% 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/FFT/prob_94__3223542_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:09:12.197

% Could-be-implicit typings (5)
thf(ty_n_t__Set__Oset_It__Nat__Onat_J, type,
    set_nat : $tType).
thf(ty_n_t__Set__Oset_Itf__a_J, type,
    set_a : $tType).
thf(ty_n_t__Num__Onum, type,
    num : $tType).
thf(ty_n_t__Nat__Onat, type,
    nat : $tType).
thf(ty_n_tf__a, type,
    a : $tType).

% Explicit typings (21)
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat, type,
    one_one_nat : nat).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat, type,
    plus_plus_nat : nat > nat > nat).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Num__Onum, type,
    plus_plus_num : num > num > num).
thf(sy_c_Groups_Oplus__class_Oplus_001tf__a, type,
    plus_plus_a : a > a > a).
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_Groups_Ozero__class_Ozero_001tf__a, type,
    zero_zero_a : a).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001tf__a, type,
    groups1145913330_nat_a : (nat > a) > set_nat > a).
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_Set_OCollect_001t__Nat__Onat, type,
    collect_nat : (nat > $o) > set_nat).
thf(sy_c_Set_OCollect_001tf__a, type,
    collect_a : (a > $o) > set_a).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Nat__Onat, type,
    image_nat_nat : (nat > nat) > set_nat > set_nat).
thf(sy_c_Set_Oimage_001tf__a_001tf__a, type,
    image_a_a : (a > a) > set_a > set_a).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Nat__Onat, type,
    set_or562006527an_nat : nat > nat > set_nat).
thf(sy_c_member_001t__Nat__Onat, type,
    member_nat : nat > set_nat > $o).
thf(sy_c_member_001tf__a, type,
    member_a : a > set_a > $o).
thf(sy_v_f, type,
    f : nat > a).
thf(sy_v_n, type,
    n : nat).

% Relevant facts (148)
thf(fact_0_one__add__one, axiom,
    (((plus_plus_nat @ one_one_nat @ one_one_nat) = (numeral_numeral_nat @ (bit0 @ one))))). % one_add_one
thf(fact_1_numeral__plus__one, axiom,
    ((![N : num]: ((plus_plus_nat @ (numeral_numeral_nat @ N) @ one_one_nat) = (numeral_numeral_nat @ (plus_plus_num @ N @ one)))))). % numeral_plus_one
thf(fact_2_one__plus__numeral, axiom,
    ((![N : num]: ((plus_plus_nat @ one_one_nat @ (numeral_numeral_nat @ N)) = (numeral_numeral_nat @ (plus_plus_num @ one @ N)))))). % one_plus_numeral
thf(fact_3_image__add__atLeastLessThan_H, axiom,
    ((![K : nat, I : nat, J : nat]: ((image_nat_nat @ (^[N2 : nat]: (plus_plus_nat @ N2 @ K)) @ (set_or562006527an_nat @ I @ J)) = (set_or562006527an_nat @ (plus_plus_nat @ I @ K) @ (plus_plus_nat @ J @ K)))))). % image_add_atLeastLessThan'
thf(fact_4_image__add__atLeastLessThan, axiom,
    ((![K : nat, I : nat, J : nat]: ((image_nat_nat @ (plus_plus_nat @ K) @ (set_or562006527an_nat @ I @ J)) = (set_or562006527an_nat @ (plus_plus_nat @ I @ K) @ (plus_plus_nat @ J @ K)))))). % image_add_atLeastLessThan
thf(fact_5_numeral__eq__one__iff, axiom,
    ((![N : num]: (((numeral_numeral_nat @ N) = one_one_nat) = (N = one))))). % numeral_eq_one_iff
thf(fact_6_one__eq__numeral__iff, axiom,
    ((![N : num]: ((one_one_nat = (numeral_numeral_nat @ N)) = (one = N))))). % one_eq_numeral_iff
thf(fact_7_distrib__left__numeral, axiom,
    ((![V : num, B : nat, C : nat]: ((times_times_nat @ (numeral_numeral_nat @ V) @ (plus_plus_nat @ B @ C)) = (plus_plus_nat @ (times_times_nat @ (numeral_numeral_nat @ V) @ B) @ (times_times_nat @ (numeral_numeral_nat @ V) @ C)))))). % distrib_left_numeral
thf(fact_8_distrib__right__numeral, axiom,
    ((![A : nat, B : nat, V : num]: ((times_times_nat @ (plus_plus_nat @ A @ B) @ (numeral_numeral_nat @ V)) = (plus_plus_nat @ (times_times_nat @ A @ (numeral_numeral_nat @ V)) @ (times_times_nat @ B @ (numeral_numeral_nat @ V))))))). % distrib_right_numeral
thf(fact_9_image__add__0, axiom,
    ((![S : set_a]: ((image_a_a @ (plus_plus_a @ zero_zero_a) @ S) = S)))). % image_add_0
thf(fact_10_image__add__0, axiom,
    ((![S : set_nat]: ((image_nat_nat @ (plus_plus_nat @ zero_zero_nat) @ S) = S)))). % image_add_0
thf(fact_11_numeral__eq__iff, axiom,
    ((![M : num, N : num]: (((numeral_numeral_nat @ M) = (numeral_numeral_nat @ N)) = (M = N))))). % numeral_eq_iff
thf(fact_12_image__eqI, axiom,
    ((![B : nat, F : nat > nat, X : nat, A2 : set_nat]: ((B = (F @ X)) => ((member_nat @ X @ A2) => (member_nat @ B @ (image_nat_nat @ F @ A2))))))). % image_eqI
thf(fact_13_image__eqI, axiom,
    ((![B : a, F : a > a, X : a, A2 : set_a]: ((B = (F @ X)) => ((member_a @ X @ A2) => (member_a @ B @ (image_a_a @ F @ A2))))))). % image_eqI
thf(fact_14_image__ident, axiom,
    ((![Y : set_nat]: ((image_nat_nat @ (^[X2 : nat]: X2) @ Y) = Y)))). % image_ident
thf(fact_15_image__ident, axiom,
    ((![Y : set_a]: ((image_a_a @ (^[X2 : a]: X2) @ Y) = Y)))). % image_ident
thf(fact_16_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_17_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_18_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_19_mult__zero__right, axiom,
    ((![A : nat]: ((times_times_nat @ A @ zero_zero_nat) = zero_zero_nat)))). % mult_zero_right
thf(fact_20_mult__zero__left, axiom,
    ((![A : nat]: ((times_times_nat @ zero_zero_nat @ A) = zero_zero_nat)))). % mult_zero_left
thf(fact_21_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_22_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_23_num__double, axiom,
    ((![N : num]: ((times_times_num @ (bit0 @ one) @ N) = (bit0 @ N))))). % num_double
thf(fact_24_numeral__plus__numeral, axiom,
    ((![M : num, N : num]: ((plus_plus_nat @ (numeral_numeral_nat @ M) @ (numeral_numeral_nat @ N)) = (numeral_numeral_nat @ (plus_plus_num @ M @ N)))))). % numeral_plus_numeral
thf(fact_25_add__numeral__left, axiom,
    ((![V : num, W : num, Z : nat]: ((plus_plus_nat @ (numeral_numeral_nat @ V) @ (plus_plus_nat @ (numeral_numeral_nat @ W) @ Z)) = (plus_plus_nat @ (numeral_numeral_nat @ (plus_plus_num @ V @ W)) @ Z))))). % add_numeral_left
thf(fact_26_add__One__commute, axiom,
    ((![N : num]: ((plus_plus_num @ one @ N) = (plus_plus_num @ N @ one))))). % add_One_commute
thf(fact_27_rev__image__eqI, axiom,
    ((![X : nat, A2 : set_nat, B : nat, F : nat > nat]: ((member_nat @ X @ A2) => ((B = (F @ X)) => (member_nat @ B @ (image_nat_nat @ F @ A2))))))). % rev_image_eqI
thf(fact_28_rev__image__eqI, axiom,
    ((![X : a, A2 : set_a, B : a, F : a > a]: ((member_a @ X @ A2) => ((B = (F @ X)) => (member_a @ B @ (image_a_a @ F @ A2))))))). % rev_image_eqI
thf(fact_29_ball__imageD, axiom,
    ((![F : nat > nat, A2 : set_nat, P : nat > $o]: ((![X3 : nat]: ((member_nat @ X3 @ (image_nat_nat @ F @ A2)) => (P @ X3))) => (![X4 : nat]: ((member_nat @ X4 @ A2) => (P @ (F @ X4)))))))). % ball_imageD
thf(fact_30_ball__imageD, axiom,
    ((![F : a > a, A2 : set_a, P : a > $o]: ((![X3 : a]: ((member_a @ X3 @ (image_a_a @ F @ A2)) => (P @ X3))) => (![X4 : a]: ((member_a @ X4 @ A2) => (P @ (F @ X4)))))))). % ball_imageD
thf(fact_31_image__cong, axiom,
    ((![M2 : set_nat, N3 : set_nat, F : nat > nat, G : nat > nat]: ((M2 = N3) => ((![X3 : nat]: ((member_nat @ X3 @ N3) => ((F @ X3) = (G @ X3)))) => ((image_nat_nat @ F @ M2) = (image_nat_nat @ G @ N3))))))). % image_cong
thf(fact_32_image__cong, axiom,
    ((![M2 : set_a, N3 : set_a, F : a > a, G : a > a]: ((M2 = N3) => ((![X3 : a]: ((member_a @ X3 @ N3) => ((F @ X3) = (G @ X3)))) => ((image_a_a @ F @ M2) = (image_a_a @ G @ N3))))))). % image_cong
thf(fact_33_bex__imageD, axiom,
    ((![F : nat > nat, A2 : set_nat, P : nat > $o]: ((?[X4 : nat]: ((member_nat @ X4 @ (image_nat_nat @ F @ A2)) & (P @ X4))) => (?[X3 : nat]: ((member_nat @ X3 @ A2) & (P @ (F @ X3)))))))). % bex_imageD
thf(fact_34_bex__imageD, axiom,
    ((![F : a > a, A2 : set_a, P : a > $o]: ((?[X4 : a]: ((member_a @ X4 @ (image_a_a @ F @ A2)) & (P @ X4))) => (?[X3 : a]: ((member_a @ X3 @ A2) & (P @ (F @ X3)))))))). % bex_imageD
thf(fact_35_image__iff, axiom,
    ((![Z : nat, F : nat > nat, A2 : set_nat]: ((member_nat @ Z @ (image_nat_nat @ F @ A2)) = (?[X2 : nat]: (((member_nat @ X2 @ A2)) & ((Z = (F @ X2))))))))). % image_iff
thf(fact_36_image__iff, axiom,
    ((![Z : a, F : a > a, A2 : set_a]: ((member_a @ Z @ (image_a_a @ F @ A2)) = (?[X2 : a]: (((member_a @ X2 @ A2)) & ((Z = (F @ X2))))))))). % image_iff
thf(fact_37_imageI, axiom,
    ((![X : nat, A2 : set_nat, F : nat > nat]: ((member_nat @ X @ A2) => (member_nat @ (F @ X) @ (image_nat_nat @ F @ A2)))))). % imageI
thf(fact_38_imageI, axiom,
    ((![X : a, A2 : set_a, F : a > a]: ((member_a @ X @ A2) => (member_a @ (F @ X) @ (image_a_a @ F @ A2)))))). % imageI
thf(fact_39_Compr__image__eq, axiom,
    ((![F : nat > nat, A2 : set_nat, P : nat > $o]: ((collect_nat @ (^[X2 : nat]: (((member_nat @ X2 @ (image_nat_nat @ F @ A2))) & ((P @ X2))))) = (image_nat_nat @ F @ (collect_nat @ (^[X2 : nat]: (((member_nat @ X2 @ A2)) & ((P @ (F @ X2))))))))))). % Compr_image_eq
thf(fact_40_Compr__image__eq, axiom,
    ((![F : a > a, A2 : set_a, P : a > $o]: ((collect_a @ (^[X2 : a]: (((member_a @ X2 @ (image_a_a @ F @ A2))) & ((P @ X2))))) = (image_a_a @ F @ (collect_a @ (^[X2 : a]: (((member_a @ X2 @ A2)) & ((P @ (F @ X2))))))))))). % Compr_image_eq
thf(fact_41_image__image, axiom,
    ((![F : nat > nat, G : nat > nat, A2 : set_nat]: ((image_nat_nat @ F @ (image_nat_nat @ G @ A2)) = (image_nat_nat @ (^[X2 : nat]: (F @ (G @ X2))) @ A2))))). % image_image
thf(fact_42_image__image, axiom,
    ((![F : a > a, G : a > a, A2 : set_a]: ((image_a_a @ F @ (image_a_a @ G @ A2)) = (image_a_a @ (^[X2 : a]: (F @ (G @ X2))) @ A2))))). % image_image
thf(fact_43_imageE, axiom,
    ((![B : nat, F : nat > nat, A2 : set_nat]: ((member_nat @ B @ (image_nat_nat @ F @ A2)) => (~ ((![X3 : nat]: ((B = (F @ X3)) => (~ ((member_nat @ X3 @ A2))))))))))). % imageE
thf(fact_44_imageE, axiom,
    ((![B : a, F : a > a, A2 : set_a]: ((member_a @ B @ (image_a_a @ F @ A2)) => (~ ((![X3 : a]: ((B = (F @ X3)) => (~ ((member_a @ X3 @ A2))))))))))). % imageE
thf(fact_45_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_46_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_47_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_48_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_49_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_50_zero__neq__numeral, axiom,
    ((![N : num]: (~ ((zero_zero_nat = (numeral_numeral_nat @ N))))))). % zero_neq_numeral
thf(fact_51_zero__neq__one, axiom,
    ((~ ((zero_zero_nat = one_one_nat))))). % zero_neq_one
thf(fact_52_combine__common__factor, axiom,
    ((![A : nat, E : nat, B : nat, C : nat]: ((plus_plus_nat @ (times_times_nat @ A @ E) @ (plus_plus_nat @ (times_times_nat @ B @ E) @ C)) = (plus_plus_nat @ (times_times_nat @ (plus_plus_nat @ A @ B) @ E) @ C))))). % combine_common_factor
thf(fact_53_distrib__right, axiom,
    ((![A : nat, B : nat, C : nat]: ((times_times_nat @ (plus_plus_nat @ A @ B) @ C) = (plus_plus_nat @ (times_times_nat @ A @ C) @ (times_times_nat @ B @ C)))))). % distrib_right
thf(fact_54_distrib__left, axiom,
    ((![A : nat, B : nat, C : nat]: ((times_times_nat @ A @ (plus_plus_nat @ B @ C)) = (plus_plus_nat @ (times_times_nat @ A @ B) @ (times_times_nat @ A @ C)))))). % distrib_left
thf(fact_55_comm__semiring__class_Odistrib, axiom,
    ((![A : nat, B : nat, C : nat]: ((times_times_nat @ (plus_plus_nat @ A @ B) @ C) = (plus_plus_nat @ (times_times_nat @ A @ C) @ (times_times_nat @ B @ C)))))). % comm_semiring_class.distrib
thf(fact_56_lambda__zero, axiom,
    (((^[H : nat]: zero_zero_nat) = (times_times_nat @ zero_zero_nat)))). % lambda_zero
thf(fact_57_lambda__one, axiom,
    (((^[X2 : nat]: X2) = (times_times_nat @ one_one_nat)))). % lambda_one
thf(fact_58_one__plus__numeral__commute, axiom,
    ((![X : num]: ((plus_plus_nat @ one_one_nat @ (numeral_numeral_nat @ X)) = (plus_plus_nat @ (numeral_numeral_nat @ X) @ one_one_nat))))). % one_plus_numeral_commute
thf(fact_59_mult__numeral__1__right, axiom,
    ((![A : nat]: ((times_times_nat @ A @ (numeral_numeral_nat @ one)) = A)))). % mult_numeral_1_right
thf(fact_60_mult__numeral__1, axiom,
    ((![A : nat]: ((times_times_nat @ (numeral_numeral_nat @ one) @ A) = A)))). % mult_numeral_1
thf(fact_61_numeral__Bit0, axiom,
    ((![N : num]: ((numeral_numeral_nat @ (bit0 @ N)) = (plus_plus_nat @ (numeral_numeral_nat @ N) @ (numeral_numeral_nat @ N)))))). % numeral_Bit0
thf(fact_62_numeral__One, axiom,
    (((numeral_numeral_nat @ one) = one_one_nat))). % numeral_One
thf(fact_63_numerals_I1_J, axiom,
    (((numeral_numeral_nat @ one) = one_one_nat))). % numerals(1)
thf(fact_64_numeral__code_I2_J, axiom,
    ((![N : num]: ((numeral_numeral_nat @ (bit0 @ N)) = (plus_plus_nat @ (numeral_numeral_nat @ N) @ (numeral_numeral_nat @ N)))))). % numeral_code(2)
thf(fact_65_sum_Oshift__bounds__nat__ivl, axiom,
    ((![G : nat > a, M : nat, K : nat, N : nat]: ((groups1145913330_nat_a @ G @ (set_or562006527an_nat @ (plus_plus_nat @ M @ K) @ (plus_plus_nat @ N @ K))) = (groups1145913330_nat_a @ (^[I2 : nat]: (G @ (plus_plus_nat @ I2 @ K))) @ (set_or562006527an_nat @ M @ N)))))). % sum.shift_bounds_nat_ivl
thf(fact_66_left__add__twice, axiom,
    ((![A : nat, B : nat]: ((plus_plus_nat @ A @ (plus_plus_nat @ A @ B)) = (plus_plus_nat @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ A) @ B))))). % left_add_twice
thf(fact_67_mult__2__right, axiom,
    ((![Z : nat]: ((times_times_nat @ Z @ (numeral_numeral_nat @ (bit0 @ one))) = (plus_plus_nat @ Z @ Z))))). % mult_2_right
thf(fact_68_mult__2, axiom,
    ((![Z : nat]: ((times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ Z) = (plus_plus_nat @ Z @ Z))))). % mult_2
thf(fact_69_nat__1__add__1, axiom,
    (((plus_plus_nat @ one_one_nat @ one_one_nat) = (numeral_numeral_nat @ (bit0 @ one))))). % nat_1_add_1
thf(fact_70_semiring__norm_I2_J, axiom,
    (((plus_plus_num @ one @ one) = (bit0 @ one)))). % semiring_norm(2)
thf(fact_71_nat__induct2, axiom,
    ((![P : nat > $o, N : nat]: ((P @ zero_zero_nat) => ((P @ one_one_nat) => ((![N4 : nat]: ((P @ N4) => (P @ (plus_plus_nat @ N4 @ (numeral_numeral_nat @ (bit0 @ one)))))) => (P @ N))))))). % nat_induct2
thf(fact_72_sum_Oneutral__const, axiom,
    ((![A2 : set_nat]: ((groups1145913330_nat_a @ (^[Uu : nat]: zero_zero_a) @ A2) = zero_zero_a)))). % sum.neutral_const
thf(fact_73_semiring__norm_I6_J, axiom,
    ((![M : num, N : num]: ((plus_plus_num @ (bit0 @ M) @ (bit0 @ N)) = (bit0 @ (plus_plus_num @ M @ N)))))). % semiring_norm(6)
thf(fact_74_nat__mult__eq__1__iff, axiom,
    ((![M : nat, N : nat]: (((times_times_nat @ M @ N) = one_one_nat) = (((M = one_one_nat)) & ((N = one_one_nat))))))). % nat_mult_eq_1_iff
thf(fact_75_nat__1__eq__mult__iff, axiom,
    ((![M : nat, N : nat]: ((one_one_nat = (times_times_nat @ M @ N)) = (((M = one_one_nat)) & ((N = one_one_nat))))))). % nat_1_eq_mult_iff
thf(fact_76_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_77_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_78_mult__0__right, axiom,
    ((![M : nat]: ((times_times_nat @ M @ zero_zero_nat) = zero_zero_nat)))). % mult_0_right
thf(fact_79_semiring__norm_I87_J, axiom,
    ((![M : num, N : num]: (((bit0 @ M) = (bit0 @ N)) = (M = N))))). % semiring_norm(87)
thf(fact_80_semiring__norm_I85_J, axiom,
    ((![M : num]: (~ (((bit0 @ M) = one)))))). % semiring_norm(85)
thf(fact_81_semiring__norm_I83_J, axiom,
    ((![N : num]: (~ ((one = (bit0 @ N))))))). % semiring_norm(83)
thf(fact_82_add__is__0, axiom,
    ((![M : nat, N : nat]: (((plus_plus_nat @ M @ N) = zero_zero_nat) = (((M = zero_zero_nat)) & ((N = zero_zero_nat))))))). % add_is_0
thf(fact_83_Nat_Oadd__0__right, axiom,
    ((![M : nat]: ((plus_plus_nat @ M @ zero_zero_nat) = M)))). % Nat.add_0_right
thf(fact_84_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_85_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_86_semiring__norm_I12_J, axiom,
    ((![N : num]: ((times_times_num @ one @ N) = N)))). % semiring_norm(12)
thf(fact_87_semiring__norm_I11_J, axiom,
    ((![M : num]: ((times_times_num @ M @ one) = M)))). % semiring_norm(11)
thf(fact_88_sum_Ocong, axiom,
    ((![A2 : set_nat, B2 : set_nat, G : nat > a, H2 : nat > a]: ((A2 = B2) => ((![X3 : nat]: ((member_nat @ X3 @ B2) => ((G @ X3) = (H2 @ X3)))) => ((groups1145913330_nat_a @ G @ A2) = (groups1145913330_nat_a @ H2 @ B2))))))). % sum.cong
thf(fact_89_sum_Oeq__general, axiom,
    ((![B2 : set_nat, A2 : set_nat, H2 : nat > nat, Gamma : nat > a, Phi : nat > a]: ((![Y2 : nat]: ((member_nat @ Y2 @ B2) => (?[X4 : nat]: (((member_nat @ X4 @ A2) & ((H2 @ X4) = Y2)) & (![Ya : nat]: (((member_nat @ Ya @ A2) & ((H2 @ Ya) = Y2)) => (Ya = X4))))))) => ((![X3 : nat]: ((member_nat @ X3 @ A2) => ((member_nat @ (H2 @ X3) @ B2) & ((Gamma @ (H2 @ X3)) = (Phi @ X3))))) => ((groups1145913330_nat_a @ Phi @ A2) = (groups1145913330_nat_a @ Gamma @ B2))))))). % sum.eq_general
thf(fact_90_sum_Oeq__general__inverses, axiom,
    ((![B2 : set_nat, K : nat > nat, A2 : set_nat, H2 : nat > nat, Gamma : nat > a, Phi : nat > a]: ((![Y2 : nat]: ((member_nat @ Y2 @ B2) => ((member_nat @ (K @ Y2) @ A2) & ((H2 @ (K @ Y2)) = Y2)))) => ((![X3 : nat]: ((member_nat @ X3 @ A2) => ((member_nat @ (H2 @ X3) @ B2) & (((K @ (H2 @ X3)) = X3) & ((Gamma @ (H2 @ X3)) = (Phi @ X3)))))) => ((groups1145913330_nat_a @ Phi @ A2) = (groups1145913330_nat_a @ Gamma @ B2))))))). % sum.eq_general_inverses
thf(fact_91_sum_Oreindex__bij__witness, axiom,
    ((![S : set_nat, I : nat > nat, J : nat > nat, T : set_nat, H2 : nat > a, G : nat > a]: ((![A3 : nat]: ((member_nat @ A3 @ S) => ((I @ (J @ A3)) = A3))) => ((![A3 : nat]: ((member_nat @ A3 @ S) => (member_nat @ (J @ A3) @ T))) => ((![B3 : nat]: ((member_nat @ B3 @ T) => ((J @ (I @ B3)) = B3))) => ((![B3 : nat]: ((member_nat @ B3 @ T) => (member_nat @ (I @ B3) @ S))) => ((![A3 : nat]: ((member_nat @ A3 @ S) => ((H2 @ (J @ A3)) = (G @ A3)))) => ((groups1145913330_nat_a @ G @ S) = (groups1145913330_nat_a @ H2 @ T)))))))))). % sum.reindex_bij_witness
thf(fact_92_sum_Oswap, axiom,
    ((![G : nat > nat > a, B2 : set_nat, A2 : set_nat]: ((groups1145913330_nat_a @ (^[I2 : nat]: (groups1145913330_nat_a @ (G @ I2) @ B2)) @ A2) = (groups1145913330_nat_a @ (^[J2 : nat]: (groups1145913330_nat_a @ (^[I2 : nat]: (G @ I2 @ J2)) @ A2)) @ B2))))). % sum.swap
thf(fact_93_sum_Oneutral, axiom,
    ((![A2 : set_nat, G : nat > a]: ((![X3 : nat]: ((member_nat @ X3 @ A2) => ((G @ X3) = zero_zero_a))) => ((groups1145913330_nat_a @ G @ A2) = zero_zero_a))))). % sum.neutral
thf(fact_94_sum_Onot__neutral__contains__not__neutral, axiom,
    ((![G : nat > a, A2 : set_nat]: ((~ (((groups1145913330_nat_a @ G @ A2) = zero_zero_a))) => (~ ((![A3 : nat]: ((member_nat @ A3 @ A2) => ((G @ A3) = zero_zero_a))))))))). % sum.not_neutral_contains_not_neutral
thf(fact_95_plus__nat_Oadd__0, axiom,
    ((![N : nat]: ((plus_plus_nat @ zero_zero_nat @ N) = N)))). % plus_nat.add_0
thf(fact_96_add__eq__self__zero, axiom,
    ((![M : nat, N : nat]: (((plus_plus_nat @ M @ N) = M) => (N = zero_zero_nat))))). % add_eq_self_zero
thf(fact_97_mult__0, axiom,
    ((![N : nat]: ((times_times_nat @ zero_zero_nat @ N) = zero_zero_nat)))). % mult_0
thf(fact_98_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_99_add__mult__distrib, axiom,
    ((![M : nat, N : nat, K : nat]: ((times_times_nat @ (plus_plus_nat @ M @ N) @ K) = (plus_plus_nat @ (times_times_nat @ M @ K) @ (times_times_nat @ N @ K)))))). % add_mult_distrib
thf(fact_100_add__mult__distrib2, axiom,
    ((![K : nat, M : nat, N : nat]: ((times_times_nat @ K @ (plus_plus_nat @ M @ N)) = (plus_plus_nat @ (times_times_nat @ K @ M) @ (times_times_nat @ K @ N)))))). % add_mult_distrib2
thf(fact_101_left__add__mult__distrib, axiom,
    ((![I : nat, U : nat, J : nat, K : nat]: ((plus_plus_nat @ (times_times_nat @ I @ U) @ (plus_plus_nat @ (times_times_nat @ J @ U) @ K)) = (plus_plus_nat @ (times_times_nat @ (plus_plus_nat @ I @ J) @ U) @ K))))). % left_add_mult_distrib
thf(fact_102_nat__mult__1, axiom,
    ((![N : nat]: ((times_times_nat @ one_one_nat @ N) = N)))). % nat_mult_1
thf(fact_103_nat__mult__1__right, axiom,
    ((![N : nat]: ((times_times_nat @ N @ one_one_nat) = N)))). % nat_mult_1_right
thf(fact_104_sum_Odistrib, axiom,
    ((![G : nat > a, H2 : nat > a, A2 : set_nat]: ((groups1145913330_nat_a @ (^[X2 : nat]: (plus_plus_a @ (G @ X2) @ (H2 @ X2))) @ A2) = (plus_plus_a @ (groups1145913330_nat_a @ G @ A2) @ (groups1145913330_nat_a @ H2 @ A2)))))). % sum.distrib
thf(fact_105_mult__eq__self__implies__10, axiom,
    ((![M : nat, N : nat]: ((M = (times_times_nat @ M @ N)) => ((N = one_one_nat) | (M = zero_zero_nat)))))). % mult_eq_self_implies_10
thf(fact_106_mult_Oleft__neutral, axiom,
    ((![A : nat]: ((times_times_nat @ one_one_nat @ A) = A)))). % mult.left_neutral
thf(fact_107_mult_Oright__neutral, axiom,
    ((![A : nat]: ((times_times_nat @ A @ one_one_nat) = A)))). % mult.right_neutral
thf(fact_108_add_Oleft__neutral, axiom,
    ((![A : nat]: ((plus_plus_nat @ zero_zero_nat @ A) = A)))). % add.left_neutral
thf(fact_109_add_Oleft__neutral, axiom,
    ((![A : a]: ((plus_plus_a @ zero_zero_a @ A) = A)))). % add.left_neutral
thf(fact_110_add_Oright__neutral, axiom,
    ((![A : nat]: ((plus_plus_nat @ A @ zero_zero_nat) = A)))). % add.right_neutral
thf(fact_111_add_Oright__neutral, axiom,
    ((![A : a]: ((plus_plus_a @ A @ zero_zero_a) = A)))). % add.right_neutral
thf(fact_112_add__right__cancel, axiom,
    ((![B : nat, A : nat, C : nat]: (((plus_plus_nat @ B @ A) = (plus_plus_nat @ C @ A)) = (B = C))))). % add_right_cancel
thf(fact_113_add__left__cancel, axiom,
    ((![A : nat, B : nat, C : nat]: (((plus_plus_nat @ A @ B) = (plus_plus_nat @ A @ C)) = (B = C))))). % add_left_cancel
thf(fact_114_zero__eq__add__iff__both__eq__0, axiom,
    ((![X : nat, Y3 : nat]: ((zero_zero_nat = (plus_plus_nat @ X @ Y3)) = (((X = zero_zero_nat)) & ((Y3 = zero_zero_nat))))))). % zero_eq_add_iff_both_eq_0
thf(fact_115_add__eq__0__iff__both__eq__0, axiom,
    ((![X : nat, Y3 : nat]: (((plus_plus_nat @ X @ Y3) = zero_zero_nat) = (((X = zero_zero_nat)) & ((Y3 = zero_zero_nat))))))). % add_eq_0_iff_both_eq_0
thf(fact_116_add__cancel__right__right, axiom,
    ((![A : nat, B : nat]: ((A = (plus_plus_nat @ A @ B)) = (B = zero_zero_nat))))). % add_cancel_right_right
thf(fact_117_add__cancel__right__left, axiom,
    ((![A : nat, B : nat]: ((A = (plus_plus_nat @ B @ A)) = (B = zero_zero_nat))))). % add_cancel_right_left
thf(fact_118_add__cancel__left__right, axiom,
    ((![A : nat, B : nat]: (((plus_plus_nat @ A @ B) = A) = (B = zero_zero_nat))))). % add_cancel_left_right
thf(fact_119_add__cancel__left__left, axiom,
    ((![B : nat, A : nat]: (((plus_plus_nat @ B @ A) = A) = (B = zero_zero_nat))))). % add_cancel_left_left
thf(fact_120_zero__reorient, axiom,
    ((![X : nat]: ((zero_zero_nat = X) = (X = zero_zero_nat))))). % zero_reorient
thf(fact_121_zero__reorient, axiom,
    ((![X : a]: ((zero_zero_a = X) = (X = zero_zero_a))))). % zero_reorient
thf(fact_122_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_123_mult_Ocommute, axiom,
    ((times_times_nat = (^[A4 : nat]: (^[B4 : nat]: (times_times_nat @ B4 @ A4)))))). % mult.commute
thf(fact_124_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_125_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_126_add__right__imp__eq, axiom,
    ((![B : nat, A : nat, C : nat]: (((plus_plus_nat @ B @ A) = (plus_plus_nat @ C @ A)) => (B = C))))). % add_right_imp_eq
thf(fact_127_add__left__imp__eq, axiom,
    ((![A : nat, B : nat, C : nat]: (((plus_plus_nat @ A @ B) = (plus_plus_nat @ A @ C)) => (B = C))))). % add_left_imp_eq
thf(fact_128_add_Oleft__commute, axiom,
    ((![B : a, A : a, C : a]: ((plus_plus_a @ B @ (plus_plus_a @ A @ C)) = (plus_plus_a @ A @ (plus_plus_a @ B @ C)))))). % add.left_commute
thf(fact_129_add_Oleft__commute, axiom,
    ((![B : nat, A : nat, C : nat]: ((plus_plus_nat @ B @ (plus_plus_nat @ A @ C)) = (plus_plus_nat @ A @ (plus_plus_nat @ B @ C)))))). % add.left_commute
thf(fact_130_add_Ocommute, axiom,
    ((plus_plus_a = (^[A4 : a]: (^[B4 : a]: (plus_plus_a @ B4 @ A4)))))). % add.commute
thf(fact_131_add_Ocommute, axiom,
    ((plus_plus_nat = (^[A4 : nat]: (^[B4 : nat]: (plus_plus_nat @ B4 @ A4)))))). % add.commute
thf(fact_132_add_Oassoc, axiom,
    ((![A : a, B : a, C : a]: ((plus_plus_a @ (plus_plus_a @ A @ B) @ C) = (plus_plus_a @ A @ (plus_plus_a @ B @ C)))))). % add.assoc
thf(fact_133_add_Oassoc, axiom,
    ((![A : nat, B : nat, C : nat]: ((plus_plus_nat @ (plus_plus_nat @ A @ B) @ C) = (plus_plus_nat @ A @ (plus_plus_nat @ B @ C)))))). % add.assoc
thf(fact_134_group__cancel_Oadd2, axiom,
    ((![B2 : a, K : a, B : a, A : a]: ((B2 = (plus_plus_a @ K @ B)) => ((plus_plus_a @ A @ B2) = (plus_plus_a @ K @ (plus_plus_a @ A @ B))))))). % group_cancel.add2
thf(fact_135_group__cancel_Oadd2, axiom,
    ((![B2 : nat, K : nat, B : nat, A : nat]: ((B2 = (plus_plus_nat @ K @ B)) => ((plus_plus_nat @ A @ B2) = (plus_plus_nat @ K @ (plus_plus_nat @ A @ B))))))). % group_cancel.add2
thf(fact_136_group__cancel_Oadd1, axiom,
    ((![A2 : a, K : a, A : a, B : a]: ((A2 = (plus_plus_a @ K @ A)) => ((plus_plus_a @ A2 @ B) = (plus_plus_a @ K @ (plus_plus_a @ A @ B))))))). % group_cancel.add1
thf(fact_137_group__cancel_Oadd1, axiom,
    ((![A2 : nat, K : nat, A : nat, B : nat]: ((A2 = (plus_plus_nat @ K @ A)) => ((plus_plus_nat @ A2 @ B) = (plus_plus_nat @ K @ (plus_plus_nat @ A @ B))))))). % group_cancel.add1
thf(fact_138_add__mono__thms__linordered__semiring_I4_J, axiom,
    ((![I : nat, J : nat, K : nat, L : nat]: (((I = J) & (K = L)) => ((plus_plus_nat @ I @ K) = (plus_plus_nat @ J @ L)))))). % add_mono_thms_linordered_semiring(4)
thf(fact_139_ab__semigroup__add__class_Oadd__ac_I1_J, axiom,
    ((![A : a, B : a, C : a]: ((plus_plus_a @ (plus_plus_a @ A @ B) @ C) = (plus_plus_a @ A @ (plus_plus_a @ B @ C)))))). % ab_semigroup_add_class.add_ac(1)
thf(fact_140_ab__semigroup__add__class_Oadd__ac_I1_J, axiom,
    ((![A : nat, B : nat, C : nat]: ((plus_plus_nat @ (plus_plus_nat @ A @ B) @ C) = (plus_plus_nat @ A @ (plus_plus_nat @ B @ C)))))). % ab_semigroup_add_class.add_ac(1)
thf(fact_141_one__reorient, axiom,
    ((![X : nat]: ((one_one_nat = X) = (X = one_one_nat))))). % one_reorient
thf(fact_142_add_Ocomm__neutral, axiom,
    ((![A : nat]: ((plus_plus_nat @ A @ zero_zero_nat) = A)))). % add.comm_neutral
thf(fact_143_add_Ocomm__neutral, axiom,
    ((![A : a]: ((plus_plus_a @ A @ zero_zero_a) = A)))). % add.comm_neutral
thf(fact_144_comm__monoid__add__class_Oadd__0, axiom,
    ((![A : nat]: ((plus_plus_nat @ zero_zero_nat @ A) = A)))). % comm_monoid_add_class.add_0
thf(fact_145_comm__monoid__add__class_Oadd__0, axiom,
    ((![A : a]: ((plus_plus_a @ zero_zero_a @ A) = A)))). % comm_monoid_add_class.add_0
thf(fact_146_mult_Ocomm__neutral, axiom,
    ((![A : nat]: ((times_times_nat @ A @ one_one_nat) = A)))). % mult.comm_neutral
thf(fact_147_comm__monoid__mult__class_Omult__1, axiom,
    ((![A : nat]: ((times_times_nat @ one_one_nat @ A) = A)))). % comm_monoid_mult_class.mult_1

% Conjectures (1)
thf(conj_0, conjecture,
    (((groups1145913330_nat_a @ f @ (set_or562006527an_nat @ zero_zero_nat @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ n))) = (plus_plus_a @ (groups1145913330_nat_a @ f @ (image_nat_nat @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one))) @ (set_or562006527an_nat @ zero_zero_nat @ n))) @ (groups1145913330_nat_a @ f @ (image_nat_nat @ (^[I2 : nat]: (plus_plus_nat @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ I2) @ one_one_nat)) @ (set_or562006527an_nat @ zero_zero_nat @ n))))))).
