% 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_182__5369348_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:27:43.535

% Could-be-implicit typings (2)
thf(ty_n_t__Num__Onum, type,
    num : $tType).
thf(ty_n_t__Nat__Onat, type,
    nat : $tType).

% Explicit typings (12)
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat, type,
    minus_minus_nat : nat > nat > nat).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat, type,
    times_times_nat : nat > nat > nat).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Num__Onum, type,
    times_times_num : num > num > num).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat, type,
    zero_zero_nat : nat).
thf(sy_c_Nat_OSuc, type,
    suc : nat > nat).
thf(sy_c_Num_Onum_OBit0, type,
    bit0 : num > num).
thf(sy_c_Num_Onum_OOne, type,
    one : num).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Nat__Onat, type,
    numeral_numeral_nat : num > nat).
thf(sy_c_Parity_Osemiring__bit__shifts__class_Opush__bit_001t__Nat__Onat, type,
    semiri2013084963it_nat : nat > nat > nat).
thf(sy_c_Rings_Odvd__class_Odvd_001t__Nat__Onat, type,
    dvd_dvd_nat : nat > nat > $o).
thf(sy_v_n, type,
    n : nat).
thf(sy_v_na____, type,
    na : nat).

% Relevant facts (134)
thf(fact_0__092_060open_062odd_An_092_060close_062, axiom,
    ((~ ((dvd_dvd_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ na))))). % \<open>odd n\<close>
thf(fact_1_n, axiom,
    ((~ ((na = zero_zero_nat))))). % n
thf(fact_2_assms_I2_J, axiom,
    ((~ ((n = zero_zero_nat))))). % assms(2)
thf(fact_3_Suc__double__not__eq__double, axiom,
    ((![M : nat, N : nat]: (~ (((suc @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ M)) = (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ N))))))). % Suc_double_not_eq_double
thf(fact_4_double__not__eq__Suc__double, axiom,
    ((![M : nat, N : nat]: (~ (((times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ M) = (suc @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ N)))))))). % double_not_eq_Suc_double
thf(fact_5_semiring__norm_I85_J, axiom,
    ((![M : num]: (~ (((bit0 @ M) = one)))))). % semiring_norm(85)
thf(fact_6_semiring__norm_I83_J, axiom,
    ((![N : num]: (~ ((one = (bit0 @ N))))))). % semiring_norm(83)
thf(fact_7_mult__numeral__left__semiring__numeral, axiom,
    ((![V : num, W : num, Z : nat]: ((times_times_nat @ (numeral_numeral_nat @ V) @ (times_times_nat @ (numeral_numeral_nat @ W) @ Z)) = (times_times_nat @ (numeral_numeral_nat @ (times_times_num @ V @ W)) @ Z))))). % mult_numeral_left_semiring_numeral
thf(fact_8_numeral__times__numeral, axiom,
    ((![M : num, N : num]: ((times_times_nat @ (numeral_numeral_nat @ M) @ (numeral_numeral_nat @ N)) = (numeral_numeral_nat @ (times_times_num @ M @ N)))))). % numeral_times_numeral
thf(fact_9_even__Suc, axiom,
    ((![N : nat]: ((dvd_dvd_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ (suc @ N)) = (~ ((dvd_dvd_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ N))))))). % even_Suc
thf(fact_10_even__Suc__Suc__iff, axiom,
    ((![N : nat]: ((dvd_dvd_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ (suc @ (suc @ N))) = (dvd_dvd_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ N))))). % even_Suc_Suc_iff
thf(fact_11_even__mult__iff, axiom,
    ((![A : nat, B : nat]: ((dvd_dvd_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ (times_times_nat @ A @ B)) = (((dvd_dvd_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ A)) | ((dvd_dvd_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ B))))))). % even_mult_iff
thf(fact_12_mult__numeral__1, axiom,
    ((![A : nat]: ((times_times_nat @ (numeral_numeral_nat @ one) @ A) = A)))). % mult_numeral_1
thf(fact_13_mult__numeral__1__right, axiom,
    ((![A : nat]: ((times_times_nat @ A @ (numeral_numeral_nat @ one)) = A)))). % mult_numeral_1_right
thf(fact_14_numeral__eq__iff, axiom,
    ((![M : num, N : num]: (((numeral_numeral_nat @ M) = (numeral_numeral_nat @ N)) = (M = N))))). % numeral_eq_iff
thf(fact_15_semiring__norm_I87_J, axiom,
    ((![M : num, N : num]: (((bit0 @ M) = (bit0 @ N)) = (M = N))))). % semiring_norm(87)
thf(fact_16_old_Onat_Oinject, axiom,
    ((![Nat : nat, Nat2 : nat]: (((suc @ Nat) = (suc @ Nat2)) = (Nat = Nat2))))). % old.nat.inject
thf(fact_17_nat_Oinject, axiom,
    ((![X2 : nat, Y2 : nat]: (((suc @ X2) = (suc @ Y2)) = (X2 = Y2))))). % nat.inject
thf(fact_18_mult__cancel2, axiom,
    ((![M : nat, K : nat, N : nat]: (((times_times_nat @ M @ K) = (times_times_nat @ N @ K)) = (((M = N)) | ((K = zero_zero_nat))))))). % mult_cancel2
thf(fact_19_mult__cancel1, axiom,
    ((![K : nat, M : nat, N : nat]: (((times_times_nat @ K @ M) = (times_times_nat @ K @ N)) = (((M = N)) | ((K = zero_zero_nat))))))). % mult_cancel1
thf(fact_20_mult__0__right, axiom,
    ((![M : nat]: ((times_times_nat @ M @ zero_zero_nat) = zero_zero_nat)))). % mult_0_right
thf(fact_21_mult__is__0, axiom,
    ((![M : nat, N : nat]: (((times_times_nat @ M @ N) = zero_zero_nat) = (((M = zero_zero_nat)) | ((N = zero_zero_nat))))))). % mult_is_0
thf(fact_22_semiring__norm_I13_J, axiom,
    ((![M : num, N : num]: ((times_times_num @ (bit0 @ M) @ (bit0 @ N)) = (bit0 @ (bit0 @ (times_times_num @ M @ N))))))). % semiring_norm(13)
thf(fact_23_semiring__norm_I11_J, axiom,
    ((![M : num]: ((times_times_num @ M @ one) = M)))). % semiring_norm(11)
thf(fact_24_semiring__norm_I12_J, axiom,
    ((![N : num]: ((times_times_num @ one @ N) = N)))). % semiring_norm(12)
thf(fact_25_one__eq__mult__iff, axiom,
    ((![M : nat, N : nat]: (((suc @ zero_zero_nat) = (times_times_nat @ M @ N)) = (((M = (suc @ zero_zero_nat))) & ((N = (suc @ zero_zero_nat)))))))). % one_eq_mult_iff
thf(fact_26_mult__eq__1__iff, axiom,
    ((![M : nat, N : nat]: (((times_times_nat @ M @ N) = (suc @ zero_zero_nat)) = (((M = (suc @ zero_zero_nat))) & ((N = (suc @ zero_zero_nat)))))))). % mult_eq_1_iff
thf(fact_27_dvd__1__iff__1, axiom,
    ((![M : nat]: ((dvd_dvd_nat @ M @ (suc @ zero_zero_nat)) = (M = (suc @ zero_zero_nat)))))). % dvd_1_iff_1
thf(fact_28_dvd__1__left, axiom,
    ((![K : nat]: (dvd_dvd_nat @ (suc @ zero_zero_nat) @ K)))). % dvd_1_left
thf(fact_29_nat__mult__dvd__cancel__disj, axiom,
    ((![K : nat, M : nat, N : nat]: ((dvd_dvd_nat @ (times_times_nat @ K @ M) @ (times_times_nat @ K @ N)) = (((K = zero_zero_nat)) | ((dvd_dvd_nat @ M @ N))))))). % nat_mult_dvd_cancel_disj
thf(fact_30_num__double, axiom,
    ((![N : num]: ((times_times_num @ (bit0 @ one) @ N) = (bit0 @ N))))). % num_double
thf(fact_31_dvd__antisym, axiom,
    ((![M : nat, N : nat]: ((dvd_dvd_nat @ M @ N) => ((dvd_dvd_nat @ N @ M) => (M = N)))))). % dvd_antisym
thf(fact_32_zero__neq__numeral, axiom,
    ((![N : num]: (~ ((zero_zero_nat = (numeral_numeral_nat @ N))))))). % zero_neq_numeral
thf(fact_33_not0__implies__Suc, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) => (?[M2 : nat]: (N = (suc @ M2))))))). % not0_implies_Suc
thf(fact_34_old_Onat_Oinducts, axiom,
    ((![P : nat > $o, Nat : nat]: ((P @ zero_zero_nat) => ((![Nat3 : nat]: ((P @ Nat3) => (P @ (suc @ Nat3)))) => (P @ Nat)))))). % old.nat.inducts
thf(fact_35_old_Onat_Oexhaust, axiom,
    ((![Y : nat]: ((~ ((Y = zero_zero_nat))) => (~ ((![Nat3 : nat]: (~ ((Y = (suc @ Nat3))))))))))). % old.nat.exhaust
thf(fact_36_Zero__not__Suc, axiom,
    ((![M : nat]: (~ ((zero_zero_nat = (suc @ M))))))). % Zero_not_Suc
thf(fact_37_Zero__neq__Suc, axiom,
    ((![M : nat]: (~ ((zero_zero_nat = (suc @ M))))))). % Zero_neq_Suc
thf(fact_38_Suc__neq__Zero, axiom,
    ((![M : nat]: (~ (((suc @ M) = zero_zero_nat)))))). % Suc_neq_Zero
thf(fact_39_zero__induct, axiom,
    ((![P : nat > $o, K : nat]: ((P @ K) => ((![N2 : nat]: ((P @ (suc @ N2)) => (P @ N2))) => (P @ zero_zero_nat)))))). % zero_induct
thf(fact_40_diff__induct, axiom,
    ((![P : nat > nat > $o, M : nat, N : nat]: ((![X : nat]: (P @ X @ zero_zero_nat)) => ((![Y3 : nat]: (P @ zero_zero_nat @ (suc @ Y3))) => ((![X : nat, Y3 : nat]: ((P @ X @ Y3) => (P @ (suc @ X) @ (suc @ Y3)))) => (P @ M @ N))))))). % diff_induct
thf(fact_41_nat__induct, axiom,
    ((![P : nat > $o, N : nat]: ((P @ zero_zero_nat) => ((![N2 : nat]: ((P @ N2) => (P @ (suc @ N2)))) => (P @ N)))))). % nat_induct
thf(fact_42_nat_OdiscI, axiom,
    ((![Nat : nat, X2 : nat]: ((Nat = (suc @ X2)) => (~ ((Nat = zero_zero_nat))))))). % nat.discI
thf(fact_43_old_Onat_Odistinct_I1_J, axiom,
    ((![Nat2 : nat]: (~ ((zero_zero_nat = (suc @ Nat2))))))). % old.nat.distinct(1)
thf(fact_44_old_Onat_Odistinct_I2_J, axiom,
    ((![Nat2 : nat]: (~ (((suc @ Nat2) = zero_zero_nat)))))). % old.nat.distinct(2)
thf(fact_45_nat_Odistinct_I1_J, axiom,
    ((![X2 : nat]: (~ ((zero_zero_nat = (suc @ X2))))))). % nat.distinct(1)
thf(fact_46_nat__mult__eq__cancel__disj, axiom,
    ((![K : nat, M : nat, N : nat]: (((times_times_nat @ K @ M) = (times_times_nat @ K @ N)) = (((K = zero_zero_nat)) | ((M = N))))))). % nat_mult_eq_cancel_disj
thf(fact_47_mult__0, axiom,
    ((![N : nat]: ((times_times_nat @ zero_zero_nat @ N) = zero_zero_nat)))). % mult_0
thf(fact_48_even__zero, axiom,
    ((dvd_dvd_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ zero_zero_nat))). % even_zero
thf(fact_49_numeral__1__eq__Suc__0, axiom,
    (((numeral_numeral_nat @ one) = (suc @ zero_zero_nat)))). % numeral_1_eq_Suc_0
thf(fact_50_numeral__2__eq__2, axiom,
    (((numeral_numeral_nat @ (bit0 @ one)) = (suc @ (suc @ zero_zero_nat))))). % numeral_2_eq_2
thf(fact_51_n__not__Suc__n, axiom,
    ((![N : nat]: (~ ((N = (suc @ N))))))). % n_not_Suc_n
thf(fact_52_Suc__inject, axiom,
    ((![X3 : nat, Y : nat]: (((suc @ X3) = (suc @ Y)) => (X3 = Y))))). % Suc_inject
thf(fact_53_even__numeral, axiom,
    ((![N : num]: (dvd_dvd_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ (numeral_numeral_nat @ (bit0 @ N)))))). % even_numeral
thf(fact_54_Suc__mult__cancel1, axiom,
    ((![K : nat, M : nat, N : nat]: (((times_times_nat @ (suc @ K) @ M) = (times_times_nat @ (suc @ K) @ N)) = (M = N))))). % Suc_mult_cancel1
thf(fact_55_evenE, axiom,
    ((![A : nat]: ((dvd_dvd_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ A) => (~ ((![B2 : nat]: (~ ((A = (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ B2))))))))))). % evenE
thf(fact_56_dvd__times__right__cancel__iff, axiom,
    ((![A : nat, B : nat, C : nat]: ((~ ((A = zero_zero_nat))) => ((dvd_dvd_nat @ (times_times_nat @ B @ A) @ (times_times_nat @ C @ A)) = (dvd_dvd_nat @ B @ C)))))). % dvd_times_right_cancel_iff
thf(fact_57_dvd__times__left__cancel__iff, axiom,
    ((![A : nat, B : nat, C : nat]: ((~ ((A = zero_zero_nat))) => ((dvd_dvd_nat @ (times_times_nat @ A @ B) @ (times_times_nat @ A @ C)) = (dvd_dvd_nat @ B @ C)))))). % dvd_times_left_cancel_iff
thf(fact_58_dvd__0__right, axiom,
    ((![A : nat]: (dvd_dvd_nat @ A @ zero_zero_nat)))). % dvd_0_right
thf(fact_59_dvd__0__left__iff, axiom,
    ((![A : nat]: ((dvd_dvd_nat @ zero_zero_nat @ A) = (A = zero_zero_nat))))). % dvd_0_left_iff
thf(fact_60_mult__cancel__right, axiom,
    ((![A : nat, C : nat, B : nat]: (((times_times_nat @ A @ C) = (times_times_nat @ B @ C)) = (((C = zero_zero_nat)) | ((A = B))))))). % mult_cancel_right
thf(fact_61_mult__cancel__left, axiom,
    ((![C : nat, A : nat, B : nat]: (((times_times_nat @ C @ A) = (times_times_nat @ C @ B)) = (((C = zero_zero_nat)) | ((A = B))))))). % mult_cancel_left
thf(fact_62_mult__eq__0__iff, axiom,
    ((![A : nat, B : nat]: (((times_times_nat @ A @ B) = zero_zero_nat) = (((A = zero_zero_nat)) | ((B = zero_zero_nat))))))). % mult_eq_0_iff
thf(fact_63_mult__zero__right, axiom,
    ((![A : nat]: ((times_times_nat @ A @ zero_zero_nat) = zero_zero_nat)))). % mult_zero_right
thf(fact_64_mult__zero__left, axiom,
    ((![A : nat]: ((times_times_nat @ zero_zero_nat @ A) = zero_zero_nat)))). % mult_zero_left
thf(fact_65_dvd__trans, axiom,
    ((![A : nat, B : nat, C : nat]: ((dvd_dvd_nat @ A @ B) => ((dvd_dvd_nat @ B @ C) => (dvd_dvd_nat @ A @ C)))))). % dvd_trans
thf(fact_66_dvd__refl, axiom,
    ((![A : nat]: (dvd_dvd_nat @ A @ A)))). % dvd_refl
thf(fact_67_mult__not__zero, axiom,
    ((![A : nat, B : nat]: ((~ (((times_times_nat @ A @ B) = zero_zero_nat))) => ((~ ((A = zero_zero_nat))) & (~ ((B = zero_zero_nat)))))))). % mult_not_zero
thf(fact_68_divisors__zero, axiom,
    ((![A : nat, B : nat]: (((times_times_nat @ A @ B) = zero_zero_nat) => ((A = zero_zero_nat) | (B = zero_zero_nat)))))). % divisors_zero
thf(fact_69_no__zero__divisors, axiom,
    ((![A : nat, B : nat]: ((~ ((A = zero_zero_nat))) => ((~ ((B = zero_zero_nat))) => (~ (((times_times_nat @ A @ B) = zero_zero_nat)))))))). % no_zero_divisors
thf(fact_70_mult__left__cancel, axiom,
    ((![C : nat, A : nat, B : nat]: ((~ ((C = zero_zero_nat))) => (((times_times_nat @ C @ A) = (times_times_nat @ C @ B)) = (A = B)))))). % mult_left_cancel
thf(fact_71_mult__right__cancel, axiom,
    ((![C : nat, A : nat, B : nat]: ((~ ((C = zero_zero_nat))) => (((times_times_nat @ A @ C) = (times_times_nat @ B @ C)) = (A = B)))))). % mult_right_cancel
thf(fact_72_dvd__0__left, axiom,
    ((![A : nat]: ((dvd_dvd_nat @ zero_zero_nat @ A) => (A = zero_zero_nat))))). % dvd_0_left
thf(fact_73_dvdE, axiom,
    ((![B : nat, A : nat]: ((dvd_dvd_nat @ B @ A) => (~ ((![K2 : nat]: (~ ((A = (times_times_nat @ B @ K2))))))))))). % dvdE
thf(fact_74_dvdI, axiom,
    ((![A : nat, B : nat, K : nat]: ((A = (times_times_nat @ B @ K)) => (dvd_dvd_nat @ B @ A))))). % dvdI
thf(fact_75_dvd__def, axiom,
    ((dvd_dvd_nat = (^[B3 : nat]: (^[A2 : nat]: (?[K3 : nat]: (A2 = (times_times_nat @ B3 @ K3)))))))). % dvd_def
thf(fact_76_dvd__mult, axiom,
    ((![A : nat, C : nat, B : nat]: ((dvd_dvd_nat @ A @ C) => (dvd_dvd_nat @ A @ (times_times_nat @ B @ C)))))). % dvd_mult
thf(fact_77_dvd__mult2, axiom,
    ((![A : nat, B : nat, C : nat]: ((dvd_dvd_nat @ A @ B) => (dvd_dvd_nat @ A @ (times_times_nat @ B @ C)))))). % dvd_mult2
thf(fact_78_dvd__mult__left, axiom,
    ((![A : nat, B : nat, C : nat]: ((dvd_dvd_nat @ (times_times_nat @ A @ B) @ C) => (dvd_dvd_nat @ A @ C))))). % dvd_mult_left
thf(fact_79_dvd__triv__left, axiom,
    ((![A : nat, B : nat]: (dvd_dvd_nat @ A @ (times_times_nat @ A @ B))))). % dvd_triv_left
thf(fact_80_mult__dvd__mono, axiom,
    ((![A : nat, B : nat, C : nat, D : nat]: ((dvd_dvd_nat @ A @ B) => ((dvd_dvd_nat @ C @ D) => (dvd_dvd_nat @ (times_times_nat @ A @ C) @ (times_times_nat @ B @ D))))))). % mult_dvd_mono
thf(fact_81_dvd__mult__right, axiom,
    ((![A : nat, B : nat, C : nat]: ((dvd_dvd_nat @ (times_times_nat @ A @ B) @ C) => (dvd_dvd_nat @ B @ C))))). % dvd_mult_right
thf(fact_82_dvd__triv__right, axiom,
    ((![A : nat, B : nat]: (dvd_dvd_nat @ A @ (times_times_nat @ B @ A))))). % dvd_triv_right
thf(fact_83_verit__eq__simplify_I8_J, axiom,
    ((![X2 : num, Y2 : num]: (((bit0 @ X2) = (bit0 @ Y2)) = (X2 = Y2))))). % verit_eq_simplify(8)
thf(fact_84_even__push__bit__iff, axiom,
    ((![N : nat, A : nat]: ((dvd_dvd_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ (semiri2013084963it_nat @ N @ A)) = (((~ ((N = zero_zero_nat)))) | ((dvd_dvd_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ A))))))). % even_push_bit_iff
thf(fact_85_push__bit__Suc, axiom,
    ((![N : nat, A : nat]: ((semiri2013084963it_nat @ (suc @ N) @ A) = (semiri2013084963it_nat @ N @ (times_times_nat @ A @ (numeral_numeral_nat @ (bit0 @ one)))))))). % push_bit_Suc
thf(fact_86_push__bit__of__0, axiom,
    ((![N : nat]: ((semiri2013084963it_nat @ N @ zero_zero_nat) = zero_zero_nat)))). % push_bit_of_0
thf(fact_87_push__bit__eq__0__iff, axiom,
    ((![N : nat, A : nat]: (((semiri2013084963it_nat @ N @ A) = zero_zero_nat) = (A = zero_zero_nat))))). % push_bit_eq_0_iff
thf(fact_88_zero__reorient, axiom,
    ((![X3 : nat]: ((zero_zero_nat = X3) = (X3 = zero_zero_nat))))). % zero_reorient
thf(fact_89_mult_Oleft__commute, axiom,
    ((![B : nat, A : nat, C : nat]: ((times_times_nat @ B @ (times_times_nat @ A @ C)) = (times_times_nat @ A @ (times_times_nat @ B @ C)))))). % mult.left_commute
thf(fact_90_mult_Ocommute, axiom,
    ((times_times_nat = (^[A2 : nat]: (^[B3 : nat]: (times_times_nat @ B3 @ A2)))))). % mult.commute
thf(fact_91_mult_Oassoc, axiom,
    ((![A : nat, B : nat, C : nat]: ((times_times_nat @ (times_times_nat @ A @ B) @ C) = (times_times_nat @ A @ (times_times_nat @ B @ C)))))). % mult.assoc
thf(fact_92_ab__semigroup__mult__class_Omult__ac_I1_J, axiom,
    ((![A : nat, B : nat, C : nat]: ((times_times_nat @ (times_times_nat @ A @ B) @ C) = (times_times_nat @ A @ (times_times_nat @ B @ C)))))). % ab_semigroup_mult_class.mult_ac(1)
thf(fact_93_push__bit__double, axiom,
    ((![N : nat, A : nat]: ((semiri2013084963it_nat @ N @ (times_times_nat @ A @ (numeral_numeral_nat @ (bit0 @ one)))) = (times_times_nat @ (semiri2013084963it_nat @ N @ A) @ (numeral_numeral_nat @ (bit0 @ one))))))). % push_bit_double
thf(fact_94_verit__eq__simplify_I10_J, axiom,
    ((![X2 : num]: (~ ((one = (bit0 @ X2))))))). % verit_eq_simplify(10)
thf(fact_95_odd__Suc__minus__one, axiom,
    ((![N : nat]: ((~ ((dvd_dvd_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ N))) => ((suc @ (minus_minus_nat @ N @ (suc @ zero_zero_nat))) = N))))). % odd_Suc_minus_one
thf(fact_96_gcd__nat_Oextremum, axiom,
    ((![A : nat]: (dvd_dvd_nat @ A @ zero_zero_nat)))). % gcd_nat.extremum
thf(fact_97_gcd__nat_Oextremum__strict, axiom,
    ((![A : nat]: (~ (((dvd_dvd_nat @ zero_zero_nat @ A) & (~ ((zero_zero_nat = A))))))))). % gcd_nat.extremum_strict
thf(fact_98_cancel__comm__monoid__add__class_Odiff__cancel, axiom,
    ((![A : nat]: ((minus_minus_nat @ A @ A) = zero_zero_nat)))). % cancel_comm_monoid_add_class.diff_cancel
thf(fact_99_diff__zero, axiom,
    ((![A : nat]: ((minus_minus_nat @ A @ zero_zero_nat) = A)))). % diff_zero
thf(fact_100_zero__diff, axiom,
    ((![A : nat]: ((minus_minus_nat @ zero_zero_nat @ A) = zero_zero_nat)))). % zero_diff
thf(fact_101_diff__0__eq__0, axiom,
    ((![N : nat]: ((minus_minus_nat @ zero_zero_nat @ N) = zero_zero_nat)))). % diff_0_eq_0
thf(fact_102_diff__self__eq__0, axiom,
    ((![M : nat]: ((minus_minus_nat @ M @ M) = zero_zero_nat)))). % diff_self_eq_0
thf(fact_103_Suc__diff__diff, axiom,
    ((![M : nat, N : nat, K : nat]: ((minus_minus_nat @ (minus_minus_nat @ (suc @ M) @ N) @ (suc @ K)) = (minus_minus_nat @ (minus_minus_nat @ M @ N) @ K))))). % Suc_diff_diff
thf(fact_104_diff__Suc__Suc, axiom,
    ((![M : nat, N : nat]: ((minus_minus_nat @ (suc @ M) @ (suc @ N)) = (minus_minus_nat @ M @ N))))). % diff_Suc_Suc
thf(fact_105_cancel__ab__semigroup__add__class_Odiff__right__commute, axiom,
    ((![A : nat, C : nat, B : nat]: ((minus_minus_nat @ (minus_minus_nat @ A @ C) @ B) = (minus_minus_nat @ (minus_minus_nat @ A @ B) @ C))))). % cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_106_bezout1__nat, axiom,
    ((![A : nat, B : nat]: (?[D2 : nat, X : nat, Y3 : nat]: ((dvd_dvd_nat @ D2 @ A) & ((dvd_dvd_nat @ D2 @ B) & (((minus_minus_nat @ (times_times_nat @ A @ X) @ (times_times_nat @ B @ Y3)) = D2) | ((minus_minus_nat @ (times_times_nat @ B @ X) @ (times_times_nat @ A @ Y3)) = D2)))))))). % bezout1_nat
thf(fact_107_diff__commute, axiom,
    ((![I : nat, J : nat, K : nat]: ((minus_minus_nat @ (minus_minus_nat @ I @ J) @ K) = (minus_minus_nat @ (minus_minus_nat @ I @ K) @ J))))). % diff_commute
thf(fact_108_dvd__diff__nat, axiom,
    ((![K : nat, M : nat, N : nat]: ((dvd_dvd_nat @ K @ M) => ((dvd_dvd_nat @ K @ N) => (dvd_dvd_nat @ K @ (minus_minus_nat @ M @ N))))))). % dvd_diff_nat
thf(fact_109_diff__mult__distrib2, axiom,
    ((![K : nat, M : nat, N : nat]: ((times_times_nat @ K @ (minus_minus_nat @ M @ N)) = (minus_minus_nat @ (times_times_nat @ K @ M) @ (times_times_nat @ K @ N)))))). % diff_mult_distrib2
thf(fact_110_diff__mult__distrib, axiom,
    ((![M : nat, N : nat, K : nat]: ((times_times_nat @ (minus_minus_nat @ M @ N) @ K) = (minus_minus_nat @ (times_times_nat @ M @ K) @ (times_times_nat @ N @ K)))))). % diff_mult_distrib
thf(fact_111_zero__induct__lemma, axiom,
    ((![P : nat > $o, K : nat, I : nat]: ((P @ K) => ((![N2 : nat]: ((P @ (suc @ N2)) => (P @ N2))) => (P @ (minus_minus_nat @ K @ I))))))). % zero_induct_lemma
thf(fact_112_diffs0__imp__equal, axiom,
    ((![M : nat, N : nat]: (((minus_minus_nat @ M @ N) = zero_zero_nat) => (((minus_minus_nat @ N @ M) = zero_zero_nat) => (M = N)))))). % diffs0_imp_equal
thf(fact_113_minus__nat_Odiff__0, axiom,
    ((![M : nat]: ((minus_minus_nat @ M @ zero_zero_nat) = M)))). % minus_nat.diff_0
thf(fact_114_right__diff__distrib_H, axiom,
    ((![A : nat, B : nat, C : nat]: ((times_times_nat @ A @ (minus_minus_nat @ B @ C)) = (minus_minus_nat @ (times_times_nat @ A @ B) @ (times_times_nat @ A @ C)))))). % right_diff_distrib'
thf(fact_115_left__diff__distrib_H, axiom,
    ((![B : nat, C : nat, A : nat]: ((times_times_nat @ (minus_minus_nat @ B @ C) @ A) = (minus_minus_nat @ (times_times_nat @ B @ A) @ (times_times_nat @ C @ A)))))). % left_diff_distrib'
thf(fact_116_gcd__nat_Onot__eq__order__implies__strict, axiom,
    ((![A : nat, B : nat]: ((~ ((A = B))) => ((dvd_dvd_nat @ A @ B) => ((dvd_dvd_nat @ A @ B) & (~ ((A = B))))))))). % gcd_nat.not_eq_order_implies_strict
thf(fact_117_gcd__nat_Ostrict__implies__not__eq, axiom,
    ((![A : nat, B : nat]: (((dvd_dvd_nat @ A @ B) & (~ ((A = B)))) => (~ ((A = B))))))). % gcd_nat.strict_implies_not_eq
thf(fact_118_gcd__nat_Ostrict__implies__order, axiom,
    ((![A : nat, B : nat]: (((dvd_dvd_nat @ A @ B) & (~ ((A = B)))) => (dvd_dvd_nat @ A @ B))))). % gcd_nat.strict_implies_order
thf(fact_119_gcd__nat_Ostrict__iff__order, axiom,
    ((![A : nat, B : nat]: ((((dvd_dvd_nat @ A @ B)) & ((~ ((A = B))))) = (((dvd_dvd_nat @ A @ B)) & ((~ ((A = B))))))))). % gcd_nat.strict_iff_order
thf(fact_120_gcd__nat_Oorder__iff__strict, axiom,
    ((dvd_dvd_nat = (^[A2 : nat]: (^[B3 : nat]: (((((dvd_dvd_nat @ A2 @ B3)) & ((~ ((A2 = B3)))))) | ((A2 = B3)))))))). % gcd_nat.order_iff_strict
thf(fact_121_gcd__nat_Ostrict__trans2, axiom,
    ((![A : nat, B : nat, C : nat]: (((dvd_dvd_nat @ A @ B) & (~ ((A = B)))) => ((dvd_dvd_nat @ B @ C) => ((dvd_dvd_nat @ A @ C) & (~ ((A = C))))))))). % gcd_nat.strict_trans2
thf(fact_122_gcd__nat_Ostrict__trans1, axiom,
    ((![A : nat, B : nat, C : nat]: ((dvd_dvd_nat @ A @ B) => (((dvd_dvd_nat @ B @ C) & (~ ((B = C)))) => ((dvd_dvd_nat @ A @ C) & (~ ((A = C))))))))). % gcd_nat.strict_trans1
thf(fact_123_gcd__nat_Ostrict__trans, axiom,
    ((![A : nat, B : nat, C : nat]: (((dvd_dvd_nat @ A @ B) & (~ ((A = B)))) => (((dvd_dvd_nat @ B @ C) & (~ ((B = C)))) => ((dvd_dvd_nat @ A @ C) & (~ ((A = C))))))))). % gcd_nat.strict_trans
thf(fact_124_gcd__nat_Oantisym, axiom,
    ((![A : nat, B : nat]: ((dvd_dvd_nat @ A @ B) => ((dvd_dvd_nat @ B @ A) => (A = B)))))). % gcd_nat.antisym
thf(fact_125_gcd__nat_Oirrefl, axiom,
    ((![A : nat]: (~ (((dvd_dvd_nat @ A @ A) & (~ ((A = A))))))))). % gcd_nat.irrefl
thf(fact_126_gcd__nat_Oeq__iff, axiom,
    (((^[Y4 : nat]: (^[Z2 : nat]: (Y4 = Z2))) = (^[A2 : nat]: (^[B3 : nat]: (((dvd_dvd_nat @ A2 @ B3)) & ((dvd_dvd_nat @ B3 @ A2)))))))). % gcd_nat.eq_iff
thf(fact_127_gcd__nat_Otrans, axiom,
    ((![A : nat, B : nat, C : nat]: ((dvd_dvd_nat @ A @ B) => ((dvd_dvd_nat @ B @ C) => (dvd_dvd_nat @ A @ C)))))). % gcd_nat.trans
thf(fact_128_gcd__nat_Orefl, axiom,
    ((![A : nat]: (dvd_dvd_nat @ A @ A)))). % gcd_nat.refl
thf(fact_129_gcd__nat_Oasym, axiom,
    ((![A : nat, B : nat]: (((dvd_dvd_nat @ A @ B) & (~ ((A = B)))) => (~ (((dvd_dvd_nat @ B @ A) & (~ ((B = A)))))))))). % gcd_nat.asym
thf(fact_130_dvd__productE, axiom,
    ((![P2 : nat, A : nat, B : nat]: ((dvd_dvd_nat @ P2 @ (times_times_nat @ A @ B)) => (~ ((![X : nat, Y3 : nat]: ((P2 = (times_times_nat @ X @ Y3)) => ((dvd_dvd_nat @ X @ A) => (~ ((dvd_dvd_nat @ Y3 @ B)))))))))))). % dvd_productE
thf(fact_131_division__decomp, axiom,
    ((![A : nat, B : nat, C : nat]: ((dvd_dvd_nat @ A @ (times_times_nat @ B @ C)) => (?[B4 : nat, C2 : nat]: ((A = (times_times_nat @ B4 @ C2)) & ((dvd_dvd_nat @ B4 @ B) & (dvd_dvd_nat @ C2 @ C)))))))). % division_decomp
thf(fact_132_gcd__nat_Oextremum__uniqueI, axiom,
    ((![A : nat]: ((dvd_dvd_nat @ zero_zero_nat @ A) => (A = zero_zero_nat))))). % gcd_nat.extremum_uniqueI
thf(fact_133_gcd__nat_Onot__eq__extremum, axiom,
    ((![A : nat]: ((~ ((A = zero_zero_nat))) = (((dvd_dvd_nat @ A @ zero_zero_nat)) & ((~ ((A = zero_zero_nat))))))))). % gcd_nat.not_eq_extremum

% Conjectures (1)
thf(conj_0, conjecture,
    ((?[M3 : nat]: (na = (suc @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ M3)))))).
