% 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_169__5369240_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:27:36.153

% Could-be-implicit typings (3)
thf(ty_n_t__Num__Onum, type,
    num : $tType).
thf(ty_n_t__Nat__Onat, type,
    nat : $tType).
thf(ty_n_t__Int__Oint, type,
    int : $tType).

% Explicit typings (27)
thf(sy_c_Groups_Otimes__class_Otimes_001t__Int__Oint, type,
    times_times_int : int > int > int).
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__Int__Oint, type,
    zero_zero_int : int).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat, type,
    zero_zero_nat : nat).
thf(sy_c_If_001t__Nat__Onat, type,
    if_nat : $o > nat > nat > nat).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint, type,
    semiri2019852685at_int : nat > int).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat, type,
    semiri1382578993at_nat : 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__Int__Oint, type,
    numeral_numeral_int : num > int).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Nat__Onat, type,
    numeral_numeral_nat : num > nat).
thf(sy_c_Num_Opred__numeral, type,
    pred_numeral : num > nat).
thf(sy_c_Parity_Osemiring__bit__shifts__class_Opush__bit_001t__Int__Oint, type,
    semiri502875007it_int : nat > int > int).
thf(sy_c_Parity_Osemiring__bit__shifts__class_Opush__bit_001t__Nat__Onat, type,
    semiri2013084963it_nat : nat > nat > nat).
thf(sy_c_Parity_Osemiring__bit__shifts__class_Otake__bit_001t__Int__Oint, type,
    semiri1605039314it_int : nat > int > int).
thf(sy_c_Parity_Osemiring__bit__shifts__class_Otake__bit_001t__Nat__Onat, type,
    semiri967765622it_nat : nat > nat > nat).
thf(sy_c_Parity_Osemiring__bits__class_Obit_001t__Int__Oint, type,
    semiri1309798031it_int : int > nat > $o).
thf(sy_c_Parity_Osemiring__bits__class_Obit_001t__Nat__Onat, type,
    semiri672524339it_nat : nat > nat > $o).
thf(sy_c_Power_Opower__class_Opower_001t__Int__Oint, type,
    power_power_int : int > nat > int).
thf(sy_c_Power_Opower__class_Opower_001t__Nat__Onat, type,
    power_power_nat : nat > nat > nat).
thf(sy_c_Rings_Odvd__class_Odvd_001t__Int__Oint, type,
    dvd_dvd_int : int > int > $o).
thf(sy_c_Rings_Odvd__class_Odvd_001t__Nat__Onat, type,
    dvd_dvd_nat : nat > nat > $o).
thf(sy_c_Rings_Omodulo__class_Omodulo_001t__Int__Oint, type,
    modulo_modulo_int : int > int > int).
thf(sy_c_Rings_Omodulo__class_Omodulo_001t__Nat__Onat, type,
    modulo_modulo_nat : nat > nat > nat).
thf(sy_v_n, type,
    n : nat).
thf(sy_v_na____, type,
    na : nat).

% Relevant facts (226)
thf(fact_0__092_060open_062even_An_092_060close_062, axiom,
    ((dvd_dvd_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ na))). % \<open>even 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_semiring__norm_I85_J, axiom,
    ((![M : num]: (~ (((bit0 @ M) = one)))))). % semiring_norm(85)
thf(fact_4_semiring__norm_I83_J, axiom,
    ((![N : num]: (~ ((one = (bit0 @ N))))))). % semiring_norm(83)
thf(fact_5_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_6_mult__numeral__left__semiring__numeral, axiom,
    ((![V : num, W : num, Z : int]: ((times_times_int @ (numeral_numeral_int @ V) @ (times_times_int @ (numeral_numeral_int @ W) @ Z)) = (times_times_int @ (numeral_numeral_int @ (times_times_num @ V @ W)) @ Z))))). % mult_numeral_left_semiring_numeral
thf(fact_7_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_8_numeral__times__numeral, axiom,
    ((![M : num, N : num]: ((times_times_int @ (numeral_numeral_int @ M) @ (numeral_numeral_int @ N)) = (numeral_numeral_int @ (times_times_num @ M @ N)))))). % numeral_times_numeral
thf(fact_9_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_10_even__mult__iff, axiom,
    ((![A : int, B : int]: ((dvd_dvd_int @ (numeral_numeral_int @ (bit0 @ one)) @ (times_times_int @ A @ B)) = (((dvd_dvd_int @ (numeral_numeral_int @ (bit0 @ one)) @ A)) | ((dvd_dvd_int @ (numeral_numeral_int @ (bit0 @ one)) @ B))))))). % even_mult_iff
thf(fact_11_mult__numeral__1, axiom,
    ((![A : nat]: ((times_times_nat @ (numeral_numeral_nat @ one) @ A) = A)))). % mult_numeral_1
thf(fact_12_mult__numeral__1, axiom,
    ((![A : int]: ((times_times_int @ (numeral_numeral_int @ 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_mult__numeral__1__right, axiom,
    ((![A : int]: ((times_times_int @ A @ (numeral_numeral_int @ one)) = A)))). % mult_numeral_1_right
thf(fact_15_verit__eq__simplify_I8_J, axiom,
    ((![X2 : num, Y2 : num]: (((bit0 @ X2) = (bit0 @ Y2)) = (X2 = Y2))))). % verit_eq_simplify(8)
thf(fact_16_semiring__norm_I87_J, axiom,
    ((![M : num, N : num]: (((bit0 @ M) = (bit0 @ N)) = (M = N))))). % semiring_norm(87)
thf(fact_17_numeral__eq__iff, axiom,
    ((![M : num, N : num]: (((numeral_numeral_nat @ M) = (numeral_numeral_nat @ N)) = (M = N))))). % numeral_eq_iff
thf(fact_18_numeral__eq__iff, axiom,
    ((![M : num, N : num]: (((numeral_numeral_int @ M) = (numeral_numeral_int @ N)) = (M = N))))). % numeral_eq_iff
thf(fact_19_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_20_evenE, axiom,
    ((![A : int]: ((dvd_dvd_int @ (numeral_numeral_int @ (bit0 @ one)) @ A) => (~ ((![B2 : int]: (~ ((A = (times_times_int @ (numeral_numeral_int @ (bit0 @ one)) @ B2))))))))))). % evenE
thf(fact_21_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_22_semiring__norm_I11_J, axiom,
    ((![M : num]: ((times_times_num @ M @ one) = M)))). % semiring_norm(11)
thf(fact_23_semiring__norm_I12_J, axiom,
    ((![N : num]: ((times_times_num @ one @ N) = N)))). % semiring_norm(12)
thf(fact_24_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_25_num__double, axiom,
    ((![N : num]: ((times_times_num @ (bit0 @ one) @ N) = (bit0 @ N))))). % num_double
thf(fact_26_zero__neq__numeral, axiom,
    ((![N : num]: (~ ((zero_zero_nat = (numeral_numeral_nat @ N))))))). % zero_neq_numeral
thf(fact_27_zero__neq__numeral, axiom,
    ((![N : num]: (~ ((zero_zero_int = (numeral_numeral_int @ N))))))). % zero_neq_numeral
thf(fact_28_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_29_even__zero, axiom,
    ((dvd_dvd_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ zero_zero_nat))). % even_zero
thf(fact_30_even__zero, axiom,
    ((dvd_dvd_int @ (numeral_numeral_int @ (bit0 @ one)) @ zero_zero_int))). % even_zero
thf(fact_31_even__numeral, axiom,
    ((![N : num]: (dvd_dvd_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ (numeral_numeral_nat @ (bit0 @ N)))))). % even_numeral
thf(fact_32_even__numeral, axiom,
    ((![N : num]: (dvd_dvd_int @ (numeral_numeral_int @ (bit0 @ one)) @ (numeral_numeral_int @ (bit0 @ N)))))). % even_numeral
thf(fact_33_verit__eq__simplify_I10_J, axiom,
    ((![X2 : num]: (~ ((one = (bit0 @ X2))))))). % verit_eq_simplify(10)
thf(fact_34_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_35_dvd__times__right__cancel__iff, axiom,
    ((![A : int, B : int, C : int]: ((~ ((A = zero_zero_int))) => ((dvd_dvd_int @ (times_times_int @ B @ A) @ (times_times_int @ C @ A)) = (dvd_dvd_int @ B @ C)))))). % dvd_times_right_cancel_iff
thf(fact_36_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_37_dvd__times__left__cancel__iff, axiom,
    ((![A : int, B : int, C : int]: ((~ ((A = zero_zero_int))) => ((dvd_dvd_int @ (times_times_int @ A @ B) @ (times_times_int @ A @ C)) = (dvd_dvd_int @ B @ C)))))). % dvd_times_left_cancel_iff
thf(fact_38_dvd__mult__cancel__right, axiom,
    ((![A : int, C : int, B : int]: ((dvd_dvd_int @ (times_times_int @ A @ C) @ (times_times_int @ B @ C)) = (((C = zero_zero_int)) | ((dvd_dvd_int @ A @ B))))))). % dvd_mult_cancel_right
thf(fact_39_dvd__mult__cancel__left, axiom,
    ((![C : int, A : int, B : int]: ((dvd_dvd_int @ (times_times_int @ C @ A) @ (times_times_int @ C @ B)) = (((C = zero_zero_int)) | ((dvd_dvd_int @ A @ B))))))). % dvd_mult_cancel_left
thf(fact_40_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_41_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_42_mult__0__right, axiom,
    ((![M : nat]: ((times_times_nat @ M @ zero_zero_nat) = zero_zero_nat)))). % mult_0_right
thf(fact_43_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_44_dvd__0__right, axiom,
    ((![A : nat]: (dvd_dvd_nat @ A @ zero_zero_nat)))). % dvd_0_right
thf(fact_45_dvd__0__right, axiom,
    ((![A : int]: (dvd_dvd_int @ A @ zero_zero_int)))). % dvd_0_right
thf(fact_46_dvd__0__left__iff, axiom,
    ((![A : nat]: ((dvd_dvd_nat @ zero_zero_nat @ A) = (A = zero_zero_nat))))). % dvd_0_left_iff
thf(fact_47_dvd__0__left__iff, axiom,
    ((![A : int]: ((dvd_dvd_int @ zero_zero_int @ A) = (A = zero_zero_int))))). % dvd_0_left_iff
thf(fact_48_mult__zero__left, axiom,
    ((![A : nat]: ((times_times_nat @ zero_zero_nat @ A) = zero_zero_nat)))). % mult_zero_left
thf(fact_49_mult__zero__left, axiom,
    ((![A : int]: ((times_times_int @ zero_zero_int @ A) = zero_zero_int)))). % mult_zero_left
thf(fact_50_mult__zero__right, axiom,
    ((![A : nat]: ((times_times_nat @ A @ zero_zero_nat) = zero_zero_nat)))). % mult_zero_right
thf(fact_51_mult__zero__right, axiom,
    ((![A : int]: ((times_times_int @ A @ zero_zero_int) = zero_zero_int)))). % mult_zero_right
thf(fact_52_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_53_mult__eq__0__iff, axiom,
    ((![A : int, B : int]: (((times_times_int @ A @ B) = zero_zero_int) = (((A = zero_zero_int)) | ((B = zero_zero_int))))))). % mult_eq_0_iff
thf(fact_54_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_55_mult__cancel__left, axiom,
    ((![C : int, A : int, B : int]: (((times_times_int @ C @ A) = (times_times_int @ C @ B)) = (((C = zero_zero_int)) | ((A = B))))))). % mult_cancel_left
thf(fact_56_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_57_mult__cancel__right, axiom,
    ((![A : int, C : int, B : int]: (((times_times_int @ A @ C) = (times_times_int @ B @ C)) = (((C = zero_zero_int)) | ((A = B))))))). % mult_cancel_right
thf(fact_58_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_59_dvd__trans, axiom,
    ((![A : int, B : int, C : int]: ((dvd_dvd_int @ A @ B) => ((dvd_dvd_int @ B @ C) => (dvd_dvd_int @ A @ C)))))). % dvd_trans
thf(fact_60_dvd__refl, axiom,
    ((![A : nat]: (dvd_dvd_nat @ A @ A)))). % dvd_refl
thf(fact_61_dvd__refl, axiom,
    ((![A : int]: (dvd_dvd_int @ A @ A)))). % dvd_refl
thf(fact_62_dvd__antisym, axiom,
    ((![M : nat, N : nat]: ((dvd_dvd_nat @ M @ N) => ((dvd_dvd_nat @ N @ M) => (M = N)))))). % dvd_antisym
thf(fact_63_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_64_mult__not__zero, axiom,
    ((![A : int, B : int]: ((~ (((times_times_int @ A @ B) = zero_zero_int))) => ((~ ((A = zero_zero_int))) & (~ ((B = zero_zero_int)))))))). % mult_not_zero
thf(fact_65_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_66_divisors__zero, axiom,
    ((![A : int, B : int]: (((times_times_int @ A @ B) = zero_zero_int) => ((A = zero_zero_int) | (B = zero_zero_int)))))). % divisors_zero
thf(fact_67_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_68_no__zero__divisors, axiom,
    ((![A : int, B : int]: ((~ ((A = zero_zero_int))) => ((~ ((B = zero_zero_int))) => (~ (((times_times_int @ A @ B) = zero_zero_int)))))))). % no_zero_divisors
thf(fact_69_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_70_mult__left__cancel, axiom,
    ((![C : int, A : int, B : int]: ((~ ((C = zero_zero_int))) => (((times_times_int @ C @ A) = (times_times_int @ 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_mult__right__cancel, axiom,
    ((![C : int, A : int, B : int]: ((~ ((C = zero_zero_int))) => (((times_times_int @ A @ C) = (times_times_int @ B @ C)) = (A = B)))))). % mult_right_cancel
thf(fact_73_dvd__0__left, axiom,
    ((![A : nat]: ((dvd_dvd_nat @ zero_zero_nat @ A) => (A = zero_zero_nat))))). % dvd_0_left
thf(fact_74_dvd__0__left, axiom,
    ((![A : int]: ((dvd_dvd_int @ zero_zero_int @ A) => (A = zero_zero_int))))). % dvd_0_left
thf(fact_75_dvdE, axiom,
    ((![B : nat, A : nat]: ((dvd_dvd_nat @ B @ A) => (~ ((![K2 : nat]: (~ ((A = (times_times_nat @ B @ K2))))))))))). % dvdE
thf(fact_76_dvdE, axiom,
    ((![B : int, A : int]: ((dvd_dvd_int @ B @ A) => (~ ((![K2 : int]: (~ ((A = (times_times_int @ B @ K2))))))))))). % dvdE
thf(fact_77_dvdI, axiom,
    ((![A : nat, B : nat, K : nat]: ((A = (times_times_nat @ B @ K)) => (dvd_dvd_nat @ B @ A))))). % dvdI
thf(fact_78_dvdI, axiom,
    ((![A : int, B : int, K : int]: ((A = (times_times_int @ B @ K)) => (dvd_dvd_int @ B @ A))))). % dvdI
thf(fact_79_dvd__def, axiom,
    ((dvd_dvd_nat = (^[B3 : nat]: (^[A2 : nat]: (?[K3 : nat]: (A2 = (times_times_nat @ B3 @ K3)))))))). % dvd_def
thf(fact_80_dvd__def, axiom,
    ((dvd_dvd_int = (^[B3 : int]: (^[A2 : int]: (?[K3 : int]: (A2 = (times_times_int @ B3 @ K3)))))))). % dvd_def
thf(fact_81_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_82_dvd__mult, axiom,
    ((![A : int, C : int, B : int]: ((dvd_dvd_int @ A @ C) => (dvd_dvd_int @ A @ (times_times_int @ B @ C)))))). % dvd_mult
thf(fact_83_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_84_dvd__mult2, axiom,
    ((![A : int, B : int, C : int]: ((dvd_dvd_int @ A @ B) => (dvd_dvd_int @ A @ (times_times_int @ B @ C)))))). % dvd_mult2
thf(fact_85_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_86_dvd__mult__left, axiom,
    ((![A : int, B : int, C : int]: ((dvd_dvd_int @ (times_times_int @ A @ B) @ C) => (dvd_dvd_int @ A @ C))))). % dvd_mult_left
thf(fact_87_dvd__triv__left, axiom,
    ((![A : nat, B : nat]: (dvd_dvd_nat @ A @ (times_times_nat @ A @ B))))). % dvd_triv_left
thf(fact_88_dvd__triv__left, axiom,
    ((![A : int, B : int]: (dvd_dvd_int @ A @ (times_times_int @ A @ B))))). % dvd_triv_left
thf(fact_89_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_90_mult__dvd__mono, axiom,
    ((![A : int, B : int, C : int, D : int]: ((dvd_dvd_int @ A @ B) => ((dvd_dvd_int @ C @ D) => (dvd_dvd_int @ (times_times_int @ A @ C) @ (times_times_int @ B @ D))))))). % mult_dvd_mono
thf(fact_91_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_92_dvd__mult__right, axiom,
    ((![A : int, B : int, C : int]: ((dvd_dvd_int @ (times_times_int @ A @ B) @ C) => (dvd_dvd_int @ B @ C))))). % dvd_mult_right
thf(fact_93_dvd__triv__right, axiom,
    ((![A : nat, B : nat]: (dvd_dvd_nat @ A @ (times_times_nat @ B @ A))))). % dvd_triv_right
thf(fact_94_dvd__triv__right, axiom,
    ((![A : int, B : int]: (dvd_dvd_int @ A @ (times_times_int @ B @ A))))). % dvd_triv_right
thf(fact_95_mult__0, axiom,
    ((![N : nat]: ((times_times_nat @ zero_zero_nat @ N) = zero_zero_nat)))). % mult_0
thf(fact_96_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_97_even__push__bit__iff, axiom,
    ((![N : nat, A : int]: ((dvd_dvd_int @ (numeral_numeral_int @ (bit0 @ one)) @ (semiri502875007it_int @ N @ A)) = (((~ ((N = zero_zero_nat)))) | ((dvd_dvd_int @ (numeral_numeral_int @ (bit0 @ one)) @ A))))))). % even_push_bit_iff
thf(fact_98_gcd__nat_Oextremum__uniqueI, axiom,
    ((![A : nat]: ((dvd_dvd_nat @ zero_zero_nat @ A) => (A = zero_zero_nat))))). % gcd_nat.extremum_uniqueI
thf(fact_99_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
thf(fact_100_gcd__nat_Oextremum__unique, axiom,
    ((![A : nat]: ((dvd_dvd_nat @ zero_zero_nat @ A) = (A = zero_zero_nat))))). % gcd_nat.extremum_unique
thf(fact_101_gcd__nat_Oextremum__strict, axiom,
    ((![A : nat]: (~ (((dvd_dvd_nat @ zero_zero_nat @ A) & (~ ((zero_zero_nat = A))))))))). % gcd_nat.extremum_strict
thf(fact_102_gcd__nat_Oextremum, axiom,
    ((![A : nat]: (dvd_dvd_nat @ A @ zero_zero_nat)))). % gcd_nat.extremum
thf(fact_103_push__bit__of__0, axiom,
    ((![N : nat]: ((semiri2013084963it_nat @ N @ zero_zero_nat) = zero_zero_nat)))). % push_bit_of_0
thf(fact_104_push__bit__of__0, axiom,
    ((![N : nat]: ((semiri502875007it_int @ N @ zero_zero_int) = zero_zero_int)))). % push_bit_of_0
thf(fact_105_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_106_push__bit__eq__0__iff, axiom,
    ((![N : nat, A : int]: (((semiri502875007it_int @ N @ A) = zero_zero_int) = (A = zero_zero_int))))). % push_bit_eq_0_iff
thf(fact_107_zero__reorient, axiom,
    ((![X : nat]: ((zero_zero_nat = X) = (X = zero_zero_nat))))). % zero_reorient
thf(fact_108_zero__reorient, axiom,
    ((![X : int]: ((zero_zero_int = X) = (X = zero_zero_int))))). % zero_reorient
thf(fact_109_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_110_mult_Oleft__commute, axiom,
    ((![B : int, A : int, C : int]: ((times_times_int @ B @ (times_times_int @ A @ C)) = (times_times_int @ A @ (times_times_int @ B @ C)))))). % mult.left_commute
thf(fact_111_mult_Ocommute, axiom,
    ((times_times_nat = (^[A2 : nat]: (^[B3 : nat]: (times_times_nat @ B3 @ A2)))))). % mult.commute
thf(fact_112_mult_Ocommute, axiom,
    ((times_times_int = (^[A2 : int]: (^[B3 : int]: (times_times_int @ B3 @ A2)))))). % mult.commute
thf(fact_113_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_114_mult_Oassoc, axiom,
    ((![A : int, B : int, C : int]: ((times_times_int @ (times_times_int @ A @ B) @ C) = (times_times_int @ A @ (times_times_int @ B @ C)))))). % mult.assoc
thf(fact_115_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_116_ab__semigroup__mult__class_Omult__ac_I1_J, axiom,
    ((![A : int, B : int, C : int]: ((times_times_int @ (times_times_int @ A @ B) @ C) = (times_times_int @ A @ (times_times_int @ B @ C)))))). % ab_semigroup_mult_class.mult_ac(1)
thf(fact_117_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_118_gcd__nat_Orefl, axiom,
    ((![A : nat]: (dvd_dvd_nat @ A @ A)))). % gcd_nat.refl
thf(fact_119_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_120_gcd__nat_Oeq__iff, axiom,
    (((^[Y : nat]: (^[Z2 : nat]: (Y = Z2))) = (^[A2 : nat]: (^[B3 : nat]: (((dvd_dvd_nat @ A2 @ B3)) & ((dvd_dvd_nat @ B3 @ A2)))))))). % gcd_nat.eq_iff
thf(fact_121_gcd__nat_Oirrefl, axiom,
    ((![A : nat]: (~ (((dvd_dvd_nat @ A @ A) & (~ ((A = A))))))))). % gcd_nat.irrefl
thf(fact_122_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_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_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_125_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_126_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_127_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_128_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_129_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_130_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_131_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_132_push__bit__double, axiom,
    ((![N : nat, A : int]: ((semiri502875007it_int @ N @ (times_times_int @ A @ (numeral_numeral_int @ (bit0 @ one)))) = (times_times_int @ (semiri502875007it_int @ N @ A) @ (numeral_numeral_int @ (bit0 @ one))))))). % push_bit_double
thf(fact_133_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_134_division__decomp, axiom,
    ((![A : int, B : int, C : int]: ((dvd_dvd_int @ A @ (times_times_int @ B @ C)) => (?[B4 : int, C2 : int]: ((A = (times_times_int @ B4 @ C2)) & ((dvd_dvd_int @ B4 @ B) & (dvd_dvd_int @ C2 @ C)))))))). % division_decomp
thf(fact_135_dvd__productE, axiom,
    ((![P : nat, A : nat, B : nat]: ((dvd_dvd_nat @ P @ (times_times_nat @ A @ B)) => (~ ((![X3 : nat, Y3 : nat]: ((P = (times_times_nat @ X3 @ Y3)) => ((dvd_dvd_nat @ X3 @ A) => (~ ((dvd_dvd_nat @ Y3 @ B)))))))))))). % dvd_productE
thf(fact_136_dvd__productE, axiom,
    ((![P : int, A : int, B : int]: ((dvd_dvd_int @ P @ (times_times_int @ A @ B)) => (~ ((![X3 : int, Y3 : int]: ((P = (times_times_int @ X3 @ Y3)) => ((dvd_dvd_int @ X3 @ A) => (~ ((dvd_dvd_int @ Y3 @ B)))))))))))). % dvd_productE
thf(fact_137_even__of__nat, axiom,
    ((![A : nat]: ((dvd_dvd_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ (semiri1382578993at_nat @ A)) = (dvd_dvd_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ A))))). % even_of_nat
thf(fact_138_even__of__nat, axiom,
    ((![A : nat]: ((dvd_dvd_int @ (numeral_numeral_int @ (bit0 @ one)) @ (semiri2019852685at_int @ A)) = (dvd_dvd_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ A))))). % even_of_nat
thf(fact_139_even__take__bit__eq, axiom,
    ((![N : nat, A : int]: ((dvd_dvd_int @ (numeral_numeral_int @ (bit0 @ one)) @ (semiri1605039314it_int @ N @ A)) = (((N = zero_zero_nat)) | ((dvd_dvd_int @ (numeral_numeral_int @ (bit0 @ one)) @ A))))))). % even_take_bit_eq
thf(fact_140_even__take__bit__eq, axiom,
    ((![N : nat, A : nat]: ((dvd_dvd_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ (semiri967765622it_nat @ N @ A)) = (((N = zero_zero_nat)) | ((dvd_dvd_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ A))))))). % even_take_bit_eq
thf(fact_141_bit__0, axiom,
    ((![A : nat]: ((semiri672524339it_nat @ A @ zero_zero_nat) = (~ ((dvd_dvd_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ A))))))). % bit_0
thf(fact_142_bit__0, axiom,
    ((![A : int]: ((semiri1309798031it_int @ A @ zero_zero_nat) = (~ ((dvd_dvd_int @ (numeral_numeral_int @ (bit0 @ one)) @ A))))))). % bit_0
thf(fact_143_of__nat__eq__iff, axiom,
    ((![M : nat, N : nat]: (((semiri2019852685at_int @ M) = (semiri2019852685at_int @ N)) = (M = N))))). % of_nat_eq_iff
thf(fact_144_take__bit__of__0, axiom,
    ((![N : nat]: ((semiri1605039314it_int @ N @ zero_zero_int) = zero_zero_int)))). % take_bit_of_0
thf(fact_145_take__bit__of__0, axiom,
    ((![N : nat]: ((semiri967765622it_nat @ N @ zero_zero_nat) = zero_zero_nat)))). % take_bit_of_0
thf(fact_146_of__nat__eq__0__iff, axiom,
    ((![M : nat]: (((semiri1382578993at_nat @ M) = zero_zero_nat) = (M = zero_zero_nat))))). % of_nat_eq_0_iff
thf(fact_147_of__nat__eq__0__iff, axiom,
    ((![M : nat]: (((semiri2019852685at_int @ M) = zero_zero_int) = (M = zero_zero_nat))))). % of_nat_eq_0_iff
thf(fact_148_of__nat__0__eq__iff, axiom,
    ((![N : nat]: ((zero_zero_nat = (semiri1382578993at_nat @ N)) = (zero_zero_nat = N))))). % of_nat_0_eq_iff
thf(fact_149_of__nat__0__eq__iff, axiom,
    ((![N : nat]: ((zero_zero_int = (semiri2019852685at_int @ N)) = (zero_zero_nat = N))))). % of_nat_0_eq_iff
thf(fact_150_of__nat__0, axiom,
    (((semiri1382578993at_nat @ zero_zero_nat) = zero_zero_nat))). % of_nat_0
thf(fact_151_of__nat__0, axiom,
    (((semiri2019852685at_int @ zero_zero_nat) = zero_zero_int))). % of_nat_0
thf(fact_152_of__nat__numeral, axiom,
    ((![N : num]: ((semiri1382578993at_nat @ (numeral_numeral_nat @ N)) = (numeral_numeral_nat @ N))))). % of_nat_numeral
thf(fact_153_of__nat__numeral, axiom,
    ((![N : num]: ((semiri2019852685at_int @ (numeral_numeral_nat @ N)) = (numeral_numeral_int @ N))))). % of_nat_numeral
thf(fact_154_of__nat__mult, axiom,
    ((![M : nat, N : nat]: ((semiri1382578993at_nat @ (times_times_nat @ M @ N)) = (times_times_nat @ (semiri1382578993at_nat @ M) @ (semiri1382578993at_nat @ N)))))). % of_nat_mult
thf(fact_155_of__nat__mult, axiom,
    ((![M : nat, N : nat]: ((semiri2019852685at_int @ (times_times_nat @ M @ N)) = (times_times_int @ (semiri2019852685at_int @ M) @ (semiri2019852685at_int @ N)))))). % of_nat_mult
thf(fact_156_take__bit__0, axiom,
    ((![A : int]: ((semiri1605039314it_int @ zero_zero_nat @ A) = zero_zero_int)))). % take_bit_0
thf(fact_157_take__bit__0, axiom,
    ((![A : nat]: ((semiri967765622it_nat @ zero_zero_nat @ A) = zero_zero_nat)))). % take_bit_0
thf(fact_158_bit__Numeral1__iff, axiom,
    ((![N : nat]: ((semiri672524339it_nat @ (numeral_numeral_nat @ one) @ N) = (N = zero_zero_nat))))). % bit_Numeral1_iff
thf(fact_159_bit__Numeral1__iff, axiom,
    ((![N : nat]: ((semiri1309798031it_int @ (numeral_numeral_int @ one) @ N) = (N = zero_zero_nat))))). % bit_Numeral1_iff
thf(fact_160_push__bit__of__nat, axiom,
    ((![N : nat, M : nat]: ((semiri502875007it_int @ N @ (semiri2019852685at_int @ M)) = (semiri2019852685at_int @ (semiri2013084963it_nat @ N @ M)))))). % push_bit_of_nat
thf(fact_161_of__nat__push__bit, axiom,
    ((![M : nat, N : nat]: ((semiri2019852685at_int @ (semiri2013084963it_nat @ M @ N)) = (semiri502875007it_int @ M @ (semiri2019852685at_int @ N)))))). % of_nat_push_bit
thf(fact_162_mult__of__nat__commute, axiom,
    ((![X : nat, Y4 : nat]: ((times_times_nat @ (semiri1382578993at_nat @ X) @ Y4) = (times_times_nat @ Y4 @ (semiri1382578993at_nat @ X)))))). % mult_of_nat_commute
thf(fact_163_mult__of__nat__commute, axiom,
    ((![X : nat, Y4 : int]: ((times_times_int @ (semiri2019852685at_int @ X) @ Y4) = (times_times_int @ Y4 @ (semiri2019852685at_int @ X)))))). % mult_of_nat_commute
thf(fact_164_int__ops_I1_J, axiom,
    (((semiri2019852685at_int @ zero_zero_nat) = zero_zero_int))). % int_ops(1)
thf(fact_165_int__ops_I3_J, axiom,
    ((![N : num]: ((semiri2019852685at_int @ (numeral_numeral_nat @ N)) = (numeral_numeral_int @ N))))). % int_ops(3)
thf(fact_166_int__ops_I7_J, axiom,
    ((![A : nat, B : nat]: ((semiri2019852685at_int @ (times_times_nat @ A @ B)) = (times_times_int @ (semiri2019852685at_int @ A) @ (semiri2019852685at_int @ B)))))). % int_ops(7)
thf(fact_167_bit__eq__iff, axiom,
    (((^[Y : int]: (^[Z2 : int]: (Y = Z2))) = (^[A2 : int]: (^[B3 : int]: (![N2 : nat]: ((semiri1309798031it_int @ A2 @ N2) = (semiri1309798031it_int @ B3 @ N2)))))))). % bit_eq_iff
thf(fact_168_take__bit__of__nat, axiom,
    ((![N : nat, M : nat]: ((semiri1605039314it_int @ N @ (semiri2019852685at_int @ M)) = (semiri2019852685at_int @ (semiri967765622it_nat @ N @ M)))))). % take_bit_of_nat
thf(fact_169_take__bit__of__nat, axiom,
    ((![N : nat, M : nat]: ((semiri967765622it_nat @ N @ (semiri1382578993at_nat @ M)) = (semiri1382578993at_nat @ (semiri967765622it_nat @ N @ M)))))). % take_bit_of_nat
thf(fact_170_bit__of__nat__iff__bit, axiom,
    ((![M : nat, N : nat]: ((semiri1309798031it_int @ (semiri2019852685at_int @ M) @ N) = (semiri672524339it_nat @ M @ N))))). % bit_of_nat_iff_bit
thf(fact_171_even__of__nat__iff, axiom,
    ((![N : nat]: ((dvd_dvd_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ (semiri1382578993at_nat @ N)) = (dvd_dvd_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ N))))). % even_of_nat_iff
thf(fact_172_even__of__nat__iff, axiom,
    ((![N : nat]: ((dvd_dvd_int @ (numeral_numeral_int @ (bit0 @ one)) @ (semiri2019852685at_int @ N)) = (dvd_dvd_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ N))))). % even_of_nat_iff
thf(fact_173_of__nat__dvd__iff, axiom,
    ((![M : nat, N : nat]: ((dvd_dvd_nat @ (semiri1382578993at_nat @ M) @ (semiri1382578993at_nat @ N)) = (dvd_dvd_nat @ M @ N))))). % of_nat_dvd_iff
thf(fact_174_of__nat__dvd__iff, axiom,
    ((![M : nat, N : nat]: ((dvd_dvd_int @ (semiri2019852685at_int @ M) @ (semiri2019852685at_int @ N)) = (dvd_dvd_nat @ M @ N))))). % of_nat_dvd_iff
thf(fact_175_int__dvd__int__iff, axiom,
    ((![M : nat, N : nat]: ((dvd_dvd_int @ (semiri2019852685at_int @ M) @ (semiri2019852685at_int @ N)) = (dvd_dvd_nat @ M @ N))))). % int_dvd_int_iff
thf(fact_176_int__eq__iff__numeral, axiom,
    ((![M : nat, V : num]: (((semiri2019852685at_int @ M) = (numeral_numeral_int @ V)) = (M = (numeral_numeral_nat @ V)))))). % int_eq_iff_numeral
thf(fact_177_nat__int__comparison_I1_J, axiom,
    (((^[Y : nat]: (^[Z2 : nat]: (Y = Z2))) = (^[A2 : nat]: (^[B3 : nat]: ((semiri2019852685at_int @ A2) = (semiri2019852685at_int @ B3))))))). % nat_int_comparison(1)
thf(fact_178_int__if, axiom,
    ((![P2 : $o, A : nat, B : nat]: ((P2 => ((semiri2019852685at_int @ (if_nat @ P2 @ A @ B)) = (semiri2019852685at_int @ A))) & ((~ (P2)) => ((semiri2019852685at_int @ (if_nat @ P2 @ A @ B)) = (semiri2019852685at_int @ B))))))). % int_if
thf(fact_179_take__bit__mult, axiom,
    ((![N : nat, K : int, L : int]: ((semiri1605039314it_int @ N @ (times_times_int @ (semiri1605039314it_int @ N @ K) @ (semiri1605039314it_int @ N @ L))) = (semiri1605039314it_int @ N @ (times_times_int @ K @ L)))))). % take_bit_mult
thf(fact_180_int__int__eq, axiom,
    ((![M : nat, N : nat]: (((semiri2019852685at_int @ M) = (semiri2019852685at_int @ N)) = (M = N))))). % int_int_eq
thf(fact_181_times__int__code_I2_J, axiom,
    ((![L : int]: ((times_times_int @ zero_zero_int @ L) = zero_zero_int)))). % times_int_code(2)
thf(fact_182_times__int__code_I1_J, axiom,
    ((![K : int]: ((times_times_int @ K @ zero_zero_int) = zero_zero_int)))). % times_int_code(1)
thf(fact_183_zdvd__mult__cancel, axiom,
    ((![K : int, M : int, N : int]: ((dvd_dvd_int @ (times_times_int @ K @ M) @ (times_times_int @ K @ N)) => ((~ ((K = zero_zero_int))) => (dvd_dvd_int @ M @ N)))))). % zdvd_mult_cancel
thf(fact_184_zdvd__mono, axiom,
    ((![K : int, M : int, T : int]: ((~ ((K = zero_zero_int))) => ((dvd_dvd_int @ M @ T) = (dvd_dvd_int @ (times_times_int @ K @ M) @ (times_times_int @ K @ T))))))). % zdvd_mono
thf(fact_185_take__bit__numeral__bit0, axiom,
    ((![L : num, K : num]: ((semiri1605039314it_int @ (numeral_numeral_nat @ L) @ (numeral_numeral_int @ (bit0 @ K))) = (times_times_int @ (semiri1605039314it_int @ (pred_numeral @ L) @ (numeral_numeral_int @ K)) @ (numeral_numeral_int @ (bit0 @ one))))))). % take_bit_numeral_bit0
thf(fact_186_take__bit__numeral__bit0, axiom,
    ((![L : num, K : num]: ((semiri967765622it_nat @ (numeral_numeral_nat @ L) @ (numeral_numeral_nat @ (bit0 @ K))) = (times_times_nat @ (semiri967765622it_nat @ (pred_numeral @ L) @ (numeral_numeral_nat @ K)) @ (numeral_numeral_nat @ (bit0 @ one))))))). % take_bit_numeral_bit0
thf(fact_187_pred__numeral__simps_I1_J, axiom,
    (((pred_numeral @ one) = zero_zero_nat))). % pred_numeral_simps(1)
thf(fact_188_bit__numeral__int__simps_I2_J, axiom,
    ((![W : num, N : num]: ((semiri1309798031it_int @ (numeral_numeral_int @ (bit0 @ W)) @ (numeral_numeral_nat @ N)) = (semiri1309798031it_int @ (numeral_numeral_int @ W) @ (pred_numeral @ N)))))). % bit_numeral_int_simps(2)
thf(fact_189_push__bit__numeral, axiom,
    ((![L : num, K : num]: ((semiri2013084963it_nat @ (numeral_numeral_nat @ L) @ (numeral_numeral_nat @ K)) = (semiri2013084963it_nat @ (pred_numeral @ L) @ (numeral_numeral_nat @ (bit0 @ K))))))). % push_bit_numeral
thf(fact_190_push__bit__numeral, axiom,
    ((![L : num, K : num]: ((semiri502875007it_int @ (numeral_numeral_nat @ L) @ (numeral_numeral_int @ K)) = (semiri502875007it_int @ (pred_numeral @ L) @ (numeral_numeral_int @ (bit0 @ K))))))). % push_bit_numeral
thf(fact_191_bit__of__nat__iff, axiom,
    ((![M : nat, N : nat]: ((semiri672524339it_nat @ (semiri1382578993at_nat @ M) @ N) = (((~ (((power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ N) = zero_zero_nat)))) & ((semiri672524339it_nat @ M @ N))))))). % bit_of_nat_iff
thf(fact_192_bit__of__nat__iff, axiom,
    ((![M : nat, N : nat]: ((semiri1309798031it_int @ (semiri2019852685at_int @ M) @ N) = (((~ (((power_power_int @ (numeral_numeral_int @ (bit0 @ one)) @ N) = zero_zero_int)))) & ((semiri672524339it_nat @ M @ N))))))). % bit_of_nat_iff
thf(fact_193_bit__mod__2__iff, axiom,
    ((![A : nat, N : nat]: ((semiri672524339it_nat @ (modulo_modulo_nat @ A @ (numeral_numeral_nat @ (bit0 @ one))) @ N) = (((N = zero_zero_nat)) & ((~ ((dvd_dvd_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ A))))))))). % bit_mod_2_iff
thf(fact_194_bit__mod__2__iff, axiom,
    ((![A : int, N : nat]: ((semiri1309798031it_int @ (modulo_modulo_int @ A @ (numeral_numeral_int @ (bit0 @ one))) @ N) = (((N = zero_zero_nat)) & ((~ ((dvd_dvd_int @ (numeral_numeral_int @ (bit0 @ one)) @ A))))))))). % bit_mod_2_iff
thf(fact_195_mod__mod__trivial, axiom,
    ((![A : nat, B : nat]: ((modulo_modulo_nat @ (modulo_modulo_nat @ A @ B) @ B) = (modulo_modulo_nat @ A @ B))))). % mod_mod_trivial
thf(fact_196_mod__mod__trivial, axiom,
    ((![A : int, B : int]: ((modulo_modulo_int @ (modulo_modulo_int @ A @ B) @ B) = (modulo_modulo_int @ A @ B))))). % mod_mod_trivial
thf(fact_197_bits__mod__0, axiom,
    ((![A : nat]: ((modulo_modulo_nat @ zero_zero_nat @ A) = zero_zero_nat)))). % bits_mod_0
thf(fact_198_bits__mod__0, axiom,
    ((![A : int]: ((modulo_modulo_int @ zero_zero_int @ A) = zero_zero_int)))). % bits_mod_0
thf(fact_199_mod__0, axiom,
    ((![A : nat]: ((modulo_modulo_nat @ zero_zero_nat @ A) = zero_zero_nat)))). % mod_0
thf(fact_200_mod__0, axiom,
    ((![A : int]: ((modulo_modulo_int @ zero_zero_int @ A) = zero_zero_int)))). % mod_0
thf(fact_201_mod__by__0, axiom,
    ((![A : nat]: ((modulo_modulo_nat @ A @ zero_zero_nat) = A)))). % mod_by_0
thf(fact_202_mod__by__0, axiom,
    ((![A : int]: ((modulo_modulo_int @ A @ zero_zero_int) = A)))). % mod_by_0
thf(fact_203_mod__self, axiom,
    ((![A : nat]: ((modulo_modulo_nat @ A @ A) = zero_zero_nat)))). % mod_self
thf(fact_204_mod__self, axiom,
    ((![A : int]: ((modulo_modulo_int @ A @ A) = zero_zero_int)))). % mod_self
thf(fact_205_mod__mult__self1__is__0, axiom,
    ((![B : nat, A : nat]: ((modulo_modulo_nat @ (times_times_nat @ B @ A) @ B) = zero_zero_nat)))). % mod_mult_self1_is_0
thf(fact_206_mod__mult__self1__is__0, axiom,
    ((![B : int, A : int]: ((modulo_modulo_int @ (times_times_int @ B @ A) @ B) = zero_zero_int)))). % mod_mult_self1_is_0
thf(fact_207_mod__mult__self2__is__0, axiom,
    ((![A : nat, B : nat]: ((modulo_modulo_nat @ (times_times_nat @ A @ B) @ B) = zero_zero_nat)))). % mod_mult_self2_is_0
thf(fact_208_mod__mult__self2__is__0, axiom,
    ((![A : int, B : int]: ((modulo_modulo_int @ (times_times_int @ A @ B) @ B) = zero_zero_int)))). % mod_mult_self2_is_0
thf(fact_209_dvd__imp__mod__0, axiom,
    ((![A : nat, B : nat]: ((dvd_dvd_nat @ A @ B) => ((modulo_modulo_nat @ B @ A) = zero_zero_nat))))). % dvd_imp_mod_0
thf(fact_210_dvd__imp__mod__0, axiom,
    ((![A : int, B : int]: ((dvd_dvd_int @ A @ B) => ((modulo_modulo_int @ B @ A) = zero_zero_int))))). % dvd_imp_mod_0
thf(fact_211_even__mod__2__iff, axiom,
    ((![A : nat]: ((dvd_dvd_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ (modulo_modulo_nat @ A @ (numeral_numeral_nat @ (bit0 @ one)))) = (dvd_dvd_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ A))))). % even_mod_2_iff
thf(fact_212_even__mod__2__iff, axiom,
    ((![A : int]: ((dvd_dvd_int @ (numeral_numeral_int @ (bit0 @ one)) @ (modulo_modulo_int @ A @ (numeral_numeral_int @ (bit0 @ one)))) = (dvd_dvd_int @ (numeral_numeral_int @ (bit0 @ one)) @ A))))). % even_mod_2_iff
thf(fact_213_take__bit__nat__def, axiom,
    ((semiri967765622it_nat = (^[N2 : nat]: (^[M2 : nat]: (modulo_modulo_nat @ M2 @ (power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ N2))))))). % take_bit_nat_def
thf(fact_214_take__bit__eq__mod, axiom,
    ((semiri1605039314it_int = (^[N2 : nat]: (^[A2 : int]: (modulo_modulo_int @ A2 @ (power_power_int @ (numeral_numeral_int @ (bit0 @ one)) @ N2))))))). % take_bit_eq_mod
thf(fact_215_take__bit__eq__mod, axiom,
    ((semiri967765622it_nat = (^[N2 : nat]: (^[A2 : nat]: (modulo_modulo_nat @ A2 @ (power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ N2))))))). % take_bit_eq_mod
thf(fact_216_take__bit__int__def, axiom,
    ((semiri1605039314it_int = (^[N2 : nat]: (^[K3 : int]: (modulo_modulo_int @ K3 @ (power_power_int @ (numeral_numeral_int @ (bit0 @ one)) @ N2))))))). % take_bit_int_def
thf(fact_217_bezw_Oinduct, axiom,
    ((![P2 : nat > nat > $o, A0 : nat, A1 : nat]: ((![X3 : nat, Y3 : nat]: (((~ ((Y3 = zero_zero_nat))) => (P2 @ Y3 @ (modulo_modulo_nat @ X3 @ Y3))) => (((~ ((Y3 = zero_zero_nat))) => (P2 @ Y3 @ (modulo_modulo_nat @ X3 @ Y3))) => (((~ ((Y3 = zero_zero_nat))) => (P2 @ Y3 @ (modulo_modulo_nat @ X3 @ Y3))) => (P2 @ X3 @ Y3))))) => (P2 @ A0 @ A1))))). % bezw.induct
thf(fact_218_dvd__mod__imp__dvd, axiom,
    ((![C : nat, A : nat, B : nat]: ((dvd_dvd_nat @ C @ (modulo_modulo_nat @ A @ B)) => ((dvd_dvd_nat @ C @ B) => (dvd_dvd_nat @ C @ A)))))). % dvd_mod_imp_dvd
thf(fact_219_dvd__mod__imp__dvd, axiom,
    ((![C : int, A : int, B : int]: ((dvd_dvd_int @ C @ (modulo_modulo_int @ A @ B)) => ((dvd_dvd_int @ C @ B) => (dvd_dvd_int @ C @ A)))))). % dvd_mod_imp_dvd
thf(fact_220_dvd__mod__iff, axiom,
    ((![C : nat, B : nat, A : nat]: ((dvd_dvd_nat @ C @ B) => ((dvd_dvd_nat @ C @ (modulo_modulo_nat @ A @ B)) = (dvd_dvd_nat @ C @ A)))))). % dvd_mod_iff
thf(fact_221_dvd__mod__iff, axiom,
    ((![C : int, B : int, A : int]: ((dvd_dvd_int @ C @ B) => ((dvd_dvd_int @ C @ (modulo_modulo_int @ A @ B)) = (dvd_dvd_int @ C @ A)))))). % dvd_mod_iff
thf(fact_222_dvd__mod, axiom,
    ((![K : nat, M : nat, N : nat]: ((dvd_dvd_nat @ K @ M) => ((dvd_dvd_nat @ K @ N) => (dvd_dvd_nat @ K @ (modulo_modulo_nat @ M @ N))))))). % dvd_mod
thf(fact_223_dvd__mod, axiom,
    ((![K : int, M : int, N : int]: ((dvd_dvd_int @ K @ M) => ((dvd_dvd_int @ K @ N) => (dvd_dvd_int @ K @ (modulo_modulo_int @ M @ N))))))). % dvd_mod
thf(fact_224_mod__mod__cancel, axiom,
    ((![C : nat, B : nat, A : nat]: ((dvd_dvd_nat @ C @ B) => ((modulo_modulo_nat @ (modulo_modulo_nat @ A @ B) @ C) = (modulo_modulo_nat @ A @ C)))))). % mod_mod_cancel
thf(fact_225_mod__mod__cancel, axiom,
    ((![C : int, B : int, A : int]: ((dvd_dvd_int @ C @ B) => ((modulo_modulo_int @ (modulo_modulo_int @ A @ B) @ C) = (modulo_modulo_int @ A @ C)))))). % mod_mod_cancel

% Helper facts (3)
thf(help_If_3_1_If_001t__Nat__Onat_T, axiom,
    ((![P2 : $o]: ((P2 = $true) | (P2 = $false))))).
thf(help_If_2_1_If_001t__Nat__Onat_T, axiom,
    ((![X : nat, Y4 : nat]: ((if_nat @ $false @ X @ Y4) = Y4)))).
thf(help_If_1_1_If_001t__Nat__Onat_T, axiom,
    ((![X : nat, Y4 : nat]: ((if_nat @ $true @ X @ Y4) = X)))).

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