% TIMEFORMAT='%3R'; { time (exec 2>&1; /home/martin/bin/satallax -E /home/martin/.isabelle/contrib/e-2.5-1/x86_64-linux/eprover -p tstp -t 5 /home/martin/judgement-day/tptp-thf/tptp/Fundamental_Theorem_Algebra/prob_826__5375936_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:33:28.064

% Could-be-implicit typings (4)
thf(ty_n_t__Polynomial__Opoly_It__Polynomial__Opoly_It__Complex__Ocomplex_J_J, type,
    poly_poly_complex : $tType).
thf(ty_n_t__Polynomial__Opoly_It__Complex__Ocomplex_J, type,
    poly_complex : $tType).
thf(ty_n_t__Complex__Ocomplex, type,
    complex : $tType).
thf(ty_n_t__Nat__Onat, type,
    nat : $tType).

% Explicit typings (34)
thf(sy_c_Fundamental__Theorem__Algebra__Mirabelle__sywschxjbb_Oconstant_001t__Complex__Ocomplex_001t__Complex__Ocomplex, type,
    fundam1158420650omplex : (complex > complex) > $o).
thf(sy_c_Fundamental__Theorem__Algebra__Mirabelle__sywschxjbb_Ooffset__poly_001t__Complex__Ocomplex, type,
    fundam1201687030omplex : poly_complex > complex > poly_complex).
thf(sy_c_Groups_Oone__class_Oone_001t__Complex__Ocomplex, type,
    one_one_complex : complex).
thf(sy_c_Groups_Oone__class_Oone_001t__Polynomial__Opoly_It__Complex__Ocomplex_J, type,
    one_one_poly_complex : poly_complex).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Complex__Ocomplex, type,
    plus_plus_complex : complex > complex > complex).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Polynomial__Opoly_It__Complex__Ocomplex_J, type,
    plus_p1547158847omplex : poly_complex > poly_complex > poly_complex).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Complex__Ocomplex_J_J, type,
    plus_p138939463omplex : poly_poly_complex > poly_poly_complex > poly_poly_complex).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Complex__Ocomplex, type,
    times_times_complex : complex > complex > complex).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Polynomial__Opoly_It__Complex__Ocomplex_J, type,
    times_1246143675omplex : poly_complex > poly_complex > poly_complex).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Complex__Ocomplex_J_J, type,
    times_1460995011omplex : poly_poly_complex > poly_poly_complex > poly_poly_complex).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Complex__Ocomplex, type,
    uminus1204672759omplex : complex > complex).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Polynomial__Opoly_It__Complex__Ocomplex_J, type,
    uminus1138659839omplex : poly_complex > poly_complex).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Complex__Ocomplex, type,
    zero_zero_complex : complex).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat, type,
    zero_zero_nat : nat).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Complex__Ocomplex_J, type,
    zero_z1746442943omplex : poly_complex).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Polynomial__Opoly_It__Complex__Ocomplex_J_J, type,
    zero_z1040703943omplex : poly_poly_complex).
thf(sy_c_Polynomial_Ois__zero_001t__Complex__Ocomplex, type,
    is_zero_complex : poly_complex > $o).
thf(sy_c_Polynomial_OpCons_001t__Complex__Ocomplex, type,
    pCons_complex : complex > poly_complex > poly_complex).
thf(sy_c_Polynomial_OpCons_001t__Polynomial__Opoly_It__Complex__Ocomplex_J, type,
    pCons_poly_complex : poly_complex > poly_poly_complex > poly_poly_complex).
thf(sy_c_Polynomial_Opcompose_001t__Complex__Ocomplex, type,
    pcompose_complex : poly_complex > poly_complex > poly_complex).
thf(sy_c_Polynomial_Opcompose_001t__Polynomial__Opoly_It__Complex__Ocomplex_J, type,
    pcompo1411605209omplex : poly_poly_complex > poly_poly_complex > poly_poly_complex).
thf(sy_c_Polynomial_Opoly_001t__Complex__Ocomplex, type,
    poly_complex2 : poly_complex > complex > complex).
thf(sy_c_Polynomial_Opoly_001t__Polynomial__Opoly_It__Complex__Ocomplex_J, type,
    poly_poly_complex2 : poly_poly_complex > poly_complex > poly_complex).
thf(sy_c_Polynomial_Opoly__cutoff_001t__Complex__Ocomplex, type,
    poly_cutoff_complex : nat > poly_complex > poly_complex).
thf(sy_c_Polynomial_Opseudo__mod_001t__Complex__Ocomplex, type,
    pseudo_mod_complex : poly_complex > poly_complex > poly_complex).
thf(sy_c_Polynomial_Opseudo__mod_001t__Polynomial__Opoly_It__Complex__Ocomplex_J, type,
    pseudo2092547131omplex : poly_poly_complex > poly_poly_complex > poly_poly_complex).
thf(sy_c_Polynomial_Oreflect__poly_001t__Complex__Ocomplex, type,
    reflect_poly_complex : poly_complex > poly_complex).
thf(sy_c_Polynomial_Oreflect__poly_001t__Polynomial__Opoly_It__Complex__Ocomplex_J, type,
    reflec309385472omplex : poly_poly_complex > poly_poly_complex).
thf(sy_c_Polynomial_Osmult_001t__Complex__Ocomplex, type,
    smult_complex : complex > poly_complex > poly_complex).
thf(sy_c_Polynomial_Osmult_001t__Polynomial__Opoly_It__Complex__Ocomplex_J, type,
    smult_poly_complex : poly_complex > poly_poly_complex > poly_poly_complex).
thf(sy_c_Polynomial_Osynthetic__div_001t__Complex__Ocomplex, type,
    synthe151143547omplex : poly_complex > complex > poly_complex).
thf(sy_v_c____, type,
    c : complex).
thf(sy_v_cs____, type,
    cs : poly_complex).
thf(sy_v_p, type,
    p : poly_complex).

% Relevant facts (136)
thf(fact_0_False, axiom,
    ((~ ((c = zero_zero_complex))))). % False
thf(fact_1_constant__def, axiom,
    ((fundam1158420650omplex = (^[F : complex > complex]: (![X : complex]: (![Y : complex]: ((F @ X) = (F @ Y)))))))). % constant_def
thf(fact_2_pCons_Oprems, axiom,
    ((~ ((?[A : complex, L : poly_complex]: ((~ ((A = zero_zero_complex))) & ((L = zero_z1746442943omplex) & ((pCons_complex @ c @ cs) = (pCons_complex @ A @ L))))))))). % pCons.prems
thf(fact_3_pCons__eq__iff, axiom,
    ((![A2 : complex, P : poly_complex, B : complex, Q : poly_complex]: (((pCons_complex @ A2 @ P) = (pCons_complex @ B @ Q)) = (((A2 = B)) & ((P = Q))))))). % pCons_eq_iff
thf(fact_4_pCons_Ohyps_I2_J, axiom,
    (((~ ((?[A3 : complex, L2 : poly_complex]: ((~ ((A3 = zero_zero_complex))) & ((L2 = zero_z1746442943omplex) & (cs = (pCons_complex @ A3 @ L2))))))) => (?[Z : complex]: ((poly_complex2 @ cs @ Z) = zero_zero_complex))))). % pCons.hyps(2)
thf(fact_5_pCons_Ohyps_I1_J, axiom,
    (((~ ((c = zero_zero_complex))) | (~ ((cs = zero_z1746442943omplex)))))). % pCons.hyps(1)
thf(fact_6_fundamental__theorem__of__algebra, axiom,
    ((![P : poly_complex]: ((~ ((fundam1158420650omplex @ (poly_complex2 @ P)))) => (?[Z : complex]: ((poly_complex2 @ P @ Z) = zero_zero_complex)))))). % fundamental_theorem_of_algebra
thf(fact_7_poly__eq__poly__eq__iff, axiom,
    ((![P : poly_complex, Q : poly_complex]: (((poly_complex2 @ P) = (poly_complex2 @ Q)) = (P = Q))))). % poly_eq_poly_eq_iff
thf(fact_8_pCons__cases, axiom,
    ((![P : poly_complex]: (~ ((![A3 : complex, Q2 : poly_complex]: (~ ((P = (pCons_complex @ A3 @ Q2)))))))))). % pCons_cases
thf(fact_9_pderiv_Ocases, axiom,
    ((![X2 : poly_complex]: (~ ((![A3 : complex, P2 : poly_complex]: (~ ((X2 = (pCons_complex @ A3 @ P2)))))))))). % pderiv.cases
thf(fact_10_synthetic__div__pCons, axiom,
    ((![A2 : complex, P : poly_complex, C : complex]: ((synthe151143547omplex @ (pCons_complex @ A2 @ P) @ C) = (pCons_complex @ (poly_complex2 @ P @ C) @ (synthe151143547omplex @ P @ C)))))). % synthetic_div_pCons
thf(fact_11_nc, axiom,
    ((~ ((?[A : complex, L : poly_complex]: ((~ ((A = zero_zero_complex))) & ((L = zero_z1746442943omplex) & (p = (pCons_complex @ A @ L))))))))). % nc
thf(fact_12_poly__pCons, axiom,
    ((![A2 : complex, P : poly_complex, X2 : complex]: ((poly_complex2 @ (pCons_complex @ A2 @ P) @ X2) = (plus_plus_complex @ A2 @ (times_times_complex @ X2 @ (poly_complex2 @ P @ X2))))))). % poly_pCons
thf(fact_13_offset__poly__single, axiom,
    ((![A2 : complex, H : complex]: ((fundam1201687030omplex @ (pCons_complex @ A2 @ zero_z1746442943omplex) @ H) = (pCons_complex @ A2 @ zero_z1746442943omplex))))). % offset_poly_single
thf(fact_14_add__pCons, axiom,
    ((![A2 : complex, P : poly_complex, B : complex, Q : poly_complex]: ((plus_p1547158847omplex @ (pCons_complex @ A2 @ P) @ (pCons_complex @ B @ Q)) = (pCons_complex @ (plus_plus_complex @ A2 @ B) @ (plus_p1547158847omplex @ P @ Q)))))). % add_pCons
thf(fact_15_poly__mult, axiom,
    ((![P : poly_complex, Q : poly_complex, X2 : complex]: ((poly_complex2 @ (times_1246143675omplex @ P @ Q) @ X2) = (times_times_complex @ (poly_complex2 @ P @ X2) @ (poly_complex2 @ Q @ X2)))))). % poly_mult
thf(fact_16_poly__add, axiom,
    ((![P : poly_complex, Q : poly_complex, X2 : complex]: ((poly_complex2 @ (plus_p1547158847omplex @ P @ Q) @ X2) = (plus_plus_complex @ (poly_complex2 @ P @ X2) @ (poly_complex2 @ Q @ X2)))))). % poly_add
thf(fact_17_synthetic__div__0, axiom,
    ((![C : complex]: ((synthe151143547omplex @ zero_z1746442943omplex @ C) = zero_z1746442943omplex)))). % synthetic_div_0
thf(fact_18_pCons__0__0, axiom,
    (((pCons_complex @ zero_zero_complex @ zero_z1746442943omplex) = zero_z1746442943omplex))). % pCons_0_0
thf(fact_19_pCons__0__0, axiom,
    (((pCons_poly_complex @ zero_z1746442943omplex @ zero_z1040703943omplex) = zero_z1040703943omplex))). % pCons_0_0
thf(fact_20_pCons__eq__0__iff, axiom,
    ((![A2 : poly_complex, P : poly_poly_complex]: (((pCons_poly_complex @ A2 @ P) = zero_z1040703943omplex) = (((A2 = zero_z1746442943omplex)) & ((P = zero_z1040703943omplex))))))). % pCons_eq_0_iff
thf(fact_21_pCons__eq__0__iff, axiom,
    ((![A2 : complex, P : poly_complex]: (((pCons_complex @ A2 @ P) = zero_z1746442943omplex) = (((A2 = zero_zero_complex)) & ((P = zero_z1746442943omplex))))))). % pCons_eq_0_iff
thf(fact_22_poly__0, axiom,
    ((![X2 : poly_complex]: ((poly_poly_complex2 @ zero_z1040703943omplex @ X2) = zero_z1746442943omplex)))). % poly_0
thf(fact_23_poly__0, axiom,
    ((![X2 : complex]: ((poly_complex2 @ zero_z1746442943omplex @ X2) = zero_zero_complex)))). % poly_0
thf(fact_24_pCons__induct, axiom,
    ((![P3 : poly_poly_complex > $o, P : poly_poly_complex]: ((P3 @ zero_z1040703943omplex) => ((![A3 : poly_complex, P2 : poly_poly_complex]: (((~ ((A3 = zero_z1746442943omplex))) | (~ ((P2 = zero_z1040703943omplex)))) => ((P3 @ P2) => (P3 @ (pCons_poly_complex @ A3 @ P2))))) => (P3 @ P)))))). % pCons_induct
thf(fact_25_pCons__induct, axiom,
    ((![P3 : poly_complex > $o, P : poly_complex]: ((P3 @ zero_z1746442943omplex) => ((![A3 : complex, P2 : poly_complex]: (((~ ((A3 = zero_zero_complex))) | (~ ((P2 = zero_z1746442943omplex)))) => ((P3 @ P2) => (P3 @ (pCons_complex @ A3 @ P2))))) => (P3 @ P)))))). % pCons_induct
thf(fact_26_mult__poly__0__left, axiom,
    ((![Q : poly_complex]: ((times_1246143675omplex @ zero_z1746442943omplex @ Q) = zero_z1746442943omplex)))). % mult_poly_0_left
thf(fact_27_poly__all__0__iff__0, axiom,
    ((![P : poly_complex]: ((![X : complex]: ((poly_complex2 @ P @ X) = zero_zero_complex)) = (P = zero_z1746442943omplex))))). % poly_all_0_iff_0
thf(fact_28_poly__all__0__iff__0, axiom,
    ((![P : poly_poly_complex]: ((![X : poly_complex]: ((poly_poly_complex2 @ P @ X) = zero_z1746442943omplex)) = (P = zero_z1040703943omplex))))). % poly_all_0_iff_0
thf(fact_29_mult__poly__0__right, axiom,
    ((![P : poly_complex]: ((times_1246143675omplex @ P @ zero_z1746442943omplex) = zero_z1746442943omplex)))). % mult_poly_0_right
thf(fact_30_offset__poly__0, axiom,
    ((![H : complex]: ((fundam1201687030omplex @ zero_z1746442943omplex @ H) = zero_z1746442943omplex)))). % offset_poly_0
thf(fact_31_offset__poly__eq__0__iff, axiom,
    ((![P : poly_complex, H : complex]: (((fundam1201687030omplex @ P @ H) = zero_z1746442943omplex) = (P = zero_z1746442943omplex))))). % offset_poly_eq_0_iff
thf(fact_32_pderiv_Oinduct, axiom,
    ((![P3 : poly_complex > $o, A0 : poly_complex]: ((![A3 : complex, P2 : poly_complex]: (((~ ((P2 = zero_z1746442943omplex))) => (P3 @ P2)) => (P3 @ (pCons_complex @ A3 @ P2)))) => (P3 @ A0))))). % pderiv.induct
thf(fact_33_poly__induct2, axiom,
    ((![P3 : poly_complex > poly_complex > $o, P : poly_complex, Q : poly_complex]: ((P3 @ zero_z1746442943omplex @ zero_z1746442943omplex) => ((![A3 : complex, P2 : poly_complex, B2 : complex, Q2 : poly_complex]: ((P3 @ P2 @ Q2) => (P3 @ (pCons_complex @ A3 @ P2) @ (pCons_complex @ B2 @ Q2)))) => (P3 @ P @ Q)))))). % poly_induct2
thf(fact_34_poly__offset__poly, axiom,
    ((![P : poly_complex, H : complex, X2 : complex]: ((poly_complex2 @ (fundam1201687030omplex @ P @ H) @ X2) = (poly_complex2 @ P @ (plus_plus_complex @ H @ X2)))))). % poly_offset_poly
thf(fact_35_add_Oleft__neutral, axiom,
    ((![A2 : complex]: ((plus_plus_complex @ zero_zero_complex @ A2) = A2)))). % add.left_neutral
thf(fact_36_add_Oleft__neutral, axiom,
    ((![A2 : poly_complex]: ((plus_p1547158847omplex @ zero_z1746442943omplex @ A2) = A2)))). % add.left_neutral
thf(fact_37_add_Oright__neutral, axiom,
    ((![A2 : complex]: ((plus_plus_complex @ A2 @ zero_zero_complex) = A2)))). % add.right_neutral
thf(fact_38_add_Oright__neutral, axiom,
    ((![A2 : poly_complex]: ((plus_p1547158847omplex @ A2 @ zero_z1746442943omplex) = A2)))). % add.right_neutral
thf(fact_39_add__cancel__left__left, axiom,
    ((![B : complex, A2 : complex]: (((plus_plus_complex @ B @ A2) = A2) = (B = zero_zero_complex))))). % add_cancel_left_left
thf(fact_40_add__cancel__left__left, axiom,
    ((![B : poly_complex, A2 : poly_complex]: (((plus_p1547158847omplex @ B @ A2) = A2) = (B = zero_z1746442943omplex))))). % add_cancel_left_left
thf(fact_41_add__cancel__left__right, axiom,
    ((![A2 : complex, B : complex]: (((plus_plus_complex @ A2 @ B) = A2) = (B = zero_zero_complex))))). % add_cancel_left_right
thf(fact_42_add__cancel__left__right, axiom,
    ((![A2 : poly_complex, B : poly_complex]: (((plus_p1547158847omplex @ A2 @ B) = A2) = (B = zero_z1746442943omplex))))). % add_cancel_left_right
thf(fact_43_add__cancel__right__left, axiom,
    ((![A2 : complex, B : complex]: ((A2 = (plus_plus_complex @ B @ A2)) = (B = zero_zero_complex))))). % add_cancel_right_left
thf(fact_44_add__cancel__right__left, axiom,
    ((![A2 : poly_complex, B : poly_complex]: ((A2 = (plus_p1547158847omplex @ B @ A2)) = (B = zero_z1746442943omplex))))). % add_cancel_right_left
thf(fact_45_add__cancel__right__right, axiom,
    ((![A2 : complex, B : complex]: ((A2 = (plus_plus_complex @ A2 @ B)) = (B = zero_zero_complex))))). % add_cancel_right_right
thf(fact_46_add__cancel__right__right, axiom,
    ((![A2 : poly_complex, B : poly_complex]: ((A2 = (plus_p1547158847omplex @ A2 @ B)) = (B = zero_z1746442943omplex))))). % add_cancel_right_right
thf(fact_47_zero__reorient, axiom,
    ((![X2 : complex]: ((zero_zero_complex = X2) = (X2 = zero_zero_complex))))). % zero_reorient
thf(fact_48_zero__reorient, axiom,
    ((![X2 : poly_complex]: ((zero_z1746442943omplex = X2) = (X2 = zero_z1746442943omplex))))). % zero_reorient
thf(fact_49_add_Ogroup__left__neutral, axiom,
    ((![A2 : complex]: ((plus_plus_complex @ zero_zero_complex @ A2) = A2)))). % add.group_left_neutral
thf(fact_50_add_Ogroup__left__neutral, axiom,
    ((![A2 : poly_complex]: ((plus_p1547158847omplex @ zero_z1746442943omplex @ A2) = A2)))). % add.group_left_neutral
thf(fact_51_add_Ocomm__neutral, axiom,
    ((![A2 : complex]: ((plus_plus_complex @ A2 @ zero_zero_complex) = A2)))). % add.comm_neutral
thf(fact_52_add_Ocomm__neutral, axiom,
    ((![A2 : poly_complex]: ((plus_p1547158847omplex @ A2 @ zero_z1746442943omplex) = A2)))). % add.comm_neutral
thf(fact_53_comm__monoid__add__class_Oadd__0, axiom,
    ((![A2 : complex]: ((plus_plus_complex @ zero_zero_complex @ A2) = A2)))). % comm_monoid_add_class.add_0
thf(fact_54_comm__monoid__add__class_Oadd__0, axiom,
    ((![A2 : poly_complex]: ((plus_p1547158847omplex @ zero_z1746442943omplex @ A2) = A2)))). % comm_monoid_add_class.add_0
thf(fact_55_mult__cancel__right, axiom,
    ((![A2 : complex, C : complex, B : complex]: (((times_times_complex @ A2 @ C) = (times_times_complex @ B @ C)) = (((C = zero_zero_complex)) | ((A2 = B))))))). % mult_cancel_right
thf(fact_56_mult__cancel__right, axiom,
    ((![A2 : poly_complex, C : poly_complex, B : poly_complex]: (((times_1246143675omplex @ A2 @ C) = (times_1246143675omplex @ B @ C)) = (((C = zero_z1746442943omplex)) | ((A2 = B))))))). % mult_cancel_right
thf(fact_57_mult__cancel__left, axiom,
    ((![C : complex, A2 : complex, B : complex]: (((times_times_complex @ C @ A2) = (times_times_complex @ C @ B)) = (((C = zero_zero_complex)) | ((A2 = B))))))). % mult_cancel_left
thf(fact_58_mult__cancel__left, axiom,
    ((![C : poly_complex, A2 : poly_complex, B : poly_complex]: (((times_1246143675omplex @ C @ A2) = (times_1246143675omplex @ C @ B)) = (((C = zero_z1746442943omplex)) | ((A2 = B))))))). % mult_cancel_left
thf(fact_59_mult__eq__0__iff, axiom,
    ((![A2 : complex, B : complex]: (((times_times_complex @ A2 @ B) = zero_zero_complex) = (((A2 = zero_zero_complex)) | ((B = zero_zero_complex))))))). % mult_eq_0_iff
thf(fact_60_mult__eq__0__iff, axiom,
    ((![A2 : poly_complex, B : poly_complex]: (((times_1246143675omplex @ A2 @ B) = zero_z1746442943omplex) = (((A2 = zero_z1746442943omplex)) | ((B = zero_z1746442943omplex))))))). % mult_eq_0_iff
thf(fact_61_mult__zero__right, axiom,
    ((![A2 : complex]: ((times_times_complex @ A2 @ zero_zero_complex) = zero_zero_complex)))). % mult_zero_right
thf(fact_62_mult__zero__right, axiom,
    ((![A2 : poly_complex]: ((times_1246143675omplex @ A2 @ zero_z1746442943omplex) = zero_z1746442943omplex)))). % mult_zero_right
thf(fact_63_mult__zero__left, axiom,
    ((![A2 : complex]: ((times_times_complex @ zero_zero_complex @ A2) = zero_zero_complex)))). % mult_zero_left
thf(fact_64_mult__zero__left, axiom,
    ((![A2 : poly_complex]: ((times_1246143675omplex @ zero_z1746442943omplex @ A2) = zero_z1746442943omplex)))). % mult_zero_left
thf(fact_65_add__scale__eq__noteq, axiom,
    ((![R : complex, A2 : complex, B : complex, C : complex, D : complex]: ((~ ((R = zero_zero_complex))) => (((A2 = B) & (~ ((C = D)))) => (~ (((plus_plus_complex @ A2 @ (times_times_complex @ R @ C)) = (plus_plus_complex @ B @ (times_times_complex @ R @ D)))))))))). % add_scale_eq_noteq
thf(fact_66_add__scale__eq__noteq, axiom,
    ((![R : poly_complex, A2 : poly_complex, B : poly_complex, C : poly_complex, D : poly_complex]: ((~ ((R = zero_z1746442943omplex))) => (((A2 = B) & (~ ((C = D)))) => (~ (((plus_p1547158847omplex @ A2 @ (times_1246143675omplex @ R @ C)) = (plus_p1547158847omplex @ B @ (times_1246143675omplex @ R @ D)))))))))). % add_scale_eq_noteq
thf(fact_67_mult__not__zero, axiom,
    ((![A2 : complex, B : complex]: ((~ (((times_times_complex @ A2 @ B) = zero_zero_complex))) => ((~ ((A2 = zero_zero_complex))) & (~ ((B = zero_zero_complex)))))))). % mult_not_zero
thf(fact_68_mult__not__zero, axiom,
    ((![A2 : poly_complex, B : poly_complex]: ((~ (((times_1246143675omplex @ A2 @ B) = zero_z1746442943omplex))) => ((~ ((A2 = zero_z1746442943omplex))) & (~ ((B = zero_z1746442943omplex)))))))). % mult_not_zero
thf(fact_69_divisors__zero, axiom,
    ((![A2 : complex, B : complex]: (((times_times_complex @ A2 @ B) = zero_zero_complex) => ((A2 = zero_zero_complex) | (B = zero_zero_complex)))))). % divisors_zero
thf(fact_70_divisors__zero, axiom,
    ((![A2 : poly_complex, B : poly_complex]: (((times_1246143675omplex @ A2 @ B) = zero_z1746442943omplex) => ((A2 = zero_z1746442943omplex) | (B = zero_z1746442943omplex)))))). % divisors_zero
thf(fact_71_no__zero__divisors, axiom,
    ((![A2 : complex, B : complex]: ((~ ((A2 = zero_zero_complex))) => ((~ ((B = zero_zero_complex))) => (~ (((times_times_complex @ A2 @ B) = zero_zero_complex)))))))). % no_zero_divisors
thf(fact_72_no__zero__divisors, axiom,
    ((![A2 : poly_complex, B : poly_complex]: ((~ ((A2 = zero_z1746442943omplex))) => ((~ ((B = zero_z1746442943omplex))) => (~ (((times_1246143675omplex @ A2 @ B) = zero_z1746442943omplex)))))))). % no_zero_divisors
thf(fact_73_mult__left__cancel, axiom,
    ((![C : complex, A2 : complex, B : complex]: ((~ ((C = zero_zero_complex))) => (((times_times_complex @ C @ A2) = (times_times_complex @ C @ B)) = (A2 = B)))))). % mult_left_cancel
thf(fact_74_mult__left__cancel, axiom,
    ((![C : poly_complex, A2 : poly_complex, B : poly_complex]: ((~ ((C = zero_z1746442943omplex))) => (((times_1246143675omplex @ C @ A2) = (times_1246143675omplex @ C @ B)) = (A2 = B)))))). % mult_left_cancel
thf(fact_75_mult__right__cancel, axiom,
    ((![C : complex, A2 : complex, B : complex]: ((~ ((C = zero_zero_complex))) => (((times_times_complex @ A2 @ C) = (times_times_complex @ B @ C)) = (A2 = B)))))). % mult_right_cancel
thf(fact_76_mult__right__cancel, axiom,
    ((![C : poly_complex, A2 : poly_complex, B : poly_complex]: ((~ ((C = zero_z1746442943omplex))) => (((times_1246143675omplex @ A2 @ C) = (times_1246143675omplex @ B @ C)) = (A2 = B)))))). % mult_right_cancel
thf(fact_77_add__0__iff, axiom,
    ((![B : complex, A2 : complex]: ((B = (plus_plus_complex @ B @ A2)) = (A2 = zero_zero_complex))))). % add_0_iff
thf(fact_78_add__0__iff, axiom,
    ((![B : poly_complex, A2 : poly_complex]: ((B = (plus_p1547158847omplex @ B @ A2)) = (A2 = zero_z1746442943omplex))))). % add_0_iff
thf(fact_79_verit__sum__simplify, axiom,
    ((![A2 : complex]: ((plus_plus_complex @ A2 @ zero_zero_complex) = A2)))). % verit_sum_simplify
thf(fact_80_verit__sum__simplify, axiom,
    ((![A2 : poly_complex]: ((plus_p1547158847omplex @ A2 @ zero_z1746442943omplex) = A2)))). % verit_sum_simplify
thf(fact_81_is__zero__null, axiom,
    ((is_zero_complex = (^[P4 : poly_complex]: (P4 = zero_z1746442943omplex))))). % is_zero_null
thf(fact_82_pcompose__pCons, axiom,
    ((![A2 : complex, P : poly_complex, Q : poly_complex]: ((pcompose_complex @ (pCons_complex @ A2 @ P) @ Q) = (plus_p1547158847omplex @ (pCons_complex @ A2 @ zero_z1746442943omplex) @ (times_1246143675omplex @ Q @ (pcompose_complex @ P @ Q))))))). % pcompose_pCons
thf(fact_83_poly__cutoff__0, axiom,
    ((![N : nat]: ((poly_cutoff_complex @ N @ zero_z1746442943omplex) = zero_z1746442943omplex)))). % poly_cutoff_0
thf(fact_84_pcompose__0, axiom,
    ((![Q : poly_complex]: ((pcompose_complex @ zero_z1746442943omplex @ Q) = zero_z1746442943omplex)))). % pcompose_0
thf(fact_85_pcompose__const, axiom,
    ((![A2 : complex, Q : poly_complex]: ((pcompose_complex @ (pCons_complex @ A2 @ zero_z1746442943omplex) @ Q) = (pCons_complex @ A2 @ zero_z1746442943omplex))))). % pcompose_const
thf(fact_86_poly__pcompose, axiom,
    ((![P : poly_complex, Q : poly_complex, X2 : complex]: ((poly_complex2 @ (pcompose_complex @ P @ Q) @ X2) = (poly_complex2 @ P @ (poly_complex2 @ Q @ X2)))))). % poly_pcompose
thf(fact_87_pcompose__idR, axiom,
    ((![P : poly_complex]: ((pcompose_complex @ P @ (pCons_complex @ zero_zero_complex @ (pCons_complex @ one_one_complex @ zero_z1746442943omplex))) = P)))). % pcompose_idR
thf(fact_88_pcompose__idR, axiom,
    ((![P : poly_poly_complex]: ((pcompo1411605209omplex @ P @ (pCons_poly_complex @ zero_z1746442943omplex @ (pCons_poly_complex @ one_one_poly_complex @ zero_z1040703943omplex))) = P)))). % pcompose_idR
thf(fact_89_reflect__poly__at__0__eq__0__iff, axiom,
    ((![P : poly_complex]: (((poly_complex2 @ (reflect_poly_complex @ P) @ zero_zero_complex) = zero_zero_complex) = (P = zero_z1746442943omplex))))). % reflect_poly_at_0_eq_0_iff
thf(fact_90_reflect__poly__at__0__eq__0__iff, axiom,
    ((![P : poly_poly_complex]: (((poly_poly_complex2 @ (reflec309385472omplex @ P) @ zero_z1746442943omplex) = zero_z1746442943omplex) = (P = zero_z1040703943omplex))))). % reflect_poly_at_0_eq_0_iff
thf(fact_91_mult__pCons__right, axiom,
    ((![P : poly_complex, A2 : complex, Q : poly_complex]: ((times_1246143675omplex @ P @ (pCons_complex @ A2 @ Q)) = (plus_p1547158847omplex @ (smult_complex @ A2 @ P) @ (pCons_complex @ zero_zero_complex @ (times_1246143675omplex @ P @ Q))))))). % mult_pCons_right
thf(fact_92_mult__pCons__right, axiom,
    ((![P : poly_poly_complex, A2 : poly_complex, Q : poly_poly_complex]: ((times_1460995011omplex @ P @ (pCons_poly_complex @ A2 @ Q)) = (plus_p138939463omplex @ (smult_poly_complex @ A2 @ P) @ (pCons_poly_complex @ zero_z1746442943omplex @ (times_1460995011omplex @ P @ Q))))))). % mult_pCons_right
thf(fact_93_poly__1, axiom,
    ((![X2 : complex]: ((poly_complex2 @ one_one_poly_complex @ X2) = one_one_complex)))). % poly_1
thf(fact_94_smult__0__right, axiom,
    ((![A2 : complex]: ((smult_complex @ A2 @ zero_z1746442943omplex) = zero_z1746442943omplex)))). % smult_0_right
thf(fact_95_reflect__poly__0, axiom,
    (((reflect_poly_complex @ zero_z1746442943omplex) = zero_z1746442943omplex))). % reflect_poly_0
thf(fact_96_poly__cutoff__1, axiom,
    ((![N : nat]: (((N = zero_zero_nat) => ((poly_cutoff_complex @ N @ one_one_poly_complex) = zero_z1746442943omplex)) & ((~ ((N = zero_zero_nat))) => ((poly_cutoff_complex @ N @ one_one_poly_complex) = one_one_poly_complex)))))). % poly_cutoff_1
thf(fact_97_mult__cancel__left1, axiom,
    ((![C : complex, B : complex]: ((C = (times_times_complex @ C @ B)) = (((C = zero_zero_complex)) | ((B = one_one_complex))))))). % mult_cancel_left1
thf(fact_98_mult__cancel__left1, axiom,
    ((![C : poly_complex, B : poly_complex]: ((C = (times_1246143675omplex @ C @ B)) = (((C = zero_z1746442943omplex)) | ((B = one_one_poly_complex))))))). % mult_cancel_left1
thf(fact_99_mult__cancel__left2, axiom,
    ((![C : complex, A2 : complex]: (((times_times_complex @ C @ A2) = C) = (((C = zero_zero_complex)) | ((A2 = one_one_complex))))))). % mult_cancel_left2
thf(fact_100_mult__cancel__left2, axiom,
    ((![C : poly_complex, A2 : poly_complex]: (((times_1246143675omplex @ C @ A2) = C) = (((C = zero_z1746442943omplex)) | ((A2 = one_one_poly_complex))))))). % mult_cancel_left2
thf(fact_101_mult__cancel__right1, axiom,
    ((![C : complex, B : complex]: ((C = (times_times_complex @ B @ C)) = (((C = zero_zero_complex)) | ((B = one_one_complex))))))). % mult_cancel_right1
thf(fact_102_mult__cancel__right1, axiom,
    ((![C : poly_complex, B : poly_complex]: ((C = (times_1246143675omplex @ B @ C)) = (((C = zero_z1746442943omplex)) | ((B = one_one_poly_complex))))))). % mult_cancel_right1
thf(fact_103_mult__cancel__right2, axiom,
    ((![A2 : complex, C : complex]: (((times_times_complex @ A2 @ C) = C) = (((C = zero_zero_complex)) | ((A2 = one_one_complex))))))). % mult_cancel_right2
thf(fact_104_mult__cancel__right2, axiom,
    ((![A2 : poly_complex, C : poly_complex]: (((times_1246143675omplex @ A2 @ C) = C) = (((C = zero_z1746442943omplex)) | ((A2 = one_one_poly_complex))))))). % mult_cancel_right2
thf(fact_105_one__poly__eq__simps_I1_J, axiom,
    ((one_one_poly_complex = (pCons_complex @ one_one_complex @ zero_z1746442943omplex)))). % one_poly_eq_simps(1)
thf(fact_106_one__poly__eq__simps_I2_J, axiom,
    (((pCons_complex @ one_one_complex @ zero_z1746442943omplex) = one_one_poly_complex))). % one_poly_eq_simps(2)
thf(fact_107_smult__0__left, axiom,
    ((![P : poly_complex]: ((smult_complex @ zero_zero_complex @ P) = zero_z1746442943omplex)))). % smult_0_left
thf(fact_108_smult__0__left, axiom,
    ((![P : poly_poly_complex]: ((smult_poly_complex @ zero_z1746442943omplex @ P) = zero_z1040703943omplex)))). % smult_0_left
thf(fact_109_smult__eq__0__iff, axiom,
    ((![A2 : poly_complex, P : poly_poly_complex]: (((smult_poly_complex @ A2 @ P) = zero_z1040703943omplex) = (((A2 = zero_z1746442943omplex)) | ((P = zero_z1040703943omplex))))))). % smult_eq_0_iff
thf(fact_110_smult__eq__0__iff, axiom,
    ((![A2 : complex, P : poly_complex]: (((smult_complex @ A2 @ P) = zero_z1746442943omplex) = (((A2 = zero_zero_complex)) | ((P = zero_z1746442943omplex))))))). % smult_eq_0_iff
thf(fact_111_smult__pCons, axiom,
    ((![A2 : complex, B : complex, P : poly_complex]: ((smult_complex @ A2 @ (pCons_complex @ B @ P)) = (pCons_complex @ (times_times_complex @ A2 @ B) @ (smult_complex @ A2 @ P)))))). % smult_pCons
thf(fact_112_poly__smult, axiom,
    ((![A2 : complex, P : poly_complex, X2 : complex]: ((poly_complex2 @ (smult_complex @ A2 @ P) @ X2) = (times_times_complex @ A2 @ (poly_complex2 @ P @ X2)))))). % poly_smult
thf(fact_113_smult__one, axiom,
    ((![C : complex]: ((smult_complex @ C @ one_one_poly_complex) = (pCons_complex @ C @ zero_z1746442943omplex))))). % smult_one
thf(fact_114_reflect__poly__const, axiom,
    ((![A2 : complex]: ((reflect_poly_complex @ (pCons_complex @ A2 @ zero_z1746442943omplex)) = (pCons_complex @ A2 @ zero_z1746442943omplex))))). % reflect_poly_const
thf(fact_115_mult__pCons__left, axiom,
    ((![A2 : complex, P : poly_complex, Q : poly_complex]: ((times_1246143675omplex @ (pCons_complex @ A2 @ P) @ Q) = (plus_p1547158847omplex @ (smult_complex @ A2 @ Q) @ (pCons_complex @ zero_zero_complex @ (times_1246143675omplex @ P @ Q))))))). % mult_pCons_left
thf(fact_116_mult__pCons__left, axiom,
    ((![A2 : poly_complex, P : poly_poly_complex, Q : poly_poly_complex]: ((times_1460995011omplex @ (pCons_poly_complex @ A2 @ P) @ Q) = (plus_p138939463omplex @ (smult_poly_complex @ A2 @ Q) @ (pCons_poly_complex @ zero_z1746442943omplex @ (times_1460995011omplex @ P @ Q))))))). % mult_pCons_left
thf(fact_117_pCons__one, axiom,
    (((pCons_complex @ one_one_complex @ zero_z1746442943omplex) = one_one_poly_complex))). % pCons_one
thf(fact_118_zero__neq__one, axiom,
    ((~ ((zero_zero_complex = one_one_complex))))). % zero_neq_one
thf(fact_119_zero__neq__one, axiom,
    ((~ ((zero_z1746442943omplex = one_one_poly_complex))))). % zero_neq_one
thf(fact_120_synthetic__div__unique__lemma, axiom,
    ((![C : complex, P : poly_complex, A2 : complex]: (((smult_complex @ C @ P) = (pCons_complex @ A2 @ P)) => (P = zero_z1746442943omplex))))). % synthetic_div_unique_lemma
thf(fact_121_offset__poly__eq__0__lemma, axiom,
    ((![C : complex, P : poly_complex, A2 : complex]: (((plus_p1547158847omplex @ (smult_complex @ C @ P) @ (pCons_complex @ A2 @ P)) = zero_z1746442943omplex) => (P = zero_z1746442943omplex))))). % offset_poly_eq_0_lemma
thf(fact_122_offset__poly__pCons, axiom,
    ((![A2 : complex, P : poly_complex, H : complex]: ((fundam1201687030omplex @ (pCons_complex @ A2 @ P) @ H) = (plus_p1547158847omplex @ (smult_complex @ H @ (fundam1201687030omplex @ P @ H)) @ (pCons_complex @ A2 @ (fundam1201687030omplex @ P @ H))))))). % offset_poly_pCons
thf(fact_123_synthetic__div__correct, axiom,
    ((![P : poly_complex, C : complex]: ((plus_p1547158847omplex @ P @ (smult_complex @ C @ (synthe151143547omplex @ P @ C))) = (pCons_complex @ (poly_complex2 @ P @ C) @ (synthe151143547omplex @ P @ C)))))). % synthetic_div_correct
thf(fact_124_synthetic__div__unique, axiom,
    ((![P : poly_complex, C : complex, Q : poly_complex, R : complex]: (((plus_p1547158847omplex @ P @ (smult_complex @ C @ Q)) = (pCons_complex @ R @ Q)) => ((R = (poly_complex2 @ P @ C)) & (Q = (synthe151143547omplex @ P @ C))))))). % synthetic_div_unique
thf(fact_125_pseudo__mod_I1_J, axiom,
    ((![G : poly_poly_complex, F2 : poly_poly_complex]: ((~ ((G = zero_z1040703943omplex))) => (?[A3 : poly_complex, Q2 : poly_poly_complex]: ((~ ((A3 = zero_z1746442943omplex))) & ((smult_poly_complex @ A3 @ F2) = (plus_p138939463omplex @ (times_1460995011omplex @ G @ Q2) @ (pseudo2092547131omplex @ F2 @ G))))))))). % pseudo_mod(1)
thf(fact_126_pseudo__mod_I1_J, axiom,
    ((![G : poly_complex, F2 : poly_complex]: ((~ ((G = zero_z1746442943omplex))) => (?[A3 : complex, Q2 : poly_complex]: ((~ ((A3 = zero_zero_complex))) & ((smult_complex @ A3 @ F2) = (plus_p1547158847omplex @ (times_1246143675omplex @ G @ Q2) @ (pseudo_mod_complex @ F2 @ G))))))))). % pseudo_mod(1)
thf(fact_127_synthetic__div__correct_H, axiom,
    ((![C : complex, P : poly_complex]: ((plus_p1547158847omplex @ (times_1246143675omplex @ (pCons_complex @ (uminus1204672759omplex @ C) @ (pCons_complex @ one_one_complex @ zero_z1746442943omplex)) @ (synthe151143547omplex @ P @ C)) @ (pCons_complex @ (poly_complex2 @ P @ C) @ zero_z1746442943omplex)) = P)))). % synthetic_div_correct'
thf(fact_128_poly__root__induct, axiom,
    ((![Q3 : poly_poly_complex > $o, P3 : poly_complex > $o, P : poly_poly_complex]: ((Q3 @ zero_z1040703943omplex) => ((![P2 : poly_poly_complex]: ((![A : poly_complex]: ((P3 @ A) => (~ (((poly_poly_complex2 @ P2 @ A) = zero_z1746442943omplex))))) => (Q3 @ P2))) => ((![A3 : poly_complex, P2 : poly_poly_complex]: ((P3 @ A3) => ((Q3 @ P2) => (Q3 @ (times_1460995011omplex @ (pCons_poly_complex @ A3 @ (pCons_poly_complex @ (uminus1138659839omplex @ one_one_poly_complex) @ zero_z1040703943omplex)) @ P2))))) => (Q3 @ P))))))). % poly_root_induct
thf(fact_129_poly__root__induct, axiom,
    ((![Q3 : poly_complex > $o, P3 : complex > $o, P : poly_complex]: ((Q3 @ zero_z1746442943omplex) => ((![P2 : poly_complex]: ((![A : complex]: ((P3 @ A) => (~ (((poly_complex2 @ P2 @ A) = zero_zero_complex))))) => (Q3 @ P2))) => ((![A3 : complex, P2 : poly_complex]: ((P3 @ A3) => ((Q3 @ P2) => (Q3 @ (times_1246143675omplex @ (pCons_complex @ A3 @ (pCons_complex @ (uminus1204672759omplex @ one_one_complex) @ zero_z1746442943omplex)) @ P2))))) => (Q3 @ P))))))). % poly_root_induct
thf(fact_130_add_Oinverse__neutral, axiom,
    (((uminus1204672759omplex @ zero_zero_complex) = zero_zero_complex))). % add.inverse_neutral
thf(fact_131_add_Oinverse__neutral, axiom,
    (((uminus1138659839omplex @ zero_z1746442943omplex) = zero_z1746442943omplex))). % add.inverse_neutral
thf(fact_132_neg__0__equal__iff__equal, axiom,
    ((![A2 : complex]: ((zero_zero_complex = (uminus1204672759omplex @ A2)) = (zero_zero_complex = A2))))). % neg_0_equal_iff_equal
thf(fact_133_neg__0__equal__iff__equal, axiom,
    ((![A2 : poly_complex]: ((zero_z1746442943omplex = (uminus1138659839omplex @ A2)) = (zero_z1746442943omplex = A2))))). % neg_0_equal_iff_equal
thf(fact_134_neg__equal__0__iff__equal, axiom,
    ((![A2 : complex]: (((uminus1204672759omplex @ A2) = zero_zero_complex) = (A2 = zero_zero_complex))))). % neg_equal_0_iff_equal
thf(fact_135_neg__equal__0__iff__equal, axiom,
    ((![A2 : poly_complex]: (((uminus1138659839omplex @ A2) = zero_z1746442943omplex) = (A2 = zero_z1746442943omplex))))). % neg_equal_0_iff_equal

% Conjectures (1)
thf(conj_0, conjecture,
    ((~ ((fundam1158420650omplex @ (poly_complex2 @ (pCons_complex @ c @ cs))))))).
