wneuper/isa: comparison src/HOL/Library/Fundamental_Theorem

equal deleted inserted replaced

-:5e9248e8e2f8
+:5c4c3a9e9102
 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 \<ge> 0"
 then have ?thesis using y0 xy real_sqrt_pow2[OF x0]
 	by (simp add: csqrt_def power2_eq_square)}
 moreover
 {assume "\<not> x \<ge> 0" hence x0: "- x \<ge> 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\<noteq>0"
 {fix x y
 let ?z = "Complex x y"
 from abs_Re_le_cmod[of ?z] have tha: "abs x \<le> cmod ?z" by auto
 hence "cmod ?z - x \<ge> 0" "cmod ?z + x \<ge> 0" by arith+
 hence "(sqrt (x * x + y * y) + x) / 2 \<ge> 0" "(sqrt (x * x + y * y) - x) / 2 \<ge> 0" by (simp_all add: power2_eq_square) }
 note th = this
 have sq4: "\<And>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)) / \<bar>y\<bar> = 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
 subsection{* Basic lemmas about complex polynomials *}
 lemma poly_bound_exists:
 shows "\<exists>m. m > 0 \<and> (\<forall>z. cmod z <= r \<longrightarrow> cmod (poly p z) \<le> 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: "\<forall>z. cmod z \<le> r \<longrightarrow> cmod (poly cs z) \<le> m"
 by blast
 let ?k = " 1 + cmod c + \<bar>r * m\<bar>"
 lemma real_sup_exists: assumes ex: "\<exists>x. P x" and bz: "\<exists>z. \<forall>x. P x \<longrightarrow> x < z"
 shows "\<exists>(s::real). \<forall>y. (\<exists>x. P x \<and> y < x) \<longleftrightarrow> y < s"
 proof-
 from ex bz obtain x Y where x: "P x" and Y: "\<And>x. P x \<Longrightarrow> x < Y"  by blast
 from ex have thx:"\<exists>x. x \<in> Collect P" by blast
 from bz have thY: "\<exists>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': "\<forall>x. P x \<longrightarrow> x \<le> L" by (auto simp add: isUb_def isLub_def setge_def setle_def leastP_def Ball_def)
 from Y have Y': "\<forall>x. P x \<longrightarrow> x \<le> Y"
 apply (clarsimp, atomize (full)) by auto
 from L Y' have "L \<le> 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" "\<forall>z. P z \<longrightarrow> \<not> y < z"
 hence nox: "\<forall>z. P z \<longrightarrow> y \<ge> z" by auto
 from nox L have "y \<ge> 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 "(\<exists>x. P x \<and> y < x) \<longleftrightarrow> y < L" by blast}
 thus ?thesis by blast
 qed
 from C z xy have "2*x \<le> 1" "2*x \<ge> -1" "2*y \<le> 1" "2*y \<ge> -1"
 by (simp_all add: cmod_def power2_eq_square algebra_simps)
 hence "abs (2*x) \<le> 1" "abs (2*y) \<le> 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 \<le> 1" "4*y^2 \<le> 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
 hence "\<exists>z. ?P z n" ..}
 moreover
 {assume o: "odd n"
 from b have b': "b^2 \<noteq> 0" unfolding power2_eq_square by simp
 have "Im (inverse b) * (Im (inverse b) * \<bar>Im b * Im b + Re b * Re b\<bar>) +
 Re (inverse b) * (Re (inverse b) * \<bar>Im b * Im b + Re b * Re b\<bar>) =
 ((Re (inverse b))^2 + (Im (inverse b))^2) * \<bar>Im b * Im b + Re b * Re b\<bar>" by algebra
 also have "\<dots> = 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) * \<bar>Im b * Im b + Re b * Re b\<bar>) +
 Re (inverse b) * (Re (inverse b) * \<bar>Im b * Im b + Re b * Re b\<bar>) =
 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 "\<exists>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 "\<exists>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 (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) \<ge> 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 "\<exists>z. ?P z n" .. }
 ultimately show "\<exists>z. ?P z n" by blast
 qed
 text{* Bolzano-Weierstrass type property for closed disc in complex plane. *}
 lemma bolzano_weierstrass_complex_disc:
 assumes r: "\<forall>n. cmod (s n) \<le> r"
 shows "\<exists>f z. subseq f \<and> (\<forall>e >0. \<exists>N. \<forall>n \<ge> N. cmod (s (f n) - z) < e)"
 proof-
 from seq_monosub[of "Re o s"]
 obtain f g where f: "subseq f" "monoseq (\<lambda>n. 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 (\<lambda>n. Im (s(f(g n))))" unfolding o_def by blast
 let ?h = "f o g"
 from r[rule_format, of 0] have rp: "r \<ge> 0" using norm_ge_zero[of "s 0"] by arith
 have th:"\<forall>n. r + 1 \<ge> \<bar> Re (s n)\<bar>"
 proof
 fix n
 from abs_Re_le_cmod[of "s n"] r[rule_format, of n]  show "\<bar>Re (s n)\<bar> \<le> r + 1" by arith
 qed
 have conv1: "convergent (\<lambda>n. 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:"\<forall>n. r + 1 \<ge> \<bar> Im (s n)\<bar>"
 proof
 fix n
 from abs_Im_le_cmod[of "s n"] r[rule_format, of n]  show "\<bar>Im (s n)\<bar> \<le> r + 1" by arith
 qed
 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 (\<lambda>n. Re (s (f n))) x"
 by blast
 hence  x: "\<forall>r>0. \<exists>n0. \<forall>n\<ge>n0. \<bar> Re (s (f n)) - x \<bar> < r"
 unfolding LIMSEQ_def real_norm_def .
 from conv2[unfolded convergent_def] obtain y where "LIMSEQ (\<lambda>n. Im (s (f (g n)))) y"
 by blast
 hence  y: "\<forall>r>0. \<exists>n0. \<forall>n\<ge>n0. \<bar> Im (s (f (g n))) - y \<bar> < 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: "\<forall>n\<ge>N1. \<bar>Re (s (f n)) - x\<bar> < e / 2" and N2: "\<forall>n\<ge>N2. \<bar>Im (s (f (g n))) - y\<bar> < e / 2" by blast
 {fix n assume nN12: "n \<ge> N1 + N2"
 hence nN1: "g n \<ge> N1" and nN2: "n \<ge> 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 "\<exists>N. \<forall>n\<ge>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 "\<exists>d >0. \<forall>w. 0 < cmod (w - z) \<and> cmod (w - z) < d \<longrightarrow> cmod (poly p w - poly p z) < e"
 proof-
 obtain q where q: "degree q = degree p" "\<And>x. poly q x = poly p (z + x)"
 proof
 show "degree (offset_poly p z) = degree p"
 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" "\<And>z. cmod z \<le> 1 \<Longrightarrow> cmod (poly cs z) \<le> 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\<noteq>z" "cmod (w-z) < d"
 	hence d1: "cmod (w-z) \<le> 1" "d \<ge> 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) \<le> 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:
 "\<exists>z. \<forall>w. cmod w \<le> r \<longrightarrow> cmod (poly p z) \<le> cmod (poly p w)"
 proof-
 {assume "\<not> r \<ge> 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 \<ge> 0"
 from rp have "cmod 0 \<le> r \<and> cmod (poly p 0) = - (- cmod (poly p 0))" by simp
 hence mth1: "\<exists>x z. cmod z \<le> r \<and> cmod (poly p z) = - x"  by blast
 {fix x z
 assume H: "cmod z \<le> r" "cmod (poly p z) = - x" "\<not>x < 1"
 hence "- x < 0 " by arith
 with H(2) norm_ge_zero[of "poly p z"]  have False by simp }
 then have mth2: "\<exists>z. \<forall>x. (\<exists>z. cmod z \<le> r \<and> cmod (poly p z) = - x) \<longrightarrow> x < z" by blast
 from real_sup_exists[OF mth1 mth2] obtain s where
 s: "\<forall>y. (\<exists>x. (\<exists>z. cmod z \<le> r \<and> cmod (poly p z) = - x) \<and> y < x) \<longleftrightarrow>(y < s)" by blast
 let ?m = "-s"
 {fix y
 from s[rule_format, of "-y"] have
 "(\<exists>z x. cmod z \<le> r \<and> -(- cmod (poly p z)) < y) \<longleftrightarrow> ?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: "\<And>z x. cmod z \<le> r \<Longrightarrow> cmod (poly p z) \<ge> ?m"
 by auto
 {fix n::nat
 from s1[rule_format, of "?m + 1/real (Suc n)"]
 have "\<exists>z. cmod z \<le> r \<and> cmod (poly p z) < - s + 1 / real (Suc n)"
 	by simp}
 hence th: "\<forall>n. \<exists>z. cmod z \<le> r \<and> cmod (poly p z) < - s + 1 / real (Suc n)" ..
 from choice[OF th] obtain g where
 g: "\<forall>n. cmod (g n) \<le> r" "\<forall>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" "\<forall>e>0. \<exists>N. \<forall>n\<ge>N. cmod (g (f n) - z) < e"
 by blast
 {fix w
 assume wr: "cmod w \<le> r"
 let ?e = "\<bar>cmod (poly p z) - ?m\<bar>"
 {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" "\<forall>w. 0<cmod (w - z)\<and> cmod(w - z) < d \<longrightarrow> 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: "\<forall>n\<ge>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 \<Longrightarrow> abs(b - m) < e2 \<Longrightarrow> 2 * e2 <= abs(b - m) + a
 ==> False" by arith}
 note th0 = this
 have ath:
 	"\<And>m x e. m <= x \<Longrightarrow>  x < m + e ==> abs(x - m::real) < e" by arith
 from s1m[OF g(1)[rule_format]]
 have th31: "?m \<le> cmod(poly p (g (f (N1 + N2))))" .
 from seq_suble[OF fz(1), of "N1+N2"]
 have th00: "real (Suc (N1+N2)) \<le> real (Suc (f (N1+N2)))" by simp
 have th000: "0 \<le> (1::real)" "(1::real) \<le> 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))) \<le> ?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:"\<bar>cmod(poly p (g (f (N1 + N2)))) - ?m\<bar>< ?e/2" by arith
 have ath2: "\<And>(a::real) b c m. \<bar>a - b\<bar> <= c ==> \<bar>b - m\<bar> <= \<bar>a - m\<bar> + c"
 	by arith
 have th22: "\<bar>cmod (poly p (g (f (N1 + N2)))) - cmod (poly p z)\<bar>
 \<le> 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) \<le> \<bar>cmod(poly p (g (f (N1 + N2)))) - ?m\<bar> + 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) \<le> cmod (poly p w)" by simp }
 hence ?thesis by blast}
 ultimately show ?thesis by blast
 qed
 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
 lemma poly_infinity:
 assumes ex: "p \<noteq> 0"
 shows "\<exists>r. \<forall>z. r \<le> cmod z \<longrightarrow> d \<le> cmod (poly (pCons a p) z)"
 using ex
 proof(induct p arbitrary: a d)
 case (pCons c cs a d)
 {assume H: "cs \<noteq> 0"
 with pCons.hyps obtain r where r: "\<forall>z. r \<le> cmod z \<longrightarrow> d + cmod a \<le> cmod (poly (pCons c cs) z)" by blast
 let ?r = "1 + \<bar>r\<bar>"
 {fix z assume h: "1 + \<bar>r\<bar> \<le> cmod z"
 have r0: "r \<le> cmod z" using h by arith
 from h have z1: "cmod z \<ge> 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 \<le> 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 \<le> cmod (poly (pCons a (pCons c cs)) z)"
 	by (simp add: diff_le_eq algebra_simps)
 from th1 th2 have "d \<le> cmod (poly (pCons a (pCons c cs)) z)"  by arith}
 hence ?case by blast}
 moreover
 {assume cs0: "\<not> (cs \<noteq> 0)"
 with pCons.prems have c0: "c \<noteq> 0" by simp
 from cs0 have cs0': "cs = 0" by simp
 {fix z
 assume h: "(\<bar>d\<bar> + cmod a) / cmod c \<le> cmod z"
 from c0 have "cmod c > 0" by simp
 from h c0 have th0: "\<bar>d\<bar> + cmod a \<le> cmod (z*c)"
 	by (simp add: field_simps norm_mult)
 have ath: "\<And>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) \<le> cmod (a + z * c) + cmod a"
 	by (simp add: algebra_simps)
 from ath[OF th1 th0] have "d \<le> 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:
 "\<exists>z.\<forall>w. cmod (poly p z) \<le> cmod (poly p w)"
 proof(induct p)
 case (pCons c cs)
 {assume cs0: "cs \<noteq> 0"
 from poly_infinity[OF cs0, of "cmod (poly (pCons c cs) 0)" c]
 obtain r where r: "\<And>z. r \<le> cmod z \<Longrightarrow> cmod (poly (pCons c cs) 0) \<le> cmod (poly (pCons c cs) z)" by blast
 have ath: "\<And>z r. r \<le> cmod z \<or> cmod z \<le> \<bar>r\<bar>" by arith
 from poly_minimum_modulus_disc[of "\<bar>r\<bar>" "pCons c cs"]
 obtain v where v: "\<And>w. cmod w \<le> \<bar>r\<bar> \<Longrightarrow> cmod (poly (pCons c cs) v) \<le> cmod (poly (pCons c cs) w)" by blast
 {fix z assume z: "r \<le> cmod z"
 from v[of 0] r[OF z]
 have "cmod (poly (pCons c cs) v) \<le> cmod (poly (pCons c cs) z)"
 	by simp }
 note v0 = this
 from v0 v ath[of r] have ?case by blast}
 moreover
 lemma nonconstant_length: "\<not> (constant (poly p)) \<Longrightarrow> psize p \<ge> 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: "\<not>(\<forall>z. z\<noteq>0 \<longrightarrow> poly p z = (0::'a::{recpower,idom}))"
 shows "\<exists>k a q. a\<noteq>0 \<and> Suc (psize q + k) = psize p \<and>
 (\<forall>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
 qed
 lemma poly_decompose:
 assumes nc: "~constant(poly p)"
 shows "\<exists>k a q. a\<noteq>(0::'a::{recpower,idom}) \<and> k\<noteq>0 \<and>
 psize q + k + 1 = psize p \<and>
 (\<forall>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:"\<forall>z. z \<noteq> 0 \<longrightarrow> 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: "\<not> (\<forall>z. z \<noteq> 0 \<longrightarrow> 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)
 let ?p = "poly p"
 assume H: "\<forall>m<n. \<forall>p. \<not> constant (poly p) \<longrightarrow> m = psize p \<longrightarrow> (\<exists>(z::complex). poly p z = 0)" and nc: "\<not> constant ?p" and n: "n = psize p"
 let ?ths = "\<exists>z. ?p z = 0"
 from nonconstant_length[OF nc] have n2: "n\<ge> 2" by (simp add: n)
 from poly_minimum_modulus obtain c where
 c: "\<forall>w. cmod (?p c) \<le> cmod (?p w)" by blast
 {assume pc: "?p c = 0" hence ?ths by blast}
 moreover
 {assume pc0: "?p c \<noteq> 0"
 from poly_offset[of p c] obtain q where
 q: "psize q = psize p" "\<forall>x. poly q x = ?p (c+x)" by blast
 {assume h: "constant (poly q)"
 from q(2) have th: "\<forall>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 "\<dots> = poly q (y - c)"
 	  using h unfolding constant_def by blast
 	also have "\<dots> = ?p y" using th by auto
 	finally have "?p x = ?p y" .}
 with nc have False unfolding constant_def by blast }
 hence qnc: "\<not> constant (poly q)" by blast
 from q(2) have pqc0: "?p c = poly q 0" by simp
 from c pqc0 have cq0: "\<forall>w. cmod (poly q 0) \<le> cmod (?p w)" by simp
 let ?a0 = "poly q 0"
 from pc0 pqc0 have a00: "?a0 \<noteq> 0" by simp
 from a00
 have qr: "\<forall>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: "\<And>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 "\<dots> = poly ?r y * ?a0" using h by simp
 	also have "\<dots> = 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: "\<not> 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 \<longleftrightarrow> cmod (poly q w / ?a0) < 1"
 	using qr[rule_format, of w] a00 by (simp add: divide_inverse mult_ac)
 also have "\<dots> \<longleftrightarrow> cmod (poly q w) < cmod ?a0"
 	using a00 unfolding norm_divide by (simp add: field_simps)
 finally have "cmod (poly ?r w) < 1 \<longleftrightarrow> cmod (poly q w) < cmod ?a0" .}
 note mrmq_eq = this
 from poly_decompose[OF rnc] obtain k a s where
 kas: "a\<noteq>0" "k\<noteq>0" "psize s + k + 1 = psize ?r"
 "\<forall>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 "\<exists>w. cmod (poly ?r w) < 1" unfolding hth by blast}
 moreover
 {assume kn: "k+1 \<noteq> n"
 from kn kas(3) q(1) n[symmetric] lgqr have k1n: "k + 1 < n" by simp
 have th01: "\<not> 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" "\<forall>z. cmod z \<le> cmod w \<longrightarrow> cmod (poly s z) \<le> m" by blast
 have w0: "w\<noteq>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 "\<dots> = 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)) \<le> \<bar>1 - t^k\<bar> + cmod (?w^k * ?w * poly s ?w)" unfolding norm_of_real by simp
 have ath: "\<And>x (t::real). 0\<le> x \<Longrightarrow> x < t \<Longrightarrow> t\<le>1 \<Longrightarrow> \<bar>1 - t\<bar> + x < 1" by arith
 have "t *cmod w \<le> 1 * cmod w" apply (rule mult_mono) using t(1,2) by auto
 then have tw: "cmod ?w \<le> 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) \<le> 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 \<le> 1"
 	by auto
 from ath[OF norm_ge_zero[of "?w^k * ?w * poly s ?w"] th120 th121]
 have th12: "\<bar>1 - t^k\<bar> + 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 "\<exists>w. cmod (poly ?r w) < 1" by blast}
 ultimately have cr0_contr: "\<exists>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
 case (pCons c cs)
 {assume "c=0" hence ?case by auto}
 moreover
 {assume c0: "c\<noteq>0"
 {assume nc: "constant (poly (pCons c cs))"
 from nc[unfolded constant_def, rule_format, of 0]
 have "\<forall>w. w \<noteq> 0 \<longrightarrow> 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\<noteq>0"
 	    from poly_bound_exists[of 1 ds] obtain m where
 	      m: "m > 0" "\<forall>z. \<forall>z. cmod z \<le> 1 \<longrightarrow> cmod (poly ds z) \<le> 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 \<noteq> 0"  "cmod ?x \<le> 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])
 	    have th0: "cmod (?x*poly ds ?x) \<le> 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) \<noteq> cmod d" by auto
 	    with cth  have ?case by blast}
 	  ultimately show ?case by blast
 	qed simp}
 then have nc: "\<not> 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 *}
 proof(induct n arbitrary: p q rule: nat_less_induct)
 fix n::nat fix p q :: "complex poly"
 assume IH: "\<forall>m<n. \<forall>p q.
 (\<forall>x. poly p x = (0::complex) \<longrightarrow> poly q x = 0) \<longrightarrow>
 degree p = m \<longrightarrow> m \<noteq> 0 \<longrightarrow> p dvd (q ^ m)"
 and pq0: "\<forall>x. poly p x = 0 \<longrightarrow> poly q x = 0"
 and dpn: "degree p = n" and n0: "n \<noteq> 0"
 from dpn n0 have pne: "p \<noteq> 0" by auto
 let ?ths = "p dvd (q ^ n)"
 {fix a assume a: "poly p a = 0"
 {assume oa: "order a p \<noteq> 0"
 let ?op = "order a p"
 from pne have ap: "([:- a, 1:] ^ ?op) dvd p"
 	"\<not> [:- 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
 then have ?ths using a order_root pne by blast}
 moreover
 {assume exa: "\<not> (\<exists>a. poly p a = 0)"
 from fundamental_theorem_of_algebra_alt[of p] exa obtain c where
 ccs: "c\<noteq>0" "p = pCons c 0" by blast
 then have pp: "\<And>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:
 "(\<forall>x. poly p x = (0::complex) \<longrightarrow> poly q x = 0) \<longleftrightarrow>
 p dvd (q ^ (degree p)) \<or> (p = 0 \<and> q = 0)"
 proof-
 {assume pe: "p = 0"
 hence eq: "(\<forall>x. poly p x = (0::complex) \<longrightarrow> poly q x = 0) \<longleftrightarrow> q = 0"
 apply 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 \<noteq> 0"
 	  hence "poly (q ^ (Suc n)) x \<noteq> 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
 apply (erule dvd_imp_degree_le [OF pq])
 done
 (* Arithmetic operations on multivariate polynomials.                        *)
 lemma mpoly_base_conv:
 "(0::complex) \<equiv> poly 0 x" "c \<equiv> poly [:c:] x" "x \<equiv> poly [:0,1:] x" by simp_all
 lemma mpoly_norm_conv:
 "poly [:0:] (x::complex) \<equiv> poly 0 x" "poly [:poly 0 y:] x \<equiv> poly 0 x" by simp_all
 lemma mpoly_sub_conv:
 "poly p (x::complex) - poly q x \<equiv> 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) \<Longrightarrow> a \<noteq> 0 \<Longrightarrow> (q = 0) \<equiv> (a * q - b * p = 0)" apply (atomize (full)) by auto
 lemma resolve_eq_raw:  "poly 0 x \<equiv> 0" "poly [:c:] x \<equiv> (c::complex)" by auto
 lemma  resolve_eq_then: "(P \<Longrightarrow> (Q \<equiv> Q1)) \<Longrightarrow> (\<not>P \<Longrightarrow> (Q \<equiv> Q2))
 \<Longrightarrow> Q \<equiv> P \<and> Q1 \<or> \<not>P\<and> 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 \<noteq> 0"
 shows "p dvd q \<equiv> q = 0" (is "?lhs \<equiv> ?rhs")
 proof-
 {assume r: ?rhs
 hence "q = p * 0" by simp
 hence ?lhs ..}
 moreover
 {assume l: ?lhs
 {assume q0: "q = 0"
 {assume q0: "q \<noteq> 0"
 from l q0 have "degree p \<le> degree q"
 by (rule dvd_imp_degree_le)
 with lgpq have ?rhs by simp }
 ultimately have ?rhs by blast }
 ultimately show "?lhs \<equiv> ?rhs" by - (atomize (full), blast)
 qed
 lemma poly_divides_conv1:
 assumes a0: "a\<noteq> (0::complex)" and pp': "(p::complex poly) dvd p'"
 and qrp': "smult a q - p' \<equiv> r"
 shows "p dvd q \<equiv> p dvd (r::complex poly)" (is "?lhs \<equiv> ?rhs")
 proof-
 {
 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 \<equiv> ?rhs"  by - (atomize(full), blast)
 qed
 lemma basic_cqe_conv1:
 "(\<exists>x. poly p x = 0 \<and> poly 0 x \<noteq> 0) \<equiv> False"
 "(\<exists>x. poly 0 x \<noteq> 0) \<equiv> False"
 "(\<exists>x. poly [:c:] x \<noteq> 0) \<equiv> c\<noteq>0"
 "(\<exists>x. poly 0 x = 0) \<equiv> True"
 "(\<exists>x. poly [:c:] x = 0) \<equiv> c = 0" by simp_all
 lemma basic_cqe_conv2:
 assumes l:"p \<noteq> 0"
 shows "(\<exists>x. poly (pCons a (pCons b p)) x = (0::complex)) \<equiv> True"
 proof-
 {fix h t
 assume h: "h\<noteq>0" "t=0"  "pCons a (pCons b p) = pCons h t"
 with l have False by simp}
 hence th: "\<not> (\<exists> h t. h\<noteq>0 \<and> t=0 \<and> pCons a (pCons b p) = pCons h t)"
 by blast
 from fundamental_theorem_of_algebra_alt[OF th]
 show "(\<exists>x. poly (pCons a (pCons b p)) x = (0::complex)) \<equiv> True" by auto
 qed
 lemma  basic_cqe_conv_2b: "(\<exists>x. poly p x \<noteq> (0::complex)) \<equiv> (p \<noteq> 0)"
 proof-
 by - (atomize (full), blast)
 qed
 lemma basic_cqe_conv3:
 fixes p q :: "complex poly"
 assumes l: "p \<noteq> 0"
 shows "(\<exists>x. poly (pCons a p) x = 0 \<and> poly q x \<noteq> 0) \<equiv> \<not> ((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 "(\<exists>x. poly (pCons a p) x = 0 \<and> poly q x \<noteq> 0) \<equiv> \<not> ((pCons a p) dvd (q ^ (psize p)))"
 lemma elim_neg_conv: "- z \<equiv> (-1) * (z::complex)" by simp
 lemma eqT_intr: "PROP P \<Longrightarrow> (True \<Longrightarrow> PROP P )" "PROP P \<Longrightarrow> True" by blast+
 lemma negate_negate_rule: "Trueprop P \<equiv> \<not> P \<equiv> False" by (atomize (full), auto)
 lemma complex_entire: "(z::complex) \<noteq> 0 \<and> w \<noteq> 0 \<equiv> z*w \<noteq> 0" by simp
 lemma resolve_eq_ne: "(P \<equiv> True) \<equiv> (\<not>P \<equiv> False)" "(P \<equiv> False) \<equiv> (\<not>P \<equiv> True)"
 by (atomize (full)) simp_all
 lemma cqe_conv1: "poly 0 x = 0 \<longleftrightarrow> True"  by simp
 lemma cqe_conv2: "(p \<Longrightarrow> (q \<equiv> r)) \<equiv> ((p \<and> q) \<equiv> (p \<and> r))"  (is "?l \<equiv> ?r")
 proof
 assume "p \<Longrightarrow> q \<equiv> r" thus "p \<and> q \<equiv> p \<and> r" apply - apply (atomize (full)) by blast

changeset 30474	5c4c3a9e9102
parent 30242	aea5d7fa7ef5
child 31021	53642251a04f