ∧ cmod (poly p z) = - x" by blast have False if "cmod z ≤ r" "cmod (poly p z) = - x" "¬ x < 1" for x z proof - from that have "- x < 0 " by arith with that(2) norm_ge_zero[of "poly p z"] show ?thesis by simp qed then have mth2: "∃z. ∀x. (∃z. cmod z ≤ r ∧ cmod (poly p z) = - x) ⟶ x < z" by blast from real_sup_exists[OF mth1 mth2] obtain s where s: "∀y. (∃x. (∃z. cmod z ≤ r ∧ cmod (poly p z) = - x) ∧ y < x) ⟷ y < s" by blast let ?m = "- s" have s1[unfolded minus_minus]: "(∃z x. cmod z ≤ r ∧ - (- cmod (poly p z)) < y) ⟷ ?m < y" for y using s[rule_format, of "-y"] unfolding minus_less_iff[of y] equation_minus_iff by blast from s1[of ?m] have s1m: "⋀z x. cmod z ≤ r ⟹ cmod (poly p z) ≥ ?m" by auto have "∃z. cmod z ≤ r ∧ cmod (poly p z) < - s + 1 / real (Suc n)" for n using s1[rule_format, of "?m + 1/real (Suc n)"] by simp then have th: "∀n. ∃z. cmod z ≤ r ∧ cmod (poly p z) < - s + 1 / real (Suc n)" .. from choice[OF th] obtain g where g: "∀n. cmod (g n) ≤ r" "∀n. cmod (poly p (g n)) <?m + 1 /real(Suc n)" by blast from bolzano_weierstrass_complex_disc[OF g(1)] obtain f z where fz: "strict_mono (f :: nat ⇒ nat)" "∀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 from poly_cont[OF e2, of z p] obtain d where d: "d > 0" "∀w. 0<cmod (w - z)∧ cmod(w - z) < d ⟶ cmod(poly p w - poly p z) < ?e/2" by blast have th1: "cmod(poly p w - poly p z) < ?e / 2" if w: "cmod (w - z) < d" for w using d(2)[rule_format, of w] w e by (cases "w = z") simp_all from fz(2) d(1) obtain N1 where N1: "∀n≥N1. cmod (g (f n) - z) < d" by blast from reals_Archimedean2[of "2/?e"] obtain N2 :: nat where N2: "2/?e < real N2" by blast have th2: "cmod (poly p (g (f (N1 + N2))) - poly p z) < ?e/2" using N1[rule_format, of "N1 + N2"] th1 by simp have th0: "a < e2 ⟹ ¦b - m¦ < e2 ⟹ 2 * e2 ≤ ¦b - m¦ + a ⟹ False" for a b e2 m :: real by arith have ath: "m ≤ x ⟹ x < m + e ⟹ ¦x - m¦ < e" for m x e :: real by arith from s1m[OF g(1)[rule_format]] have th31: "?m ≤ cmod(poly p (g (f (N1 + N2))))" . from seq_suble[OF fz(1), of "N1 + N2"] have th00: "real (Suc (N1 + N2)) ≤ real (Suc (f (N1 + N2)))" by simp have th000: "0 ≤ (1::real)" "(1::real) ≤ 1" "real (Suc (N1 + N2)) > 0" using N2 by auto from frac_le[OF th000 th00] have th00: "?m + 1 / real (Suc (f (N1 + N2))) ≤ ?m + 1 / real (Suc (N1 + N2))" by simp from g(2)[rule_format, of "f (N1 + N2)"] have th01:"cmod (poly p (g (f (N1 + N2)))) < - s + 1 / real (Suc (f (N1 + N2)))" . from order_less_le_trans[OF th01 th00] have th32: "cmod (poly p (g (f (N1 + N2)))) < ?m + (1/ real(Suc (N1 + N2)))" . from N2 have "2/?e < real (Suc (N1 + N2))" by arith with 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 ath[OF th31 th32] have thc1: "¦cmod (poly p (g (f (N1 + N2)))) - ?m¦ < ?e/2" by arith have ath2: "¦a - b¦ ≤ c ⟹ ¦b - m¦ ≤ ¦a - m¦ + c" for a b c m :: real by arith have th22: "¦cmod (poly p (g (f (N1 + N2)))) - cmod (poly p z)¦ ≤ cmod (poly p (g (f (N1 + N2))) - poly p z)" by (simp add: norm_triangle_ineq3) from ath2[OF th22, of ?m] have thc2: "2 * (?e/2) ≤ ¦cmod(poly p (g (f (N1 + N2)))) - ?m¦ + cmod (poly p (g (f (N1 + N2))) - poly p z)" by simp from th0[OF th2 thc1 thc2] have False . } then have "?e = 0" by auto then have "cmod (poly p z) = ?m" by simp 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 r0: "r ≤ norm z" using that by arith from r[rule_format, OF r0] have th0: "d + norm a ≤ 1 * norm(poly (pCons c cs) z)" by arith from that have z1: "norm z ≥ 1" by arith from order_trans[OF th0 mult_right_mono[OF z1 norm_ge_zero[of "poly (pCons c cs) z"]]] have th1: "d ≤ norm(z * poly (pCons c cs) z) - norm a" unfolding norm_mult by (simp add: algebra_simps) from norm_diff_ineq[of "z * poly (pCons c cs) z" a] have th2: "norm (z * poly (pCons c cs) z) - norm a ≤ norm (poly (pCons a (pCons c cs)) z)" by (simp add: algebra_simps) from th1 th2 show ?thesis by arith qed then show ?thesis by blast next case True with pCons.prems have c0: "c ≠ 0" by simp have "d ≤ norm (poly (pCons a (pCons c cs)) z)" if h: "(¦d¦ + norm a) / norm c ≤ norm z" for z :: 'a proof - from c0 have "norm c > 0" by simp from h c0 have th0: "¦d¦ + norm a ≤ norm (z * c)" by (simp add: field_simps norm_mult) have ath: "⋀mzh mazh ma. mzh ≤ mazh + ma ⟹ ¦d¦ + ma ≤ mzh ⟹ d ≤ mazh" by arith from norm_diff_ineq[of "z * c" a] have th1: "norm (z * c) ≤ norm (a + z * c) + norm a" by (simp add: algebra_simps) from ath[OF th1 th0] show ?thesis using True by simp 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 have ath: "⋀z r. r ≤ cmod z ∨ cmod z ≤ ¦r¦" by arith from poly_minimum_modulus_disc[of "¦r¦" "pCons c cs"] obtain v where v: "cmod (poly (pCons c cs) v) ≤ cmod (poly (pCons c cs) w)" if "cmod w ≤ ¦r¦" for w by blast have "cmod (poly (pCons c cs) v) ≤ cmod (poly (pCons c cs) z)" if z: "r ≤ cmod z" for z using v[of 0] r[OF z] by simp with v ath[of r] show ?thesis by blast next case True with pCons.hyps show ?thesis by simp qed 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 next case False show ?thesis apply (rule exI[where x=0]) apply (rule exI[where x=c]) apply (auto simp: False) done qed qed lemma poly_decompose: assumes nc: "¬ constant (poly p)" shows "∃k a q. a ≠ (0::'a::idom) ∧ 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)" proof assume "∀z. z ≠ 0 ⟶ poly cs z = 0" then have "poly (pCons c cs) x = poly (pCons c cs) y" for x y by (cases "x = 0") auto with pCons.prems show False by (auto simp add: constant_def) qed from poly_decompose_lemma[OF this] show ?case apply clarsimp apply (rule_tac x="k+1" in exI) apply (rule_tac x="a" in exI) apply simp apply (rule_tac x="q" in exI) apply (auto simp add: psize_def split: if_splits) done qed text ‹Fundamental theorem of algebra› lemma 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 from poly_offset[of p c] obtain q where q: "psize q = psize p" "∀x. poly q x = ?p (c + x)" by blast have False if h: "constant (poly q)" proof - from q(2) have th: "∀x. poly q (x - c) = ?p x" by auto have "?p x = ?p y" for x y proof - from th have "?p x = poly q (x - c)" by auto also have "… = poly q (y - c)" using h unfolding constant_def by blast also have "… = ?p y" using th by auto finally show ?thesis . qed with less(2) show ?thesis unfolding constant_def by blast qed then have qnc: "¬ constant (poly q)" by blast 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" using a00 unfolding psize_def degree_def by (simp add: poly_eq_iff) have False if h: "⋀x y. poly ?r x = poly ?r y" proof - have "poly q x = poly q y" for x y proof - from qr[rule_format, of x] have "poly q x = poly ?r x * ?a0" by auto also have "… = poly ?r y * ?a0" using h by simp also have "… = poly q y" using qr[rule_format, of y] by simp finally show ?thesis . qed with qnc show ?thesis unfolding constant_def by blast qed then have rnc: "¬ constant (poly ?r)" unfolding constant_def by blast from qr[rule_format, of 0] a00 have r01: "poly ?r 0 = 1" by auto have mrmq_eq: "cmod (poly ?r w) < 1 ⟷ cmod (poly q w) < cmod ?a0" for w proof - have "cmod (poly ?r w) < 1 ⟷ cmod (poly q w / ?a0) < 1" using qr[rule_format, of w] a00 by (simp add: divide_inverse ac_simps) also have "… ⟷ cmod (poly q w) < cmod ?a0" using a00 unfolding norm_divide by (simp add: field_simps) finally show ?thesis . qed 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(3) lgqr[symmetric] q(1) have s0: "s = 0" by auto have hth[symmetric]: "cmod (poly ?r w) = cmod (1 + a * w ^ k)" for w using kas(4)[rule_format, of w] s0 r01 by (simp add: algebra_simps) from reduce_poly_simple[OF kas(1,2)] show ?thesis unfolding hth by blast next case False note kn = this from kn kas(3) q(1) lgqr have k1n: "k + 1 < psize p" by simp have th01: "¬ constant (poly (pCons 1 (monom a (k - 1))))" unfolding constant_def poly_pCons poly_monom using kas(1) apply simp apply (rule exI[where x=0]) apply (rule exI[where x=1]) apply simp done from kas(1) kas(2) have th02: "k + 1 = psize (pCons 1 (monom a (k - 1)))" by (simp add: psize_def degree_monom_eq) from less(1) [OF k1n [simplified th02] th01] obtain w where w: "1 + w^k * a = 0" unfolding poly_pCons poly_monom using kas(2) by (cases k) (auto simp add: algebra_simps) 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 w0: "w ≠ 0" using kas(2) w by (auto simp add: power_0_left) from w have "(1 + w ^ k * a) - 1 = 0 - 1" by simp then have wm1: "w^k * a = - 1" by simp have inv0: "0 < inverse (cmod w ^ (k + 1) * m)" using norm_ge_zero[of w] w0 m(1) by (simp add: inverse_eq_divide zero_less_mult_iff) 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 th11: "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 "t * cmod w ≤ 1 * cmod w" apply (rule mult_mono) using t(1,2) apply auto done then have tw: "cmod ?w ≤ cmod w" using t(1) by (simp add: norm_mult) from t inv0 have "t * (cmod w ^ (k + 1) * m) < 1" by (simp add: field_simps) with zero_less_power[OF t(1), of k] have th30: "t^k * (t* (cmod w ^ (k + 1) * m)) < t^k * 1" by simp have "cmod (?w^k * ?w * poly s ?w) = t^k * (t* (cmod w ^ (k + 1) * cmod (poly s ?w)))" using w0 t(1) by (simp add: algebra_simps power_mult_distrib norm_power norm_mult) then have "cmod (?w^k * ?w * poly s ?w) ≤ t^k * (t* (cmod w ^ (k + 1) * m))" using t(1,2) m(2)[rule_format, OF tw] w0 by auto with th30 have th120: "cmod (?w^k * ?w * poly s ?w) < t^k" by simp from power_strict_mono[OF t(2), of k] t(1) kas(2) have th121: "t^k ≤ 1" by auto from ath[OF norm_ge_zero[of "?w^k * ?w * poly s ?w"] th120 th121] have th12: "¦1 - t^k¦ + cmod (?w^k * ?w * poly s ?w) < 1" . from th11 th12 have "cmod (1 + ?w^k * (a + ?w * poly s ?w)) < 1" by arith then have "cmod (poly ?r ?w) < 1" unfolding kas(4)[rule_format, of ?w] r01 by simp then show ?thesis by blast qed with cq0 q(2) show ?thesis unfolding mrmq_eq not_less[symmetric] by auto 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)" using nc proof (induct p) case 0 then show ?case by simp next case (pCons c cs) show ?case proof (cases "c = 0") case True then show ?thesis by auto next case False have "¬ constant (poly (pCons c cs))" proof assume nc: "constant (poly (pCons c cs))" from nc[unfolded constant_def, rule_format, of 0] have "∀w. w ≠ 0 ⟶ poly cs w = 0" by auto then have "cs = 0" proof (induct cs) case 0 then show ?case by simp next case (pCons d ds) show ?case proof (cases "d = 0") case True then show ?thesis using pCons.prems pCons.hyps by simp next case False from poly_bound_exists[of 1 ds] obtain m where m: "m > 0" "∀z. ∀z. cmod z ≤ 1 ⟶ cmod (poly ds z) ≤ m" by blast have dm: "cmod d / m > 0" using False m(1) by (simp add: field_simps) from field_lbound_gt_zero[OF dm zero_less_one] obtain x where x: "x > 0" "x < cmod d / m" "x < 1" by blast let ?x = "complex_of_real x" from x have cx: "?x ≠ 0" "cmod ?x ≤ 1" by simp_all from pCons.prems[rule_format, OF cx(1)] have cth: "cmod (?x*poly ds ?x) = cmod d" by (simp add: eq_diff_eq[symmetric]) from m(2)[rule_format, OF cx(2)] x(1) have th0: "cmod (?x*poly ds ?x) ≤ x*m" by (simp add: norm_mult) from x(2) m(1) have "x * m < cmod d" by (simp add: field_simps) with th0 have "cmod (?x*poly ds ?x) ≠ cmod d" by auto with cth show ?thesis by blast qed qed then show False using pCons.prems False by blast qed then show ?thesis by (rule fundamental_theorem_of_algebra) qed 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 = p * ?w" apply (subst r) apply (subst s) apply (subst kpn) using k oop [of a] apply (subst power_mult_distrib) apply simp apply (subst power_add [symmetric]) apply simp done then show ?thesis unfolding dvd_def by blast next case False with sne dpn s oa have dsn: "degree s < n" apply auto apply (erule ssubst) apply (simp add: degree_mult_eq degree_linear_power) done have "poly r x = 0" if h: "poly s x = 0" for x proof - have xa: "x ≠ a" proof assume "x = a" from h[unfolded this poly_eq_0_iff_dvd] obtain u where u: "s = [:- a, 1:] * u" by (rule dvdE) have "p = [:- a, 1:] ^ (Suc ?op) * u" apply (subst s) apply (subst u) apply (simp only: power_Suc ac_simps) done with ap(2)[unfolded dvd_def] show False by blast qed from h have "poly p x = 0" by (subst s) simp with pq0 have "poly q x = 0" by blast with r xa show ?thesis by auto qed with IH[rule_format, OF dsn, of s r] False have "s dvd (r ^ (degree s))" by blast then obtain u where u: "r ^ (degree s) = s * u" .. then have u': "⋀x. poly s x * poly u x = poly r x ^ degree s" by (simp only: poly_mult[symmetric] poly_power[symmetric]) let ?w = "(u * ([:-a,1:] ^ (n - ?op))) * (r ^ (n - degree s))" from oop[of a] dsn have "q ^ n = p * ?w" apply - apply (subst s) apply (subst r) apply (simp only: power_mult_distrib) apply (subst mult.assoc [where b=s]) apply (subst mult.assoc [where a=u]) apply (subst mult.assoc [where b=u, symmetric]) apply (subst u [symmetric]) apply (simp add: ac_simps power_add [symmetric]) done then show ?thesis unfolding dvd_def by blast qed qed qed then show ?thesis using a order_root pne by blast next case False with fundamental_theorem_of_algebra_alt[of p] obtain c where ccs: "c ≠ 0" "p = pCons c 0" by blast then have pp: "poly p x = c" for x by simp let ?w = "[:1/c:] * (q ^ n)" from ccs have "(q ^ n) = (p * ?w)" by simp then show ?thesis unfolding dvd_def by blast 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 eq: "(∀x. poly p x = (0::complex) ⟶ poly q x = 0) ⟷ q = 0" by (auto simp add: poly_all_0_iff_0) { assume "p dvd (q ^ (degree p))" then obtain r where r: "q ^ (degree p) = p * r" .. from r p have False by simp } with eq p show ?thesis by blast next case dp: 2 then obtain k where k: "p = [:k:]" "k ≠ 0" by (cases p) (simp split: if_splits) then have th1: "∀x. poly p x ≠ 0" by simp from k dp(2) have "q ^ (degree p) = p * [:1/k:]" by simp then have th2: "p dvd (q ^ (degree p))" .. from dp(1) th1 th2 show ?thesis by blast next case dp: 3 have False if dvd: "p dvd (q ^ (Suc n))" and h: "poly p x = 0" "poly q x ≠ 0" for x proof - from dvd obtain u where u: "q ^ (Suc n) = p * u" .. from h have "poly (q ^ (Suc n)) x ≠ 0" by simp with u h(1) show ?thesis by (simp only: poly_mult) simp qed 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 th: "poly p = poly [:poly p 0:]" by auto then have "p = [:poly p 0:]" by (simp add: poly_eq_poly_eq_iff) then have "degree p = degree [:poly p 0:]" by simp then show ?thesis by simp qed show ?lhs if ?rhs proof - from that obtain k where "p = [:k:]" by (cases p) (simp split: if_splits) then show ?thesis unfolding constant_def by auto qed qed 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)" proof - have "pCons 0 q = q * [:0,1:]" by simp then have "q dvd (pCons 0 q)" .. with pq show ?thesis by (rule dvd_trans) qed 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" (is "?lhs ⟷ ?rhs") proof assume ?rhs then have "q = p * 0" by simp then show ?lhs .. next assume l: ?lhs show ?rhs proof (cases "q = 0") case True then show ?thesis by simp next assume q0: "q ≠ 0" from l q0 have "degree p ≤ degree q" by (rule dvd_imp_degree_le) with lgpq show ?thesis by simp qed qed 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" (is "?lhs ⟷ ?rhs") proof from pp' obtain t where t: "p' = p * t" .. show ?rhs if ?lhs proof - from that obtain u where u: "q = p * u" .. have "r = p * (smult a u - t)" using u qrp' [symmetric] t by (simp add: algebra_simps) then show ?thesis .. qed show ?lhs if ?rhs proof - from that obtain u where u: "r = p * u" .. from u [symmetric] t qrp' [symmetric] a0 have "q = p * smult (1/a) (u + t)" by (simp add: algebra_simps) then show ?thesis .. qed qed 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)" proof - have False if "h ≠ 0" "t = 0" and "pCons a (pCons b p) = pCons h t" for h t using l that by simp then have th: "¬ (∃ h t. h ≠ 0 ∧ t = 0 ∧ pCons a (pCons b p) = pCons h t)" by blast from fundamental_theorem_of_algebra_alt[OF th] show ?thesis by auto qed 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)" proof - from l have dp: "degree (pCons a p) = psize p" by (simp add: psize_def) from nullstellensatz_univariate[of "pCons a p" q] l show ?thesis by (metis dp pCons_eq_0_iff) qed 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" proof - from h have "poly (q ^ n) = poly r" by auto then have "(q ^ n) = r" by (simp add: poly_eq_poly_eq_iff) then show "p dvd (q ^ n) ⟷ p dvd r" by simp qed lemma poly_const_conv: fixes x :: "'a::comm_ring_1" shows "poly [:c:] x = y ⟷ c = y" by simp end