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theory Fundamental_Theorem_Algebra(* Author: Amine Chaieb, TU Muenchen *) header{*Fundamental Theorem of Algebra*} theory Fundamental_Theorem_Algebra imports Polynomial Complex begin subsection {* Square root of complex numbers *} definition csqrt :: "complex => complex" where "csqrt z = (if Im z = 0 then if 0 ≤ Re z then Complex (sqrt(Re z)) 0 else Complex 0 (sqrt(- Re z)) else Complex (sqrt((cmod z + Re z) /2)) ((Im z / abs(Im z)) * sqrt((cmod z - Re z) /2)))" lemma csqrt[algebra]: "csqrt z ^ 2 = z" proof- obtain x y where xy: "z = Complex x y" by (cases z) {assume y0: "y = 0" {assume x0: "x ≥ 0" then have ?thesis using y0 xy real_sqrt_pow2[OF x0] by (simp add: csqrt_def power2_eq_square)} moreover {assume "¬ x ≥ 0" hence x0: "- x ≥ 0" by arith then have ?thesis using y0 xy real_sqrt_pow2[OF x0] by (simp add: csqrt_def power2_eq_square) } ultimately have ?thesis by blast} moreover {assume y0: "y≠0" {fix x y let ?z = "Complex x y" from abs_Re_le_cmod[of ?z] have tha: "abs x ≤ cmod ?z" by auto hence "cmod ?z - x ≥ 0" "cmod ?z + x ≥ 0" by arith+ hence "(sqrt (x * x + y * y) + x) / 2 ≥ 0" "(sqrt (x * x + y * y) - x) / 2 ≥ 0" by (simp_all add: power2_eq_square) } note th = this have sq4: "!!x::real. x^2 / 4 = (x / 2) ^ 2" by (simp add: power2_eq_square) from th[of x y] have sq4': "sqrt (((sqrt (x * x + y * y) + x)^2 / 4)) = (sqrt (x * x + y * y) + x) / 2" "sqrt (((sqrt (x * x + y * y) - x)^2 / 4)) = (sqrt (x * x + y * y) - x) / 2" unfolding sq4 by simp_all then have th1: "sqrt ((sqrt (x * x + y * y) + x) * (sqrt (x * x + y * y) + x) / 4) - sqrt ((sqrt (x * x + y * y) - x) * (sqrt (x * x + y * y) - x) / 4) = x" unfolding power2_eq_square by simp have "sqrt 4 = sqrt (2^2)" by simp hence sqrt4: "sqrt 4 = 2" by (simp only: real_sqrt_abs) have th2: "2 *(y * sqrt ((sqrt (x * x + y * y) - x) * (sqrt (x * x + y * y) + x) / 4)) / ¦y¦ = y" using iffD2[OF real_sqrt_pow2_iff sum_power2_ge_zero[of x y]] y0 unfolding power2_eq_square by (simp add: algebra_simps real_sqrt_divide sqrt4) from y0 xy have ?thesis apply (simp add: csqrt_def power2_eq_square) apply (simp add: real_sqrt_sum_squares_mult_ge_zero[of x y] real_sqrt_pow2[OF th(1)[of x y], unfolded power2_eq_square] real_sqrt_pow2[OF th(2)[of x y], unfolded power2_eq_square] real_sqrt_mult[symmetric]) using th1 th2 ..} ultimately show ?thesis by blast qed subsection{* More lemmas about module of complex numbers *} lemma complex_of_real_power: "complex_of_real x ^ n = complex_of_real (x^n)" by (rule of_real_power [symmetric]) lemma real_down2: "(0::real) < d1 ==> 0 < d2 ==> EX e. 0 < e & e < d1 & e < d2" apply (rule exI[where x = "min d1 d2 / 2"]) by (simp add: field_simps min_def) 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 complex polynomials *} lemma poly_bound_exists: shows "∃m. m > 0 ∧ (∀z. cmod z <= r --> cmod (poly p z) ≤ m)" proof(induct p) case 0 thus ?case by (rule exI[where x=1], simp) next case (pCons c cs) from pCons.hyps obtain m where m: "∀z. cmod z ≤ r --> cmod (poly cs z) ≤ m" by blast let ?k = " 1 + cmod c + ¦r * m¦" have kp: "?k > 0" using abs_ge_zero[of "r*m"] norm_ge_zero[of c] by arith {fix z assume H: "cmod z ≤ r" from m H have th: "cmod (poly cs z) ≤ m" by blast from H have rp: "r ≥ 0" using norm_ge_zero[of z] by arith have "cmod (poly (pCons c cs) z) ≤ cmod c + cmod (z* poly cs z)" using norm_triangle_ineq[of c "z* poly cs z"] by simp also have "… ≤ cmod 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 have "cmod (poly (pCons c cs) z) ≤ ?k" .} with kp show ?case by blast qed text{* Offsetting the variable in a polynomial gives another of same degree *} definition "offset_poly p h = poly_rec 0 (λa p q. smult h q + pCons a q) p" lemma offset_poly_0: "offset_poly 0 h = 0" unfolding offset_poly_def by (simp add: poly_rec_0) lemma offset_poly_pCons: "offset_poly (pCons a p) h = smult h (offset_poly p h) + pCons a (offset_poly p h)" unfolding offset_poly_def by (simp add: poly_rec_pCons) 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, 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: "∃ q. psize q = psize p ∧ (∀x. poly q (x::complex) = 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 ex bz obtain x Y where x: "P x" and Y: "!!x. P x ==> x < Y" by blast from ex have thx:"∃x. x ∈ Collect P" by blast from bz have thY: "∃Y. isUb UNIV (Collect P) Y" by(auto simp add: isUb_def isLub_def setge_def setle_def leastP_def Ball_def order_le_less) from reals_complete[OF thx thY] obtain L where L: "isLub UNIV (Collect P) L" by blast from Y[OF x] have xY: "x < Y" . from L have L': "∀x. P x --> x ≤ L" by (auto simp add: isUb_def isLub_def setge_def setle_def leastP_def Ball_def) from Y have Y': "∀x. P x --> x ≤ Y" apply (clarsimp, atomize (full)) by auto from L Y' have "L ≤ Y" by (auto simp add: isUb_def isLub_def setge_def setle_def leastP_def Ball_def) {fix y {fix z assume z: "P z" "y < z" from L' z have "y < L" by auto } moreover {assume yL: "y < L" "∀z. P z --> ¬ y < z" hence nox: "∀z. P z --> y ≥ z" by auto from nox L have "y ≥ L" by (auto simp add: isUb_def isLub_def setge_def setle_def leastP_def Ball_def) with yL(1) have False by arith} ultimately have "(∃x. P x ∧ y < x) <-> y < L" by blast} thus ?thesis by blast 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 + ii) < 1 ∨ cmod (z - ii) < 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) {assume C: "cmod (z + 1) ≥ 1" "cmod (z - 1) ≥ 1" "cmod (z + ii) ≥ 1" "cmod (z - ii) ≥ 1" from C 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) hence "abs (2*x) ≤ 1" "abs (2*y) ≤ 1" by simp_all hence "(abs (2 * x))^2 <= 1^2" "(abs (2 * y)) ^2 <= 1^2" by - (rule power_mono, simp, simp)+ hence th0: "4*x^2 ≤ 1" "4*y^2 ≤ 1" by (simp_all add: power2_abs power_mult_distrib) from add_mono[OF th0] xy have False by simp } thus ?thesis unfolding linorder_not_le[symmetric] by blast qed text{* Hence we can always reduce modulus of @{text "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)" and n: "n ≠ 0" let ?P = "λz n. cmod (1 + b * z ^ n) < 1" {assume e: "even n" hence "∃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 csqrt) hence "∃z. ?P z n" ..} moreover {assume o: "odd n" from b have b': "b^2 ≠ 0" unfolding power2_eq_square by simp have "Im (inverse b) * (Im (inverse b) * ¦Im b * Im b + Re b * Re b¦) + Re (inverse b) * (Re (inverse b) * ¦Im b * Im b + Re b * Re b¦) = ((Re (inverse b))^2 + (Im (inverse b))^2) * ¦Im b * Im b + Re b * Re b¦" by algebra also have "… = cmod (inverse b) ^2 * cmod b ^ 2" apply (simp add: cmod_def) using realpow_two_le_add_order[of "Re b" "Im b"] by (simp add: power2_eq_square) finally have th0: "Im (inverse b) * (Im (inverse b) * ¦Im b * Im b + Re b * Re b¦) + Re (inverse b) * (Re (inverse b) * ¦Im b * Im b + Re b * Re b¦) = 1" apply (simp add: power2_eq_square norm_mult[symmetric] norm_inverse[symmetric]) using right_inverse[OF b'] by (simp add: power2_eq_square[symmetric] power_inverse[symmetric] algebra_simps) have th0: "cmod (complex_of_real (cmod b) / b) = 1" apply (simp add: complex_Re_mult cmod_def power2_eq_square Re_complex_of_real Im_complex_of_real divide_inverse algebra_simps ) by (simp add: real_sqrt_mult[symmetric] th0) from o have "∃m. n = Suc (2*m)" by presburger+ then obtain m where m: "n = Suc (2*m)" by blast from unimodular_reduce_norm[OF th0] o have "∃v. cmod (complex_of_real (cmod b) / b + v^n) < 1" apply (cases "cmod (complex_of_real (cmod b) / b + 1) < 1", rule_tac x="1" in exI, simp) apply (cases "cmod (complex_of_real (cmod b) / b - 1) < 1", rule_tac x="-1" in exI, simp add: diff_def) apply (cases "cmod (complex_of_real (cmod b) / b + ii) < 1") apply (cases "even m", rule_tac x="ii" in exI, simp add: m power_mult) apply (rule_tac x="- ii" in exI, simp add: m power_mult) apply (cases "even m", rule_tac x="- ii" in exI, simp add: m power_mult diff_def) apply (rule_tac x="ii" in exI, simp add: m power_mult diff_def) 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 o, of "cmod b"] have th1: "?w ^ n = v^n / complex_of_real (cmod b)" by (simp add: power_divide complex_of_real_power) have th2:"cmod (complex_of_real (cmod b) / b) = 1" using b by (simp add: norm_divide) hence 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] right_distrib) using b v by (simp add: th2) from mult_less_imp_less_left[OF th4 th3] have "?P ?w n" unfolding th1 . hence "∃z. ?P z n" .. } ultimately show "∃z. ?P z n" by blast 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. subseq f ∧ (∀e >0. ∃N. ∀n ≥ N. cmod (s (f n) - z) < e)" proof- from seq_monosub[of "Re o s"] obtain f g where f: "subseq f" "monoseq (λn. Re (s (f n)))" unfolding o_def by blast from seq_monosub[of "Im o s o f"] obtain g where g: "subseq g" "monoseq (λn. Im (s(f(g n))))" unfolding o_def by blast let ?h = "f o 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 (rule exI[where x= "r + 1"]) using th rp apply simp using f(2) . 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 (rule exI[where x= "r + 1"]) using th rp apply simp using g(2) . from conv1[unfolded convergent_def] obtain x where "LIMSEQ (λn. Re (s (f n))) x" by blast hence x: "∀r>0. ∃n0. ∀n≥n0. ¦ Re (s (f n)) - x ¦ < r" unfolding LIMSEQ_def real_norm_def . from conv2[unfolded convergent_def] obtain y where "LIMSEQ (λn. Im (s (f (g n)))) y" by blast hence y: "∀r>0. ∃n0. ∀n≥n0. ¦ Im (s (f (g n))) - y ¦ < r" unfolding LIMSEQ_def real_norm_def . let ?w = "Complex x y" from f(1) g(1) have hs: "subseq ?h" unfolding subseq_def by auto {fix e assume ep: "e > (0::real)" hence 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 {fix n assume nN12: "n ≥ N1 + N2" hence 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]] have "cmod (s (?h n) - ?w) < e" using metric_bound_lemma[of "s (f (g n))" ?w] by simp } hence "∃N. ∀n≥N. cmod (s (?h n) - ?w) < e" by blast } with hs show ?thesis by blast qed text{* Polynomial is continuous. *} lemma poly_cont: assumes ep: "e > 0" shows "∃d >0. ∀w. 0 < cmod (w - z) ∧ cmod (w - z) < d --> cmod (poly p w - poly p z) < e" proof- obtain q where q: "degree q = degree p" "!!x. poly q x = poly p (z + 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 {fix w note q(2)[of "w - z", simplified]} note th = this show ?thesis unfolding th[symmetric] proof(induct q) case 0 thus ?case using ep by auto next case (pCons c cs) from poly_bound_exists[of 1 "cs"] obtain m where m: "m > 0" "!!z. cmod z ≤ 1 ==> cmod (poly cs z) ≤ m" by blast from ep m(1) have em0: "e/m > 0" by (simp add: field_simps) have one0: "1 > (0::real)" by arith from real_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 real_mult_order) show ?case proof(rule ex_forward[OF real_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" "cmod (w-z) < d" hence d1: "cmod (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: "cmod (w-z) ≤ d" by simp from mult_mono[OF th m(2)[OF d1(1)] d1(2) norm_ge_zero] dme show "cmod (w - z) * cmod (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- {assume "¬ r ≥ 0" hence ?thesis unfolding linorder_not_le apply - apply (rule exI[where x=0]) apply auto apply (subgoal_tac "cmod w < 0") apply simp apply arith done } moreover {assume rp: "r ≥ 0" from rp have "cmod 0 ≤ r ∧ cmod (poly p 0) = - (- cmod (poly p 0))" by simp hence mth1: "∃x z. cmod z ≤ r ∧ cmod (poly p z) = - x" by blast {fix x z assume H: "cmod z ≤ r" "cmod (poly p z) = - x" "¬x < 1" hence "- x < 0 " by arith with H(2) norm_ge_zero[of "poly p z"] have False by simp } 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" {fix y from s[rule_format, of "-y"] have "(∃z x. cmod z ≤ r ∧ -(- cmod (poly p z)) < y) <-> ?m < y" unfolding minus_less_iff[of y ] equation_minus_iff by blast } note s1 = this[unfolded minus_minus] from s1[of ?m] have s1m: "!!z x. cmod z ≤ r ==> cmod (poly p z) ≥ ?m" by auto {fix n::nat from s1[rule_format, of "?m + 1/real (Suc n)"] have "∃z. cmod z ≤ r ∧ cmod (poly p z) < - s + 1 / real (Suc n)" by simp} hence 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: "subseq 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" hence 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 {fix w assume w: "cmod (w - z) < d" have "cmod(poly p w - poly p z) < ?e / 2" using d(2)[rule_format, of w] w e by (cases "w=z", simp_all)} note th1 = this from fz(2)[rule_format, OF 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 {fix a b e2 m :: real have "a < e2 ==> abs(b - m) < e2 ==> 2 * e2 <= abs(b - m) + a ==> False" by arith} note th0 = this have ath: "!!m x e. m <= x ==> x < m + e ==> abs(x - m::real) < e" 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::real) b c m. ¦a - b¦ <= c ==> ¦b - m¦ <= ¦a - m¦ + c" 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 .} hence "?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 } hence ?thesis by blast} ultimately show ?thesis by blast qed lemma "(rcis (sqrt (abs r)) (a/2)) ^ 2 = rcis (abs r) a" unfolding power2_eq_square apply (simp add: rcis_mult) apply (simp add: power2_eq_square[symmetric]) done lemma cispi: "cis pi = -1" unfolding cis_def by simp lemma "(rcis (sqrt (abs r)) ((pi + a)/2)) ^ 2 = rcis (- abs r) a" unfolding power2_eq_square apply (simp add: rcis_mult add_divide_distrib) apply (simp add: power2_eq_square[symmetric] rcis_def cispi cis_mult[symmetric]) done text {* Nonzero polynomial in z goes to infinity as z does. *} lemma poly_infinity: assumes ex: "p ≠ 0" shows "∃r. ∀z. r ≤ cmod z --> d ≤ cmod (poly (pCons a p) z)" using ex proof(induct p arbitrary: a d) case (pCons c cs a d) {assume H: "cs ≠ 0" with pCons.hyps obtain r where r: "∀z. r ≤ cmod z --> d + cmod a ≤ cmod (poly (pCons c cs) z)" by blast let ?r = "1 + ¦r¦" {fix z assume h: "1 + ¦r¦ ≤ cmod z" have r0: "r ≤ cmod z" using h by arith from r[rule_format, OF r0] have th0: "d + cmod a ≤ 1 * cmod(poly (pCons c cs) z)" by arith from h have z1: "cmod 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 ≤ cmod(z * poly (pCons c cs) z) - cmod a" unfolding norm_mult by (simp add: algebra_simps) from complex_mod_triangle_sub[of "z * poly (pCons c cs) z" a] have th2: "cmod(z * poly (pCons c cs) z) - cmod a ≤ cmod (poly (pCons a (pCons c cs)) z)" by (simp add: diff_le_eq algebra_simps) from th1 th2 have "d ≤ cmod (poly (pCons a (pCons c cs)) z)" by arith} hence ?case by blast} moreover {assume cs0: "¬ (cs ≠ 0)" with pCons.prems have c0: "c ≠ 0" by simp from cs0 have cs0': "cs = 0" by simp {fix z assume h: "(¦d¦ + cmod a) / cmod c ≤ cmod z" from c0 have "cmod c > 0" by simp from h c0 have th0: "¦d¦ + cmod a ≤ cmod (z*c)" by (simp add: field_simps norm_mult) have ath: "!!mzh mazh ma. mzh <= mazh + ma ==> abs(d) + ma <= mzh ==> d <= mazh" by arith from complex_mod_triangle_sub[of "z*c" a ] have th1: "cmod (z * c) ≤ cmod (a + z * c) + cmod a" by (simp add: algebra_simps) from ath[OF th1 th0] have "d ≤ cmod (poly (pCons a (pCons c cs)) z)" using cs0' by simp} then have ?case by blast} ultimately show ?case by blast qed simp 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 (pCons c cs) {assume cs0: "cs ≠ 0" from poly_infinity[OF cs0, of "cmod (poly (pCons c cs) 0)" c] obtain r where r: "!!z. r ≤ cmod z ==> cmod (poly (pCons c cs) 0) ≤ cmod (poly (pCons c cs) 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: "!!w. cmod w ≤ ¦r¦ ==> cmod (poly (pCons c cs) v) ≤ cmod (poly (pCons c cs) w)" by blast {fix z assume z: "r ≤ cmod z" from v[of 0] r[OF z] have "cmod (poly (pCons c cs) v) ≤ cmod (poly (pCons c cs) z)" by simp } note v0 = this from v0 v ath[of r] have ?case by blast} moreover {assume cs0: "¬ (cs ≠ 0)" hence th:"cs = 0" by simp from th pCons.hyps have ?case by simp} ultimately show ?case by blast qed simp text{* Constant function (non-syntactic characterization). *} definition "constant f = (∀x y. f x = f y)" lemma nonconstant_length: "¬ (constant (poly p)) ==> psize p ≥ 2" unfolding constant_def psize_def apply (induct p, auto) done lemma poly_replicate_append: "poly (monom 1 n * p) (x::'a::{recpower, 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::{recpower,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 thus ?case by simp next case (pCons c cs) {assume c0: "c = 0" from pCons.hyps pCons.prems c0 have ?case apply auto apply (rule_tac x="k+1" in exI) apply (rule_tac x="a" in exI, clarsimp) apply (rule_tac x="q" in exI) by (auto simp add: power_Suc)} moreover {assume c0: "c≠0" hence ?case apply- apply (rule exI[where x=0]) apply (rule exI[where x=c], clarsimp) apply (rule exI[where x=cs]) apply auto done} ultimately show ?case by blast qed lemma poly_decompose: assumes nc: "~constant(poly p)" shows "∃k a q. a≠(0::'a::{recpower,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 thus ?case by (simp add: constant_def) next case (pCons c cs) {assume C:"∀z. z ≠ 0 --> poly cs z = 0" {fix x y from C have "poly (pCons c cs) x = poly (pCons c cs) y" by (cases "x=0", auto)} with pCons.prems have False by (auto simp add: constant_def)} hence th: "¬ (∀z. z ≠ 0 --> poly cs z = 0)" .. from poly_decompose_lemma[OF th] 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: power_Suc) apply (auto simp add: psize_def split: if_splits) done qed text{* Fundamental theorem of algebral *} lemma fundamental_theorem_of_algebra: assumes nc: "~constant(poly p)" shows "∃z::complex. poly p z = 0" using nc proof(induct n≡ "psize p" arbitrary: p rule: nat_less_induct) fix n fix p :: "complex poly" let ?p = "poly p" assume H: "∀m<n. ∀p. ¬ constant (poly p) --> m = psize p --> (∃(z::complex). poly p z = 0)" and nc: "¬ constant ?p" and n: "n = psize p" let ?ths = "∃z. ?p z = 0" from nonconstant_length[OF nc] have n2: "n≥ 2" by (simp add: n) from poly_minimum_modulus obtain c where c: "∀w. cmod (?p c) ≤ cmod (?p w)" by blast {assume pc: "?p c = 0" hence ?ths by blast} moreover {assume pc0: "?p c ≠ 0" from poly_offset[of p c] obtain q where q: "psize q = psize p" "∀x. poly q x = ?p (c+x)" by blast {assume h: "constant (poly q)" from q(2) have th: "∀x. poly q (x - c) = ?p x" by auto {fix x y 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 have "?p x = ?p y" .} with nc have False unfolding constant_def by blast } hence 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 pc0 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: expand_poly_eq) {assume h: "!!x y. poly ?r x = poly ?r y" {fix x y 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 have "poly q x = poly q y" .} with qnc have False unfolding constant_def by blast} hence rnc: "¬ constant (poly ?r)" unfolding constant_def by blast from qr[rule_format, of 0] a00 have r01: "poly ?r 0 = 1" by auto {fix w have "cmod (poly ?r w) < 1 <-> cmod (poly q w / ?a0) < 1" using qr[rule_format, of w] a00 by (simp add: divide_inverse mult_ac) also have "… <-> cmod (poly q w) < cmod ?a0" using a00 unfolding norm_divide by (simp add: field_simps) finally have "cmod (poly ?r w) < 1 <-> cmod (poly q w) < cmod ?a0" .} note mrmq_eq = this 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 {assume "k + 1 = n" with kas(3) lgqr[symmetric] q(1) n[symmetric] have s0:"s=0" by auto {fix w have "cmod (poly ?r w) = cmod (1 + a * w ^ k)" using kas(4)[rule_format, of w] s0 r01 by (simp add: algebra_simps)} note hth = this [symmetric] from reduce_poly_simple[OF kas(1,2)] have "∃w. cmod (poly ?r w) < 1" unfolding hth by blast} moreover {assume kn: "k+1 ≠ n" from kn kas(3) q(1) n[symmetric] lgqr have k1n: "k + 1 < n" by simp have th01: "¬ constant (poly (pCons 1 (monom a (k - 1))))" unfolding constant_def poly_pCons poly_monom using kas(1) apply simp by (rule exI[where x=0], rule exI[where x=1], simp) 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 H[rule_format, OF k1n th01 th02] 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 real_down2[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)" apply - apply (rule cong[OF refl[of cmod]]) apply assumption done 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) by auto 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: inverse_eq_divide field_simps) with zero_less_power[OF t(1), of k] have th30: "t^k * (t* (cmod w ^ (k + 1) * m)) < t^k * 1" apply - apply (rule mult_strict_left_mono) by simp_all 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_of_real 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 apply (simp only: ) apply auto apply (rule mult_mono, simp_all add: norm_ge_zero)+ apply (simp add: zero_le_mult_iff zero_le_power) done 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 have "∃w. cmod (poly ?r w) < 1" by blast} ultimately have cr0_contr: "∃w. cmod (poly ?r w) < 1" by blast from cr0_contr cq0 q(2) have ?ths unfolding mrmq_eq not_less[symmetric] by auto} ultimately show ?ths by blast 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 (pCons c cs) {assume "c=0" hence ?case by auto} moreover {assume c0: "c≠0" {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 hence "cs = 0" proof(induct cs) case (pCons d ds) {assume "d=0" hence ?case using pCons.prems pCons.hyps by simp} moreover {assume d0: "d≠0" 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 d0 m(1) by (simp add: field_simps) from real_down2[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 have ?case by blast} ultimately show ?case by blast qed simp} then have nc: "¬ constant (poly (pCons c cs))" using pCons.prems c0 by blast from fundamental_theorem_of_algebra[OF nc] have ?case .} ultimately show ?case by blast qed simp subsection{* Nullstellenstatz, 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 prems 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 let ?ths = "p dvd (q ^ n)" {fix a assume a: "poly p a = 0" {assume oa: "order a p ≠ 0" 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] {assume q0: "q = 0" hence ?ths using n0 by (simp add: power_0_left)} moreover {assume q0: "q ≠ 0" 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 {assume ds0: "degree s = 0" from ds0 have "∃k. s = [:k:]" by (cases s, simp split: if_splits) then obtain k where kpn: "s = [:k:]" by blast from sne kpn have k: "k ≠ 0" by simp let ?w = "([:1/k:] * ([:-a,1:] ^ (n - ?op))) * (r ^ n)" from k oop [of a] have "q ^ n = p * ?w" apply - apply (subst r, subst s, subst kpn) apply (subst power_mult_distrib, simp) apply (subst power_add [symmetric], simp) done hence ?ths unfolding dvd_def by blast} moreover {assume ds0: "degree s ≠ 0" from ds0 sne dpn s oa have dsn: "degree s < n" apply auto apply (erule ssubst) apply (simp add: degree_mult_eq degree_linear_power) done {fix x assume h: "poly s x = 0" {assume xa: "x = a" from h[unfolded xa poly_eq_0_iff_dvd] obtain u where u: "s = [:- a, 1:] * u" by (rule dvdE) have "p = [:- a, 1:] ^ (Suc ?op) * u" by (subst s, subst u, simp only: power_Suc mult_ac) with ap(2)[unfolded dvd_def] have False by blast} note xa = this from h have "poly p x = 0" by (subst s, simp) with pq0 have "poly q x = 0" by blast with r xa have "poly r x = 0" by (auto simp add: uminus_add_conv_diff)} note impth = this from IH[rule_format, OF dsn, of s r] impth ds0 have "s dvd (r ^ (degree s))" by blast then obtain u where u: "r ^ (degree s) = s * u" .. hence 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, 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: mult_ac power_add [symmetric]) done hence ?ths unfolding dvd_def by blast} ultimately have ?ths by blast } ultimately have ?ths by blast} then have ?ths using a order_root pne by blast} moreover {assume exa: "¬ (∃a. poly p a = 0)" from fundamental_theorem_of_algebra_alt[of p] exa obtain c where ccs: "c≠0" "p = pCons c 0" by blast then have pp: "!!x. poly p x = c" by simp let ?w = "[:1/c:] * (q ^ n)" from ccs have "(q ^ n) = (p * ?w) " by (simp add: smult_smult) hence ?ths unfolding dvd_def by blast} ultimately show ?ths by blast qed lemma nullstellensatz_univariate: "(∀x. poly p x = (0::complex) --> poly q x = 0) <-> p dvd (q ^ (degree p)) ∨ (p = 0 ∧ q = 0)" proof- {assume pe: "p = 0" hence eq: "(∀x. poly p x = (0::complex) --> poly q x = 0) <-> q = 0" apply auto apply (rule poly_zero [THEN iffD1]) by (rule ext, simp) {assume "p dvd (q ^ (degree p))" then obtain r where r: "q ^ (degree p) = p * r" .. from r pe have False by simp} with eq pe have ?thesis by blast} moreover {assume pe: "p ≠ 0" {assume dp: "degree p = 0" then obtain k where k: "p = [:k:]" "k≠0" using pe by (cases p, simp split: if_splits) hence th1: "∀x. poly p x ≠ 0" by simp from k dp have "q ^ (degree p) = p * [:1/k:]" by (simp add: one_poly_def) hence th2: "p dvd (q ^ (degree p))" .. from th1 th2 pe have ?thesis by blast} moreover {assume dp: "degree p ≠ 0" then obtain n where n: "degree p = Suc n " by (cases "degree p", auto) {assume "p dvd (q ^ (Suc n))" then obtain u where u: "q ^ (Suc n) = p * u" .. {fix x assume h: "poly p x = 0" "poly q x ≠ 0" hence "poly (q ^ (Suc n)) x ≠ 0" by simp hence False using u h(1) by (simp only: poly_mult) simp}} with n nullstellensatz_lemma[of p q "degree p"] dp have ?thesis by auto} ultimately have ?thesis by blast} ultimately show ?thesis by blast 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 assume l: ?lhs from l[unfolded constant_def, rule_format, of _ "0"] have th: "poly p = poly [:poly p 0:]" apply - by (rule ext, simp) then have "p = [:poly p 0:]" by (simp add: poly_eq_iff) then have "degree p = degree [:poly p 0:]" by simp then show ?rhs by simp next assume r: ?rhs then obtain k where "p = [:k:]" by (cases p, simp split: if_splits) then show ?lhs unfolding constant_def by auto qed lemma divides_degree: assumes pq: "p dvd (q:: complex poly)" shows "degree p ≤ degree q ∨ q = 0" apply (cases "q = 0", simp_all) apply (erule dvd_imp_degree_le [OF pq]) done (* Arithmetic operations on multivariate polynomials. *) lemma mpoly_base_conv: "(0::complex) ≡ poly 0 x" "c ≡ poly [:c:] x" "x ≡ poly [:0,1:] x" by simp_all lemma mpoly_norm_conv: "poly [:0:] (x::complex) ≡ poly 0 x" "poly [:poly 0 y:] x ≡ poly 0 x" by simp_all lemma mpoly_sub_conv: "poly p (x::complex) - poly q x ≡ poly p x + -1 * poly q x" by (simp add: diff_def) lemma poly_pad_rule: "poly p x = 0 ==> poly (pCons 0 p) x = (0::complex)" by simp lemma poly_cancel_eq_conv: "p = (0::complex) ==> a ≠ 0 ==> (q = 0) ≡ (a * q - b * p = 0)" apply (atomize (full)) by auto lemma resolve_eq_raw: "poly 0 x ≡ 0" "poly [:c:] x ≡ (c::complex)" by auto lemma resolve_eq_then: "(P ==> (Q ≡ Q1)) ==> (¬P ==> (Q ≡ Q2)) ==> Q ≡ P ∧ Q1 ∨ ¬P∧ Q2" apply (atomize (full)) by blast lemma poly_divides_pad_rule: fixes p q :: "complex poly" assumes pq: "p dvd q" shows "p dvd (pCons (0::complex) 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_pad_const_rule: fixes p q :: "complex poly" assumes pq: "p dvd q" shows "p dvd (smult a q)" proof- have "smult a q = q * [:a:]" by simp then have "q dvd smult a q" .. with pq show ?thesis by (rule dvd_trans) qed lemma poly_divides_conv0: fixes p :: "complex poly" assumes lgpq: "degree q < degree p" and lq:"p ≠ 0" shows "p dvd q ≡ q = 0" (is "?lhs ≡ ?rhs") proof- {assume r: ?rhs hence "q = p * 0" by simp hence ?lhs ..} moreover {assume l: ?lhs {assume q0: "q = 0" hence ?rhs by simp} moreover {assume q0: "q ≠ 0" from l q0 have "degree p ≤ degree q" by (rule dvd_imp_degree_le) with lgpq have ?rhs by simp } ultimately have ?rhs by blast } ultimately show "?lhs ≡ ?rhs" by - (atomize (full), blast) qed lemma poly_divides_conv1: assumes a0: "a≠ (0::complex)" and pp': "(p::complex poly) dvd p'" and qrp': "smult a q - p' ≡ r" shows "p dvd q ≡ p dvd (r::complex poly)" (is "?lhs ≡ ?rhs") proof- { from pp' obtain t where t: "p' = p * t" .. {assume l: ?lhs then obtain u where u: "q = p * u" .. have "r = p * (smult a u - t)" using u qrp' [symmetric] t by (simp add: algebra_simps mult_smult_right) then have ?rhs ..} moreover {assume r: ?rhs then 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 mult_smult_right smult_smult) hence ?lhs ..} ultimately have "?lhs = ?rhs" by blast } thus "?lhs ≡ ?rhs" by - (atomize(full), blast) 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)) ≡ True" proof- {fix h t assume h: "h≠0" "t=0" "pCons a (pCons b p) = pCons h t" with l have False by simp} hence 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 "(∃x. poly (pCons a (pCons b p)) x = (0::complex)) ≡ True" by auto qed lemma basic_cqe_conv_2b: "(∃x. poly p x ≠ (0::complex)) ≡ (p ≠ 0)" proof- have "p = 0 <-> poly p = poly 0" by (simp add: poly_zero) also have "… <-> (¬ (∃x. poly p x ≠ 0))" by (auto intro: ext) finally show "(∃x. poly p x ≠ (0::complex)) ≡ p ≠ 0" by - (atomize (full), blast) qed 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 "(∃x. poly (pCons a p) x = 0 ∧ poly q x ≠ 0) ≡ ¬ ((pCons a p) dvd (q ^ (psize p)))" unfolding dp by - (atomize (full), auto) 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 intro: ext) then have "(q ^ n) = r" by (simp add: poly_eq_iff) thus "p dvd (q ^ n) ≡ p dvd r" by simp qed lemma pmult_Cons_Cons: "(pCons (a::complex) (pCons b p) * q = (smult a q) + (pCons 0 (pCons b p * q)))" by simp lemma elim_neg_conv: "- z ≡ (-1) * (z::complex)" by simp lemma eqT_intr: "PROP P ==> (True ==> PROP P )" "PROP P ==> True" by blast+ lemma negate_negate_rule: "Trueprop P ≡ ¬ P ≡ False" by (atomize (full), auto) lemma complex_entire: "(z::complex) ≠ 0 ∧ w ≠ 0 ≡ z*w ≠ 0" by simp lemma resolve_eq_ne: "(P ≡ True) ≡ (¬P ≡ False)" "(P ≡ False) ≡ (¬P ≡ True)" by (atomize (full)) simp_all lemma cqe_conv1: "poly 0 x = 0 <-> True" by simp lemma cqe_conv2: "(p ==> (q ≡ r)) ≡ ((p ∧ q) ≡ (p ∧ r))" (is "?l ≡ ?r") proof assume "p ==> q ≡ r" thus "p ∧ q ≡ p ∧ r" apply - apply (atomize (full)) by blast next assume "p ∧ q ≡ p ∧ r" "p" thus "q ≡ r" apply - apply (atomize (full)) apply blast done qed lemma poly_const_conv: "poly [:c:] (x::complex) = y <-> c = y" by simp end