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theory Fundamental_Theorem_Algebra(* Title: Fundamental_Theorem_Algebra.thy ID: $Id: Fundamental_Theorem_Algebra.thy,v 1.3 2008/02/27 14:35:42 chaieb Exp $ Author: Amine Chaieb *) header{*Fundamental Theorem of Algebra*} theory Fundamental_Theorem_Algebra imports Univ_Poly Dense_Linear_Order Complex begin section {* 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: "csqrt z ^ 2 = z" proof- obtain x y where xy: "z = Complex x y" by (cases z, simp_all) {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 (cases "x ≥ 0", 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: ring_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 section{* More lemmas about module of complex numbers *} lemma complex_of_real_power: "complex_of_real x ^ n = complex_of_real (x^n)" by (induct n, auto) lemma cmod_pos: "cmod z ≥ 0" by simp lemma complex_mod_triangle_ineq: "cmod (z + w) ≤ cmod z + cmod w" using complex_mod_triangle_ineq2[of z w] by (simp add: ring_simps) lemma cmod_mult: "cmod (z*w) = cmod z * cmod w" proof- from rcis_Ex[of z] rcis_Ex[of w] obtain rz az rw aw where z: "z = rcis rz az" and w: "w = rcis rw aw" by blast thus ?thesis by (simp add: rcis_mult abs_mult) qed lemma cmod_divide: "cmod (z/w) = cmod z / cmod w" proof- from rcis_Ex[of z] rcis_Ex[of w] obtain rz az rw aw where z: "z = rcis rz az" and w: "w = rcis rw aw" by blast thus ?thesis by (simp add: rcis_divide) qed lemma cmod_inverse: "cmod (inverse z) = inverse (cmod z)" using cmod_divide[of 1 z] by (simp add: inverse_eq_divide) lemma cmod_uminus: "cmod (- z) = cmod z" unfolding cmod_def by simp lemma cmod_abs_norm: "¦cmod w - cmod z¦ ≤ cmod (w - z)" proof- have ath: "!!(a::real) b x. a - b <= x ==> b - a <= x ==> abs(a - b) <= x" by arith from complex_mod_triangle_ineq2[of "w - z" z] have th1: "cmod w - cmod z ≤ cmod (w - z)" by simp from complex_mod_triangle_ineq2[of "- (w - z)" "w"] have th2: "cmod z - cmod w ≤ cmod (w - z)" using cmod_uminus [of "w - z"] by simp from ath[OF th1 th2] show ?thesis . qed lemma cmod_power: "cmod (z ^n) = cmod z ^ n" by (induct n, auto simp add: cmod_mult) lemma real_down2: "(0::real) < d1 ==> 0 < d2 ==> EX e. 0 < e & e < d1 & e < d2" apply ferrack apply arith done lemma cmod_complex_of_real: "cmod (complex_of_real x) = ¦x¦" unfolding cmod_def by auto 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 section{* 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 Nil thus ?case by (rule exI[where x=1], simp) next case (Cons c cs) from Cons.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"] cmod_pos[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 cmod_pos[of z] by arith have "cmod (poly (c # cs) z) ≤ cmod c + cmod (z* poly cs z)" using complex_mod_triangle_ineq[of c "z* poly cs z"] by simp also have "… ≤ cmod c + r*m" using mult_mono[OF H th rp cmod_pos[of "poly cs z"]] by (simp add: cmod_mult) also have "… ≤ ?k" by simp finally have "cmod (poly (c # cs) z) ≤ ?k" .} with kp show ?case by blast qed text{* Offsetting the variable in a polynomial gives another of same degree *} (* FIXME : Lemma holds also in locale --- fix it later *) lemma poly_offset_lemma: shows "∃b q. (length q = length p) ∧ (∀x. poly (b#q) (x::complex) = (a + x) * poly p x)" proof(induct p) case Nil thus ?case by simp next case (Cons c cs) from Cons.hyps obtain b q where bq: "length q = length cs" "∀x. poly (b # q) x = (a + x) * poly cs x" by blast let ?b = "a*c" let ?q = "(b+c)#q" have lg: "length ?q = length (c#cs)" using bq(1) by simp {fix x from bq(2)[rule_format, of x] have "x*poly (b # q) x = x*((a + x) * poly cs x)" by simp hence "poly (?b# ?q) x = (a + x) * poly (c # cs) x" by (simp add: ring_simps)} with lg show ?case by blast qed (* FIXME : This one too*) lemma poly_offset: "∃ q. length q = length p ∧ (∀x. poly q (x::complex) = poly p (a + x))" proof (induct p) case Nil thus ?case by simp next case (Cons c cs) from Cons.hyps obtain q where q: "length q = length cs" "∀x. poly q x = poly cs (a + x)" by blast from poly_offset_lemma[of q a] obtain b p where bp: "length p = length q" "∀x. poly (b # p) x = (a + x) * poly q x" by blast thus ?case using q bp by - (rule exI[where x="(c + b)#p"], simp) 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 section{* Some theorems about Sequences*} text{* Given a binary function @{text "f:: nat => 'a => 'a"}, its values are uniquely determined by a function g *} lemma num_Axiom: "EX! g. g 0 = e ∧ (∀n. g (Suc n) = f n (g n))" unfolding Ex1_def apply (rule_tac x="nat_rec e f" in exI) apply (rule conjI)+ apply (rule def_nat_rec_0, simp) apply (rule allI, rule def_nat_rec_Suc, simp) apply (rule allI, rule impI, rule ext) apply (erule conjE) apply (induct_tac x) apply (simp add: nat_rec_0) apply (erule_tac x="n" in allE) apply (simp) done text{* An equivalent formulation of monotony -- Not used here, but might be useful *} lemma mono_Suc: "mono f = (∀n. (f n :: 'a :: order) ≤ f (Suc n))" unfolding mono_def proof auto fix A B :: nat assume H: "∀n. f n ≤ f (Suc n)" "A ≤ B" hence "∃k. B = A + k" apply - apply (thin_tac "∀n. f n ≤ f (Suc n)") by presburger then obtain k where k: "B = A + k" by blast {fix a k have "f a ≤ f (a + k)" proof (induct k) case 0 thus ?case by simp next case (Suc k) from Suc.hyps H(1)[rule_format, of "a + k"] show ?case by simp qed} with k show "f A ≤ f B" by blast qed text{* for any sequence, there is a mootonic subsequence *} lemma seq_monosub: "∃f. subseq f ∧ monoseq (λ n. (s (f n)))" proof- {assume H: "∀n. ∃p >n. ∀ m≥p. s m ≤ s p" let ?P = "λ p n. p > n ∧ (∀m ≥ p. s m ≤ s p)" from num_Axiom[of "SOME p. ?P p 0" "λp n. SOME p. ?P p n"] obtain f where f: "f 0 = (SOME p. ?P p 0)" "∀n. f (Suc n) = (SOME p. ?P p (f n))" by blast have "?P (f 0) 0" unfolding f(1) some_eq_ex[of "λp. ?P p 0"] using H apply - apply (erule allE[where x=0], erule exE, rule_tac x="p" in exI) unfolding order_le_less by blast hence f0: "f 0 > 0" "∀m ≥ f 0. s m ≤ s (f 0)" by blast+ {fix n have "?P (f (Suc n)) (f n)" unfolding f(2)[rule_format, of n] some_eq_ex[of "λp. ?P p (f n)"] using H apply - apply (erule allE[where x="f n"], erule exE, rule_tac x="p" in exI) unfolding order_le_less by blast hence "f (Suc n) > f n" "∀m ≥ f (Suc n). s m ≤ s (f (Suc n))" by blast+} note fSuc = this {fix p q assume pq: "p ≥ f q" have "s p ≤ s(f(q))" using f0(2)[rule_format, of p] pq fSuc by (cases q, simp_all) } note pqth = this {fix q have "f (Suc q) > f q" apply (induct q) using f0(1) fSuc(1)[of 0] apply simp by (rule fSuc(1))} note fss = this from fss have th1: "subseq f" unfolding subseq_Suc_iff .. {fix a b have "f a ≤ f (a + b)" proof(induct b) case 0 thus ?case by simp next case (Suc b) from fSuc(1)[of "a + b"] Suc.hyps show ?case by simp qed} note fmon0 = this have "monoseq (λn. s (f n))" proof- {fix n have "s (f n) ≥ s (f (Suc n))" proof(cases n) case 0 assume n0: "n = 0" from fSuc(1)[of 0] have th0: "f 0 ≤ f (Suc 0)" by simp from f0(2)[rule_format, OF th0] show ?thesis using n0 by simp next case (Suc m) assume m: "n = Suc m" from fSuc(1)[of n] m have th0: "f (Suc m) ≤ f (Suc (Suc m))" by simp from m fSuc(2)[rule_format, OF th0] show ?thesis by simp qed} thus "monoseq (λn. s (f n))" unfolding monoseq_Suc by blast qed with th1 have ?thesis by blast} moreover {fix N assume N: "∀p >N. ∃ m≥p. s m > s p" {fix p assume p: "p ≥ Suc N" hence pN: "p > N" by arith with N obtain m where m: "m ≥ p" "s m > s p" by blast have "m ≠ p" using m(2) by auto with m have "∃m>p. s p < s m" by - (rule exI[where x=m], auto)} note th0 = this let ?P = "λm x. m > x ∧ s x < s m" from num_Axiom[of "SOME x. ?P x (Suc N)" "λm x. SOME y. ?P y x"] obtain f where f: "f 0 = (SOME x. ?P x (Suc N))" "∀n. f (Suc n) = (SOME m. ?P m (f n))" by blast have "?P (f 0) (Suc N)" unfolding f(1) some_eq_ex[of "λp. ?P p (Suc N)"] using N apply - apply (erule allE[where x="Suc N"], clarsimp) apply (rule_tac x="m" in exI) apply auto apply (subgoal_tac "Suc N ≠ m") apply simp apply (rule ccontr, simp) done hence f0: "f 0 > Suc N" "s (Suc N) < s (f 0)" by blast+ {fix n have "f n > N ∧ ?P (f (Suc n)) (f n)" unfolding f(2)[rule_format, of n] some_eq_ex[of "λp. ?P p (f n)"] proof (induct n) case 0 thus ?case using f0 N apply auto apply (erule allE[where x="f 0"], clarsimp) apply (rule_tac x="m" in exI, simp) by (subgoal_tac "f 0 ≠ m", auto) next case (Suc n) from Suc.hyps have Nfn: "N < f n" by blast from Suc.hyps obtain m where m: "m > f n" "s (f n) < s m" by blast with Nfn have mN: "m > N" by arith note key = Suc.hyps[unfolded some_eq_ex[of "λp. ?P p (f n)", symmetric] f(2)[rule_format, of n, symmetric]] from key have th0: "f (Suc n) > N" by simp from N[rule_format, OF th0] obtain m' where m': "m' ≥ f (Suc n)" "s (f (Suc n)) < s m'" by blast have "m' ≠ f (Suc (n))" apply (rule ccontr) using m'(2) by auto hence "m' > f (Suc n)" using m'(1) by simp with key m'(2) show ?case by auto qed} note fSuc = this {fix n have "f n ≥ Suc N ∧ f(Suc n) > f n ∧ s(f n) < s(f(Suc n))" using fSuc[of n] by auto hence "f n ≥ Suc N" "f(Suc n) > f n" "s(f n) < s(f(Suc n))" by blast+} note thf = this have sqf: "subseq f" unfolding subseq_Suc_iff using thf by simp have "monoseq (λn. s (f n))" unfolding monoseq_Suc using thf apply - apply (rule disjI1) apply auto apply (rule order_less_imp_le) apply blast done then have ?thesis using sqf by blast} ultimately show ?thesis unfolding linorder_not_less[symmetric] by blast qed lemma seq_suble: assumes sf: "subseq f" shows "n ≤ f n" proof(induct n) case 0 thus ?case by simp next case (Suc n) from sf[unfolded subseq_Suc_iff, rule_format, of n] Suc.hyps have "n < f (Suc n)" by arith thus ?case by arith qed section {* 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 ring_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 cmod_mult[symmetric] cmod_inverse[symmetric]) using right_inverse[OF b'] by (simp add: power2_eq_square[symmetric] power_inverse[symmetric] ring_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 ring_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: cmod_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: cmod_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 cmod_pos[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- from poly_offset[of p z] obtain q where q: "length q = length p" "!!x. poly q x = poly p (z + x)" by blast {fix w note q(2)[of "w - z", simplified]} note th = this show ?thesis unfolding th[symmetric] proof(induct q) case Nil thus ?case using ep by auto next case (Cons 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: cmod_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) cmod_pos] 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) cmod_pos[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: cmod_abs_norm) 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. *} instance complex::idom_char_0 by (intro_classes) instance complex :: recpower_idom_char_0 by intro_classes lemma poly_infinity: assumes ex: "list_ex (λc. c ≠ 0) p" shows "∃r. ∀z. r ≤ cmod z --> d ≤ cmod (poly (a#p) z)" using ex proof(induct p arbitrary: a d) case (Cons c cs a d) {assume H: "list_ex (λc. c≠0) cs" with Cons.hyps obtain r where r: "∀z. r ≤ cmod z --> d + cmod a ≤ cmod (poly (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 (c#cs) z)" by arith from h have z1: "cmod z ≥ 1" by arith from order_trans[OF th0 mult_right_mono[OF z1 cmod_pos[of "poly (c#cs) z"]]] have th1: "d ≤ cmod(z * poly (c#cs) z) - cmod a" unfolding cmod_mult by (simp add: ring_simps) from complex_mod_triangle_sub[of "z * poly (c#cs) z" a] have th2: "cmod(z * poly (c#cs) z) - cmod a ≤ cmod (poly (a#c#cs) z)" by (simp add: diff_le_eq ring_simps) from th1 th2 have "d ≤ cmod (poly (a#c#cs) z)" by arith} hence ?case by blast} moreover {assume cs0: "¬ (list_ex (λc. c ≠ 0) cs)" with Cons.prems have c0: "c ≠ 0" by simp from cs0 have cs0': "list_all (λc. c = 0) cs" by (auto simp add: list_all_iff list_ex_iff) {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 cmod_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: ring_simps) from ath[OF th1 th0] have "d ≤ cmod (poly (a # c # cs) z)" using poly_0[OF 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 (Cons c cs) {assume cs0: "list_ex (λc. c ≠ 0) cs" from poly_infinity[OF cs0, of "cmod (poly (c#cs) 0)" c] obtain r where r: "!!z. r ≤ cmod z ==> cmod (poly (c # cs) 0) ≤ cmod (poly (c # cs) z)" by blast have ath: "!!z r. r ≤ cmod z ∨ cmod z ≤ ¦r¦" by arith from poly_minimum_modulus_disc[of "¦r¦" "c#cs"] obtain v where v: "!!w. cmod w ≤ ¦r¦ ==> cmod (poly (c # cs) v) ≤ cmod (poly (c # cs) w)" by blast {fix z assume z: "r ≤ cmod z" from v[of 0] r[OF z] have "cmod (poly (c # cs) v) ≤ cmod (poly (c # cs) z)" by simp } note v0 = this from v0 v ath[of r] have ?case by blast} moreover {assume cs0: "¬ (list_ex (λc. c≠0) cs)" hence th:"list_all (λc. c = 0) cs" by (simp add: list_all_iff list_ex_iff) from poly_0[OF th] Cons.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)) ==> length p ≥ 2" unfolding constant_def apply (induct p, auto) apply (unfold not_less[symmetric]) apply simp apply (rule ccontr) apply auto done lemma poly_replicate_append: "poly ((replicate n 0)@p) (x::'a::{recpower, comm_ring}) = x^n * poly p x" by(induct n, auto simp add: power_Suc ring_simps) 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 (length q + k) = length p ∧ (∀z. poly p z = z^k * poly (a#q) z)" using nz proof(induct p) case Nil thus ?case by simp next case (Cons c cs) {assume c0: "c = 0" from Cons.hyps Cons.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 ∧ length q + k + 1 = length p ∧ (∀z. poly p z = poly p 0 + z^k * poly (a#q) z)" using nc proof(induct p) case Nil thus ?case by (simp add: constant_def) next case (Cons c cs) {assume C:"∀z. z ≠ 0 --> poly cs z = 0" {fix x y from C have "poly (c#cs) x = poly (c#cs) y" by (cases "x=0", auto)} with Cons.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) 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≡ "length p" arbitrary: p rule: nat_less_induct) fix n fix p :: "complex list" let ?p = "poly p" assume H: "∀m<n. ∀p. ¬ constant (poly p) --> m = length p --> (∃(z::complex). poly p z = 0)" and nc: "¬ constant ?p" and n: "n = length 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: "length q = length 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 (map (op * (inverse ?a0)) q) z * ?a0" by (simp add: poly_cmult_map) let ?r = "map (op * (inverse ?a0)) q" have lgqr: "length q = length ?r" by simp {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 also have "… <-> cmod (poly q w) < cmod ?a0" using a00 unfolding cmod_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" "length s + k + 1 = length ?r" "∀z. poly ?r z = poly ?r 0 + z^k* poly (a#s) z" by blast {assume "k + 1 = n" with kas(3) lgqr[symmetric] q(1) n[symmetric] have s0:"s=[]" 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: ring_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] have k1n: "k + 1 < n" by simp have th01: "¬ constant (poly (1#((replicate (k - 1) 0)@[a])))" unfolding constant_def poly_Nil poly_Cons poly_replicate_append using kas(1) apply simp by (rule exI[where x=0], rule exI[where x=1], simp) from kas(2) have th02: "k+1 = length (1#((replicate (k - 1) 0)@[a]))" by simp from H[rule_format, OF k1n th01 th02] obtain w where w: "1 + w^k * a = 0" unfolding poly_Nil poly_Cons poly_replicate_append using kas(2) by (auto simp add: power_Suc[symmetric, of _ "k - Suc 0"] mult_assoc[of _ _ a, symmetric]) 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 cmod_pos[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: ring_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 complex_mod_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 cmod_complex_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: cmod_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: ring_simps power_mult_distrib cmod_complex_of_real cmod_power cmod_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: cmod_pos)+ 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 cmod_pos[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 ∧ list_all(λb. b = 0) l ∧ p = a#l)" shows "∃z. poly p z = (0::complex)" using nc proof(induct p) case (Cons c cs) {assume "c=0" hence ?case by auto} moreover {assume c0: "c≠0" {assume nc: "constant (poly (c#cs))" from nc[unfolded constant_def, rule_format, of 0] have "∀w. w ≠ 0 --> poly cs w = 0" by auto hence "list_all (λc. c=0) cs" proof(induct cs) case (Cons d ds) {assume "d=0" hence ?case using Cons.prems Cons.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 Cons.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: cmod_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 (c#cs))" using Cons.prems c0 by blast from fundamental_theorem_of_algebra[OF nc] have ?case .} ultimately show ?case by blast qed simp section{* Nullstellenstatz, degrees and divisibility of polynomials *} lemma nullstellensatz_lemma: fixes p :: "complex list" assumes "∀x. poly p x = 0 --> poly q x = 0" and "degree p = n" and "n ≠ 0" shows "p divides (pexp q n)" using prems proof(induct n arbitrary: p q rule: nat_less_induct) fix n::nat fix p q :: "complex list" assume IH: "∀m<n. ∀p q. (∀x. poly p x = (0::complex) --> poly q x = 0) --> degree p = m --> m ≠ 0 --> p divides (q %^ m)" and pq0: "∀x. poly p x = 0 --> poly q x = 0" and dpn: "degree p = n" and n0: "n ≠ 0" let ?ths = "p divides (q %^ n)" {fix a assume a: "poly p a = 0" {assume p0: "poly p = poly []" hence ?ths unfolding divides_def using pq0 n0 apply - apply (rule exI[where x="[]"], rule ext) by (auto simp add: poly_mult poly_exp)} moreover {assume p0: "poly p ≠ poly []" and oa: "order a p ≠ 0" from p0 have pne: "p ≠ []" by auto let ?op = "order a p" from p0 have ap: "([- a, 1] %^ ?op) divides p" "¬ pexp [- a, 1] (Suc ?op) divides p" using order by blast+ note oop = order_degree[OF p0, unfolded dpn] {assume q0: "q = []" hence ?ths using n0 unfolding divides_def apply simp apply (rule exI[where x="[]"], rule ext) by (simp add: divides_def poly_exp poly_mult)} moreover {assume q0: "q≠[]" from pq0[rule_format, OF a, unfolded poly_linear_divides] q0 obtain r where r: "q = pmult [- a, 1] r" by blast from ap[unfolded divides_def] obtain s where s: "poly p = poly (pmult (pexp [- a, 1] ?op) s)" by blast have s0: "poly s ≠ poly []" using s p0 by (simp add: poly_entire) hence pns0: "poly (pnormalize s) ≠ poly []" and sne: "s≠[]" by auto {assume ds0: "degree s = 0" from ds0 pns0 have "∃k. pnormalize s = [k]" unfolding degree_def by (cases "pnormalize s", auto) then obtain k where kpn: "pnormalize s = [k]" by blast from pns0[unfolded poly_zero] kpn have k: "k ≠0" "poly s = poly [k]" using poly_normalize[of s] by simp_all let ?w = "pmult (pmult [1/k] (pexp [-a,1] (n - ?op))) (pexp r n)" from k r s oop have "poly (pexp q n) = poly (pmult p ?w)" by - (rule ext, simp add: poly_mult poly_exp poly_cmult poly_add power_add[symmetric] ring_simps power_mult_distrib[symmetric]) hence ?ths unfolding divides_def by blast} moreover {assume ds0: "degree s ≠ 0" from ds0 s0 dpn degree_unique[OF s, unfolded linear_pow_mul_degree] oa have dsn: "degree s < n" by auto {fix x assume h: "poly s x = 0" {assume xa: "x = a" from h[unfolded xa poly_linear_divides] sne obtain u where u: "s = pmult [- a, 1] u" by blast have "poly p = poly (pmult (pexp [- a, 1] (Suc ?op)) u)" unfolding s u apply (rule ext) by (simp add: ring_simps power_mult_distrib[symmetric] poly_mult poly_cmult poly_add poly_exp) with ap(2)[unfolded divides_def] have False by blast} note xa = this from h s have "poly p x = 0" by (simp add: poly_mult) with pq0 have "poly q x = 0" by blast with r xa have "poly r x = 0" by (auto simp add: poly_mult poly_add poly_cmult eq_diff_eq[symmetric])} note impth = this from IH[rule_format, OF dsn, of s r] impth ds0 have "s divides (pexp r (degree s))" by blast then obtain u where u: "poly (pexp r (degree s)) = poly (pmult s u)" unfolding divides_def by blast hence u': "!!x. poly s x * poly u x = poly r x ^ degree s" by (simp add: poly_mult[symmetric] poly_exp[symmetric]) let ?w = "pmult (pmult u (pexp [-a,1] (n - ?op))) (pexp r (n - degree s))" from u' s r oop[of a] dsn have "poly (pexp q n) = poly (pmult p ?w)" apply - apply (rule ext) apply (simp only: power_mult_distrib power_add[symmetric] poly_add poly_mult poly_exp poly_cmult ring_simps) apply (simp add: power_mult_distrib power_add[symmetric] poly_add poly_mult poly_exp poly_cmult mult_assoc[symmetric]) done hence ?ths unfolding divides_def by blast} ultimately have ?ths by blast } ultimately have ?ths by blast} ultimately have ?ths using a order_root by blast} moreover {assume exa: "¬ (∃a. poly p a = 0)" from fundamental_theorem_of_algebra_alt[of p] exa obtain c cs where ccs: "c≠0" "list_all (λc. c = 0) cs" "p = c#cs" by blast from poly_0[OF ccs(2)] ccs(3) have pp: "!!x. poly p x = c" by simp let ?w = "pmult [1/c] (pexp q n)" from pp ccs(1) have "poly (pexp q n) = poly (pmult p ?w) " apply - apply (rule ext) unfolding poly_mult_assoc[symmetric] by (simp add: poly_mult) hence ?ths unfolding divides_def by blast} ultimately show ?ths by blast qed lemma nullstellensatz_univariate: "(∀x. poly p x = (0::complex) --> poly q x = 0) <-> p divides (q %^ (degree p)) ∨ (poly p = poly [] ∧ poly q = poly [])" proof- {assume pe: "poly p = poly []" hence eq: "(∀x. poly p x = (0::complex) --> poly q x = 0) <-> poly q = poly []" apply auto by (rule ext, simp) {assume "p divides (pexp q (degree p))" then obtain r where r: "poly (pexp q (degree p)) = poly (pmult p r)" unfolding divides_def by blast from cong[OF r refl] pe degree_unique[OF pe] have False by (simp add: poly_mult degree_def)} with eq pe have ?thesis by blast} moreover {assume pe: "poly p ≠ poly []" have p0: "poly [0] = poly []" by (rule ext, simp) {assume dp: "degree p = 0" then obtain k where "pnormalize p = [k]" using pe poly_normalize[of p] unfolding degree_def by (cases "pnormalize p", auto) hence k: "pnormalize p = [k]" "poly p = poly [k]" "k≠0" using pe poly_normalize[of p] by (auto simp add: p0) hence th1: "∀x. poly p x ≠ 0" by simp from k(2,3) dp have "poly (pexp q (degree p)) = poly (pmult p [1/k]) " by - (rule ext, simp add: poly_mult poly_exp) hence th2: "p divides (pexp q (degree p))" unfolding divides_def by blast 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 divides (pexp q (Suc n))" then obtain u where u: "poly (pexp q (Suc n)) = poly (pmult p u)" unfolding divides_def by blast hence u' :"!!x. poly (pexp q (Suc n)) x = poly (pmult p u) x" by simp_all {fix x assume h: "poly p x = 0" "poly q x ≠ 0" hence "poly (pexp q (Suc n)) x ≠ 0" by (simp only: poly_exp) simp hence False using u' h(1) by (simp only: poly_mult poly_exp) 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 (in idom_char_0) constant_degree: "constant (poly p) <-> degree p = 0" (is "?lhs = ?rhs") proof assume l: ?lhs from l[unfolded constant_def, rule_format, of _ "zero"] have th: "poly p = poly [poly p 0]" apply - by (rule ext, simp) from degree_unique[OF th] show ?rhs by (simp add: degree_def) next assume r: ?rhs from r have "pnormalize p = [] ∨ (∃k. pnormalize p = [k])" unfolding degree_def by (cases "pnormalize p", auto) then show ?lhs unfolding constant_def poly_normalize[of p, symmetric] by (auto simp del: poly_normalize) qed (* It would be nicer to prove this without using algebraic closure... *) lemma divides_degree_lemma: assumes dpn: "degree (p::complex list) = n" shows "n ≤ degree (p *** q) ∨ poly (p *** q) = poly []" using dpn proof(induct n arbitrary: p q) case 0 thus ?case by simp next case (Suc n p q) from Suc.prems fundamental_theorem_of_algebra[of p] constant_degree[of p] obtain a where a: "poly p a = 0" by auto then obtain r where r: "p = pmult [-a, 1] r" unfolding poly_linear_divides using Suc.prems by (auto simp add: degree_def) {assume h: "poly (pmult r q) = poly []" hence "poly (pmult p q) = poly []" using r apply - apply (rule ext) by (auto simp add: poly_entire poly_mult poly_add poly_cmult) hence ?case by blast} moreover {assume h: "poly (pmult r q) ≠ poly []" hence r0: "poly r ≠ poly []" and q0: "poly q ≠ poly []" by (auto simp add: poly_entire) have eq: "poly (pmult p q) = poly (pmult [-a, 1] (pmult r q))" apply - apply (rule ext) by (simp add: r poly_mult poly_add poly_cmult ring_simps) from linear_mul_degree[OF h, of "- a"] have dqe: "degree (pmult p q) = degree (pmult r q) + 1" unfolding degree_unique[OF eq] . from linear_mul_degree[OF r0, of "- a", unfolded r[symmetric]] r Suc.prems have dr: "degree r = n" by auto from Suc.hyps[OF dr, of q] have "Suc n ≤ degree (pmult p q)" unfolding dqe using h by (auto simp del: poly.simps) hence ?case by blast} ultimately show ?case by blast qed lemma divides_degree: assumes pq: "p divides (q:: complex list)" shows "degree p ≤ degree q ∨ poly q = poly []" using pq divides_degree_lemma[OF refl, of p] apply (auto simp add: divides_def poly_entire) apply atomize apply (erule_tac x="qa" in allE, auto) apply (subgoal_tac "degree q = degree (p *** qa)", simp) apply (rule degree_unique, simp) done (* Arithmetic operations on multivariate polynomials. *) lemma mpoly_base_conv: "(0::complex) ≡ poly [] x" "c ≡ poly [c] x" "x ≡ poly [0,1] x" by simp_all lemma mpoly_norm_conv: "poly [0] (x::complex) ≡ poly [] x" "poly [poly [] y] x ≡ poly [] 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 (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 [] 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 expand_ex_beta_conv: "list_ex P [c] ≡ P c" by simp lemma poly_divides_pad_rule: fixes p q :: "complex list" assumes pq: "p divides q" shows "p divides ((0::complex)#q)" proof- from pq obtain r where r: "poly q = poly (p *** r)" unfolding divides_def by blast hence "poly (0#q) = poly (p *** ([0,1] *** r))" by - (rule ext, simp add: poly_mult poly_cmult poly_add) thus ?thesis unfolding divides_def by blast qed lemma poly_divides_pad_const_rule: fixes p q :: "complex list" assumes pq: "p divides q" shows "p divides (a %* q)" proof- from pq obtain r where r: "poly q = poly (p *** r)" unfolding divides_def by blast hence "poly (a %* q) = poly (p *** (a %* r))" by - (rule ext, simp add: poly_mult poly_cmult poly_add) thus ?thesis unfolding divides_def by blast qed lemma poly_divides_conv0: fixes p :: "complex list" assumes lgpq: "length q < length p" and lq:"last p ≠ 0" shows "p divides q ≡ (¬ (list_ex (λc. c ≠ 0) q))" (is "?lhs ≡ ?rhs") proof- {assume r: ?rhs hence eq: "poly q = poly []" unfolding poly_zero by (simp add: list_all_iff list_ex_iff) hence "poly q = poly (p *** [])" by - (rule ext, simp add: poly_mult) hence ?lhs unfolding divides_def by blast} moreover {assume l: ?lhs have ath: "!!lq lp dq::nat. lq < lp ==> lq ≠ 0 ==> dq <= lq - 1 ==> dq < lp - 1" by arith {assume q0: "length q = 0" hence "q = []" by simp hence ?rhs by simp} moreover {assume lgq0: "length q ≠ 0" from pnormalize_length[of q] have dql: "degree q ≤ length q - 1" unfolding degree_def by simp from ath[OF lgpq lgq0 dql, unfolded pnormal_degree[OF lq, symmetric]] divides_degree[OF l] have "poly q = poly []" by auto hence ?rhs unfolding poly_zero by (simp add: list_all_iff list_ex_iff)} 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 list) divides p'" and qrp': "!!x. a * poly q x - poly p' x ≡ poly r x" shows "p divides q ≡ p divides (r::complex list)" (is "?lhs ≡ ?rhs") proof- { from pp' obtain t where t: "poly p' = poly (p *** t)" unfolding divides_def by blast {assume l: ?lhs then obtain u where u: "poly q = poly (p *** u)" unfolding divides_def by blast have "poly r = poly (p *** ((a %* u) +++ (-- t)))" using u qrp' t by - (rule ext, simp add: poly_add poly_mult poly_cmult poly_minus ring_simps) then have ?rhs unfolding divides_def by blast} moreover {assume r: ?rhs then obtain u where u: "poly r = poly (p *** u)" unfolding divides_def by blast from u t qrp' a0 have "poly q = poly (p *** ((1/a) %* (u +++ t)))" by - (rule ext, atomize (full), simp add: poly_mult poly_add poly_cmult field_simps) hence ?lhs unfolding divides_def by blast} ultimately have "?lhs = ?rhs" by blast } thus "?lhs ≡ ?rhs" by - (atomize(full), blast) qed lemma basic_cqe_conv1: "(∃x. poly p x = 0 ∧ poly [] x ≠ 0) ≡ False" "(∃x. poly [] x ≠ 0) ≡ False" "(∃x. poly [c] x ≠ 0) ≡ c≠0" "(∃x. poly [] x = 0) ≡ True" "(∃x. poly [c] x = 0) ≡ c = 0" by simp_all lemma basic_cqe_conv2: assumes l:"last (a#b#p) ≠ 0" shows "(∃x. poly (a#b#p) x = (0::complex)) ≡ True" proof- {fix h t assume h: "h≠0" "list_all (λc. c=(0::complex)) t" "a#b#p = h#t" hence "list_all (λc. c= 0) (b#p)" by simp moreover have "last (b#p) ∈ set (b#p)" by simp ultimately have "last (b#p) = 0" by (simp add: list_all_iff) with l have False by simp} hence th: "¬ (∃ h t. h≠0 ∧ list_all (λc. c=0) t ∧ a#b#p = h#t)" by blast from fundamental_theorem_of_algebra_alt[OF th] show "(∃x. poly (a#b#p) x = (0::complex)) ≡ True" by auto qed lemma basic_cqe_conv_2b: "(∃x. poly p x ≠ (0::complex)) ≡ (list_ex (λc. c ≠ 0) p)" proof- have "¬ (list_ex (λc. c ≠ 0) p) <-> poly p = poly []" by (simp add: poly_zero list_all_iff list_ex_iff) also have "… <-> (¬ (∃x. poly p x ≠ 0))" by (auto intro: ext) finally show "(∃x. poly p x ≠ (0::complex)) ≡ (list_ex (λc. c ≠ 0) p)" by - (atomize (full), blast) qed lemma basic_cqe_conv3: fixes p q :: "complex list" assumes l: "last (a#p) ≠ 0" shows "(∃x. poly (a#p) x =0 ∧ poly q x ≠ 0) ≡ ¬ ((a#p) divides (q %^ (length p)))" proof- note np = pnormalize_eq[OF l] {assume "poly (a#p) = poly []" hence False using l unfolding poly_zero apply (auto simp add: list_all_iff del: last.simps) apply (cases p, simp_all) done} then have p0: "poly (a#p) ≠ poly []" by blast from np have dp:"degree (a#p) = length p" by (simp add: degree_def) from nullstellensatz_univariate[of "a#p" q] p0 dp show "(∃x. poly (a#p) x =0 ∧ poly q x ≠ 0) ≡ ¬ ((a#p) divides (q %^ (length p)))" by - (atomize (full), auto) qed lemma basic_cqe_conv4: fixes p q :: "complex list" assumes h: "!!x. poly (q %^ n) x ≡ poly r x" shows "p divides (q %^ n) ≡ p divides r" proof- from h have "poly (q %^ n) = poly r" by (auto intro: ext) thus "p divides (q %^ n) ≡ p divides r" unfolding divides_def by simp qed lemma pmult_Cons_Cons: "((a::complex)#b#p) *** q = (a %*q) +++ (0#((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 last_simps: "last [x] = x" "last (x#y#ys) = last (y#ys)" by simp_all lemma length_simps: "length [] = 0" "length (x#y#xs) = length xs + 2" "length [x] = 1" by simp_all 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 [] 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
lemma csqrt:
csqrt z ^ 2 = z
lemma complex_of_real_power:
complex_of_real x ^ n = complex_of_real (x ^ n)
lemma cmod_pos:
0 ≤ cmod z
lemma complex_mod_triangle_ineq:
cmod (z + w) ≤ cmod z + cmod w
lemma cmod_mult:
cmod (z * w) = cmod z * cmod w
lemma cmod_divide:
cmod (z / w) = cmod z / cmod w
lemma cmod_inverse:
cmod (inverse z) = inverse (cmod z)
lemma cmod_uminus:
cmod (- z) = cmod z
lemma cmod_abs_norm:
¦cmod w - cmod z¦ ≤ cmod (w - z)
lemma cmod_power:
cmod (z ^ n) = cmod z ^ n
lemma real_down2:
[| 0 < d1.0; 0 < d2.0 |] ==> ∃e>0. e < d1.0 ∧ e < d2.0
lemma cmod_complex_of_real:
cmod (complex_of_real x) = ¦x¦
lemma complex_mod_triangle_sub:
cmod w ≤ cmod (w + z) + cmod z
lemma poly_bound_exists:
∃m>0. ∀z. cmod z ≤ r --> cmod (poly p z) ≤ m
lemma poly_offset_lemma:
∃b q. length q = length p ∧ (∀x. poly (b # q) x = (a + x) * poly p x)
lemma poly_offset:
∃q. length q = length p ∧ (∀x. poly q x = poly p (a + x))
lemma real_sup_exists:
[| ∃x. P x; ∃z. ∀x. P x --> x < z |] ==> ∃s. ∀y. (∃x. P x ∧ y < x) = (y < s)
lemma num_Axiom:
∃!g. g 0 = e ∧ (∀n. g (Suc n) = f n (g n))
lemma mono_Suc:
mono f = (∀n. f n ≤ f (Suc n))
lemma seq_monosub:
∃f. subseq f ∧ monoseq (λn. s (f n))
lemma seq_suble:
subseq f ==> n ≤ f n
lemma unimodular_reduce_norm:
cmod z = 1
==> cmod (z + 1) < 1 ∨
cmod (z - 1) < 1 ∨ cmod (z + \<i>) < 1 ∨ cmod (z - \<i>) < 1
lemma reduce_poly_simple:
[| b ≠ 0; n ≠ 0 |] ==> ∃z. cmod (1 + b * z ^ n) < 1
lemma metric_bound_lemma:
cmod (x - y) ≤ ¦Re x - Re y¦ + ¦Im x - Im y¦
lemma bolzano_weierstrass_complex_disc:
∀n. cmod (s n) ≤ r ==> ∃f z. subseq f ∧ (∀e>0. ∃N. ∀n≥N. cmod (s (f n) - z) < e)
lemma poly_cont:
0 < e
==> ∃d>0. ∀w. 0 < cmod (w - z) ∧ cmod (w - z) < d -->
cmod (poly p w - poly p z) < e
lemma poly_minimum_modulus_disc:
∃z. ∀w. cmod w ≤ r --> cmod (poly p z) ≤ cmod (poly p w)
lemma
rcis (sqrt ¦r¦) (a / 2) ^ 2 = rcis ¦r¦ a
lemma cispi:
cis pi = -1
lemma
rcis (sqrt ¦r¦) ((pi + a) / 2) ^ 2 = rcis (- ¦r¦) a
lemma poly_infinity:
list_ex (λc. c ≠ 0) p ==> ∃r. ∀z. r ≤ cmod z --> d ≤ cmod (poly (a # p) z)
lemma poly_minimum_modulus:
∃z. ∀w. cmod (poly p z) ≤ cmod (poly p w)
lemma nonconstant_length:
¬ constant (poly p) ==> 2 ≤ length p
lemma poly_replicate_append:
poly (replicate n (0::'a) @ p) x = x ^ n * poly p x
lemma poly_decompose_lemma:
¬ (∀z. z ≠ (0::'a) --> poly p z = (0::'a))
==> ∃k a q.
a ≠ (0::'a) ∧
Suc (length q + k) = length p ∧ (∀z. poly p z = z ^ k * poly (a # q) z)
lemma poly_decompose:
¬ constant (poly p)
==> ∃k a q.
a ≠ (0::'a) ∧
k ≠ 0 ∧
length q + k + 1 = length p ∧
(∀z. poly p z = poly p (0::'a) + z ^ k * poly (a # q) z)
lemma fundamental_theorem_of_algebra:
¬ constant (poly p) ==> ∃z. poly p z = 0
lemma fundamental_theorem_of_algebra_alt:
¬ (∃a l. a ≠ 0 ∧ list_all (λb. b = 0) l ∧ p = a # l) ==> ∃z. poly p z = 0
lemma nullstellensatz_lemma:
[| ∀x. poly p x = 0 --> poly q x = 0; degree p = n; n ≠ 0 |]
==> p divides q %^ n
lemma nullstellensatz_univariate:
(∀x. poly p x = 0 --> poly q x = 0) =
(p divides q %^ degree p ∨ poly p = poly [] ∧ poly q = poly [])
lemma constant_degree:
constant (poly p) = (degree p = 0)
lemma divides_degree_lemma:
degree p = n ==> n ≤ degree (p *** q) ∨ poly (p *** q) = poly []
lemma divides_degree:
p divides q ==> degree p ≤ degree q ∨ poly q = poly []
lemma mpoly_base_conv:
0 == poly [] x
c == poly [c] x
x == poly [0, 1] x
lemma mpoly_norm_conv:
poly [0] x == poly [] x
poly [poly [] y] x == poly [] x
lemma mpoly_sub_conv:
poly p x - poly q x == poly p x + -1 * poly q x
lemma poly_pad_rule:
poly p x = 0 ==> poly (0 # p) x = 0
lemma poly_cancel_eq_conv:
[| p = 0; a ≠ 0 |] ==> q = 0 == a * q - b * p = 0
lemma resolve_eq_raw:
poly [] x == 0
poly [c] x == c
lemma resolve_eq_then:
[| P ==> Q == Q1.0; ¬ P ==> Q == Q2.0 |] ==> Q == P ∧ Q1.0 ∨ ¬ P ∧ Q2.0
lemma expand_ex_beta_conv:
list_ex P [c] == P c
lemma poly_divides_pad_rule:
p divides q ==> p divides (0 # q)
lemma poly_divides_pad_const_rule:
p divides q ==> p divides (a %* q)
lemma poly_divides_conv0:
[| length q < length p; last p ≠ 0 |] ==> p divides q == ¬ list_ex (λc. c ≠ 0) q
lemma poly_divides_conv1:
[| a ≠ 0; p divides p'; !!x. a * poly q x - poly p' x == poly r x |]
==> p divides q == p divides r
lemma basic_cqe_conv1:
∃x. poly p x = (0::'a) ∧ poly [] x ≠ (0::'a) == False
∃x. poly [] x ≠ (0::'b) == False
∃x. poly [c] x ≠ (0::'c) == c ≠ (0::'c)
∃x. poly [] x = (0::'d) == True
∃x. poly [c] x = (0::'c) == c = (0::'c)
lemma basic_cqe_conv2:
last (a # b # p) ≠ 0 ==> ∃x. poly (a # b # p) x = 0 == True
lemma basic_cqe_conv_2b:
∃x. poly p x ≠ 0 == list_ex (λc. c ≠ 0) p
lemma basic_cqe_conv3:
last (a # p) ≠ 0
==> ∃x. poly (a # p) x = 0 ∧ poly q x ≠ 0 == ¬ (a # p) divides q %^ length p
lemma basic_cqe_conv4:
(!!x. poly (q %^ n) x == poly r x) ==> p divides q %^ n == p divides r
lemma pmult_Cons_Cons:
(a # b # p) *** q = a %* q +++ (0 # (b # p) *** q)
lemma elim_neg_conv:
- z == -1 * z
lemma eqT_intr:
[| PROP P; True |] ==> PROP P
PROP P ==> True
lemma negate_negate_rule:
P == ¬ P == False
lemma last_simps:
last [x] = x
last (x # y # ys) = last (y # ys)
lemma length_simps:
length [] = 0
length (x # y # xs) = length xs + 2
length [x] = 1
lemma complex_entire:
z ≠ 0 ∧ w ≠ 0 == z * w ≠ 0
lemma resolve_eq_ne:
(P == True) == ¬ P == False
(P == False) == ¬ P == True
lemma cqe_conv1:
(poly [] x = (0::'a)) = True
lemma cqe_conv2:
(p ==> q == r) == p ∧ q == p ∧ r
lemma poly_const_conv:
(poly [c] x = y) = (c = y)