% 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_39__3222628_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:08:53.216

% Could-be-implicit typings (4)
thf(ty_n_t__Set__Oset_It__Nat__Onat_J, type,
    set_nat : $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 (15)
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_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_OBit1, type,
    bit1 : 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_Rings_Odivide__class_Odivide_001t__Nat__Onat, type,
    divide_divide_nat : nat > nat > nat).
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_v_x, type,
    x : nat > a).

% Relevant facts (103)
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_numeral__eq__one__iff, axiom,
    ((![N : num]: (((numeral_numeral_nat @ N) = one_one_nat) = (N = one))))). % numeral_eq_one_iff
thf(fact_4_one__eq__numeral__iff, axiom,
    ((![N : num]: ((one_one_nat = (numeral_numeral_nat @ N)) = (one = N))))). % one_eq_numeral_iff
thf(fact_5_sum_Oneutral__const, axiom,
    ((![A : set_nat]: ((groups1145913330_nat_a @ (^[Uu : nat]: zero_zero_a) @ A) = zero_zero_a)))). % sum.neutral_const
thf(fact_6_semiring__norm_I86_J, axiom,
    ((![M : num]: (~ (((bit1 @ M) = one)))))). % semiring_norm(86)
thf(fact_7_semiring__norm_I84_J, axiom,
    ((![N : num]: (~ ((one = (bit1 @ N))))))). % semiring_norm(84)
thf(fact_8_semiring__norm_I89_J, axiom,
    ((![M : num, N : num]: (~ (((bit1 @ M) = (bit0 @ N))))))). % semiring_norm(89)
thf(fact_9_semiring__norm_I88_J, axiom,
    ((![M : num, N : num]: (~ (((bit0 @ M) = (bit1 @ N))))))). % semiring_norm(88)
thf(fact_10_semiring__norm_I85_J, axiom,
    ((![M : num]: (~ (((bit0 @ M) = one)))))). % semiring_norm(85)
thf(fact_11_semiring__norm_I83_J, axiom,
    ((![N : num]: (~ ((one = (bit0 @ N))))))). % semiring_norm(83)
thf(fact_12_numeral__eq__iff, axiom,
    ((![M : num, N : num]: (((numeral_numeral_nat @ M) = (numeral_numeral_nat @ N)) = (M = N))))). % numeral_eq_iff
thf(fact_13_semiring__norm_I87_J, axiom,
    ((![M : num, N : num]: (((bit0 @ M) = (bit0 @ N)) = (M = N))))). % semiring_norm(87)
thf(fact_14_semiring__norm_I90_J, axiom,
    ((![M : num, N : num]: (((bit1 @ M) = (bit1 @ N)) = (M = N))))). % semiring_norm(90)
thf(fact_15_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_16_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_17_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_18_semiring__norm_I2_J, axiom,
    (((plus_plus_num @ one @ one) = (bit0 @ one)))). % semiring_norm(2)
thf(fact_19_semiring__norm_I7_J, axiom,
    ((![M : num, N : num]: ((plus_plus_num @ (bit0 @ M) @ (bit1 @ N)) = (bit1 @ (plus_plus_num @ M @ N)))))). % semiring_norm(7)
thf(fact_20_semiring__norm_I9_J, axiom,
    ((![M : num, N : num]: ((plus_plus_num @ (bit1 @ M) @ (bit0 @ N)) = (bit1 @ (plus_plus_num @ M @ N)))))). % semiring_norm(9)
thf(fact_21_semiring__norm_I3_J, axiom,
    ((![N : num]: ((plus_plus_num @ one @ (bit0 @ N)) = (bit1 @ N))))). % semiring_norm(3)
thf(fact_22_semiring__norm_I4_J, axiom,
    ((![N : num]: ((plus_plus_num @ one @ (bit1 @ N)) = (bit0 @ (plus_plus_num @ N @ one)))))). % semiring_norm(4)
thf(fact_23_semiring__norm_I5_J, axiom,
    ((![M : num]: ((plus_plus_num @ (bit0 @ M) @ one) = (bit1 @ M))))). % semiring_norm(5)
thf(fact_24_semiring__norm_I8_J, axiom,
    ((![M : num]: ((plus_plus_num @ (bit1 @ M) @ one) = (bit0 @ (plus_plus_num @ M @ one)))))). % semiring_norm(8)
thf(fact_25_semiring__norm_I10_J, axiom,
    ((![M : num, N : num]: ((plus_plus_num @ (bit1 @ M) @ (bit1 @ N)) = (bit0 @ (plus_plus_num @ (plus_plus_num @ M @ N) @ one)))))). % semiring_norm(10)
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_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_28_sum_Oreindex__bij__witness, axiom,
    ((![S : set_nat, I : nat > nat, J : nat > nat, T : set_nat, H : nat > a, G : nat > a]: ((![A2 : nat]: ((member_nat @ A2 @ S) => ((I @ (J @ A2)) = A2))) => ((![A2 : nat]: ((member_nat @ A2 @ S) => (member_nat @ (J @ A2) @ T))) => ((![B : nat]: ((member_nat @ B @ T) => ((J @ (I @ B)) = B))) => ((![B : nat]: ((member_nat @ B @ T) => (member_nat @ (I @ B) @ S))) => ((![A2 : nat]: ((member_nat @ A2 @ S) => ((H @ (J @ A2)) = (G @ A2)))) => ((groups1145913330_nat_a @ G @ S) = (groups1145913330_nat_a @ H @ T)))))))))). % sum.reindex_bij_witness
thf(fact_29_sum_Oeq__general__inverses, axiom,
    ((![B2 : set_nat, K : nat > nat, A : set_nat, H : nat > nat, Gamma : nat > a, Phi : nat > a]: ((![Y : nat]: ((member_nat @ Y @ B2) => ((member_nat @ (K @ Y) @ A) & ((H @ (K @ Y)) = Y)))) => ((![X : nat]: ((member_nat @ X @ A) => ((member_nat @ (H @ X) @ B2) & (((K @ (H @ X)) = X) & ((Gamma @ (H @ X)) = (Phi @ X)))))) => ((groups1145913330_nat_a @ Phi @ A) = (groups1145913330_nat_a @ Gamma @ B2))))))). % sum.eq_general_inverses
thf(fact_30_sum_Oeq__general, axiom,
    ((![B2 : set_nat, A : set_nat, H : nat > nat, Gamma : nat > a, Phi : nat > a]: ((![Y : nat]: ((member_nat @ Y @ B2) => (?[X2 : nat]: (((member_nat @ X2 @ A) & ((H @ X2) = Y)) & (![Ya : nat]: (((member_nat @ Ya @ A) & ((H @ Ya) = Y)) => (Ya = X2))))))) => ((![X : nat]: ((member_nat @ X @ A) => ((member_nat @ (H @ X) @ B2) & ((Gamma @ (H @ X)) = (Phi @ X))))) => ((groups1145913330_nat_a @ Phi @ A) = (groups1145913330_nat_a @ Gamma @ B2))))))). % sum.eq_general
thf(fact_31_sum_Ocong, axiom,
    ((![A : set_nat, B2 : set_nat, G : nat > a, H : nat > a]: ((A = B2) => ((![X : nat]: ((member_nat @ X @ B2) => ((G @ X) = (H @ X)))) => ((groups1145913330_nat_a @ G @ A) = (groups1145913330_nat_a @ H @ B2))))))). % sum.cong
thf(fact_32_sum_Oswap, axiom,
    ((![G : nat > nat > a, B2 : set_nat, A : set_nat]: ((groups1145913330_nat_a @ (^[I2 : nat]: (groups1145913330_nat_a @ (G @ I2) @ B2)) @ A) = (groups1145913330_nat_a @ (^[J2 : nat]: (groups1145913330_nat_a @ (^[I2 : nat]: (G @ I2 @ J2)) @ A)) @ B2))))). % sum.swap
thf(fact_33_zero__neq__numeral, axiom,
    ((![N : num]: (~ ((zero_zero_nat = (numeral_numeral_nat @ N))))))). % zero_neq_numeral
thf(fact_34_sum_Onot__neutral__contains__not__neutral, axiom,
    ((![G : nat > a, A : set_nat]: ((~ (((groups1145913330_nat_a @ G @ A) = zero_zero_a))) => (~ ((![A2 : nat]: ((member_nat @ A2 @ A) => ((G @ A2) = zero_zero_a))))))))). % sum.not_neutral_contains_not_neutral
thf(fact_35_sum_Oneutral, axiom,
    ((![A : set_nat, G : nat > a]: ((![X : nat]: ((member_nat @ X @ A) => ((G @ X) = zero_zero_a))) => ((groups1145913330_nat_a @ G @ A) = zero_zero_a))))). % sum.neutral
thf(fact_36_sum_Odistrib, axiom,
    ((![G : nat > a, H : nat > a, A : set_nat]: ((groups1145913330_nat_a @ (^[X3 : nat]: (plus_plus_a @ (G @ X3) @ (H @ X3))) @ A) = (plus_plus_a @ (groups1145913330_nat_a @ G @ A) @ (groups1145913330_nat_a @ H @ A)))))). % sum.distrib
thf(fact_37_one__plus__numeral__commute, axiom,
    ((![X4 : num]: ((plus_plus_nat @ one_one_nat @ (numeral_numeral_nat @ X4)) = (plus_plus_nat @ (numeral_numeral_nat @ X4) @ one_one_nat))))). % one_plus_numeral_commute
thf(fact_38_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_39_numeral__One, axiom,
    (((numeral_numeral_nat @ one) = one_one_nat))). % numeral_One
thf(fact_40_num_Oexhaust, axiom,
    ((![Y2 : num]: ((~ ((Y2 = one))) => ((![X22 : num]: (~ ((Y2 = (bit0 @ X22))))) => (~ ((![X32 : num]: (~ ((Y2 = (bit1 @ X32)))))))))))). % num.exhaust
thf(fact_41_num_Oinduct, axiom,
    ((![P : num > $o, Num : num]: ((P @ one) => ((![X : num]: ((P @ X) => (P @ (bit0 @ X)))) => ((![X : num]: ((P @ X) => (P @ (bit1 @ X)))) => (P @ Num))))))). % num.induct
thf(fact_42_numerals_I1_J, axiom,
    (((numeral_numeral_nat @ one) = one_one_nat))). % numerals(1)
thf(fact_43_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_44_numeral__Bit1, axiom,
    ((![N : num]: ((numeral_numeral_nat @ (bit1 @ N)) = (plus_plus_nat @ (plus_plus_nat @ (numeral_numeral_nat @ N) @ (numeral_numeral_nat @ N)) @ one_one_nat))))). % numeral_Bit1
thf(fact_45_numeral__code_I3_J, axiom,
    ((![N : num]: ((numeral_numeral_nat @ (bit1 @ N)) = (plus_plus_nat @ (plus_plus_nat @ (numeral_numeral_nat @ N) @ (numeral_numeral_nat @ N)) @ one_one_nat))))). % numeral_code(3)
thf(fact_46_nat__induct2, axiom,
    ((![P : nat > $o, N : nat]: ((P @ zero_zero_nat) => ((P @ one_one_nat) => ((![N2 : nat]: ((P @ N2) => (P @ (plus_plus_nat @ N2 @ (numeral_numeral_nat @ (bit0 @ one)))))) => (P @ N))))))). % nat_induct2
thf(fact_47_zero__eq__add__iff__both__eq__0, axiom,
    ((![X4 : nat, Y2 : nat]: ((zero_zero_nat = (plus_plus_nat @ X4 @ Y2)) = (((X4 = zero_zero_nat)) & ((Y2 = zero_zero_nat))))))). % zero_eq_add_iff_both_eq_0
thf(fact_48_add__eq__0__iff__both__eq__0, axiom,
    ((![X4 : nat, Y2 : nat]: (((plus_plus_nat @ X4 @ Y2) = zero_zero_nat) = (((X4 = zero_zero_nat)) & ((Y2 = zero_zero_nat))))))). % add_eq_0_iff_both_eq_0
thf(fact_49_add__cancel__right__right, axiom,
    ((![A3 : nat, B3 : nat]: ((A3 = (plus_plus_nat @ A3 @ B3)) = (B3 = zero_zero_nat))))). % add_cancel_right_right
thf(fact_50_add__cancel__right__left, axiom,
    ((![A3 : nat, B3 : nat]: ((A3 = (plus_plus_nat @ B3 @ A3)) = (B3 = zero_zero_nat))))). % add_cancel_right_left
thf(fact_51_add__cancel__left__right, axiom,
    ((![A3 : nat, B3 : nat]: (((plus_plus_nat @ A3 @ B3) = A3) = (B3 = zero_zero_nat))))). % add_cancel_left_right
thf(fact_52_add__cancel__left__left, axiom,
    ((![B3 : nat, A3 : nat]: (((plus_plus_nat @ B3 @ A3) = A3) = (B3 = zero_zero_nat))))). % add_cancel_left_left
thf(fact_53_add_Oright__neutral, axiom,
    ((![A3 : nat]: ((plus_plus_nat @ A3 @ zero_zero_nat) = A3)))). % add.right_neutral
thf(fact_54_add_Oright__neutral, axiom,
    ((![A3 : a]: ((plus_plus_a @ A3 @ zero_zero_a) = A3)))). % add.right_neutral
thf(fact_55_add__left__cancel, axiom,
    ((![A3 : nat, B3 : nat, C : nat]: (((plus_plus_nat @ A3 @ B3) = (plus_plus_nat @ A3 @ C)) = (B3 = C))))). % add_left_cancel
thf(fact_56_add__right__cancel, axiom,
    ((![B3 : nat, A3 : nat, C : nat]: (((plus_plus_nat @ B3 @ A3) = (plus_plus_nat @ C @ A3)) = (B3 = C))))). % add_right_cancel
thf(fact_57_add_Oleft__neutral, axiom,
    ((![A3 : nat]: ((plus_plus_nat @ zero_zero_nat @ A3) = A3)))). % add.left_neutral
thf(fact_58_add_Oleft__neutral, axiom,
    ((![A3 : a]: ((plus_plus_a @ zero_zero_a @ A3) = A3)))). % add.left_neutral
thf(fact_59_zero__reorient, axiom,
    ((![X4 : nat]: ((zero_zero_nat = X4) = (X4 = zero_zero_nat))))). % zero_reorient
thf(fact_60_zero__reorient, axiom,
    ((![X4 : a]: ((zero_zero_a = X4) = (X4 = zero_zero_a))))). % zero_reorient
thf(fact_61_ab__semigroup__add__class_Oadd__ac_I1_J, axiom,
    ((![A3 : a, B3 : a, C : a]: ((plus_plus_a @ (plus_plus_a @ A3 @ B3) @ C) = (plus_plus_a @ A3 @ (plus_plus_a @ B3 @ C)))))). % ab_semigroup_add_class.add_ac(1)
thf(fact_62_ab__semigroup__add__class_Oadd__ac_I1_J, axiom,
    ((![A3 : nat, B3 : nat, C : nat]: ((plus_plus_nat @ (plus_plus_nat @ A3 @ B3) @ C) = (plus_plus_nat @ A3 @ (plus_plus_nat @ B3 @ C)))))). % ab_semigroup_add_class.add_ac(1)
thf(fact_63_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_64_group__cancel_Oadd1, axiom,
    ((![A : a, K : a, A3 : a, B3 : a]: ((A = (plus_plus_a @ K @ A3)) => ((plus_plus_a @ A @ B3) = (plus_plus_a @ K @ (plus_plus_a @ A3 @ B3))))))). % group_cancel.add1
thf(fact_65_group__cancel_Oadd1, axiom,
    ((![A : nat, K : nat, A3 : nat, B3 : nat]: ((A = (plus_plus_nat @ K @ A3)) => ((plus_plus_nat @ A @ B3) = (plus_plus_nat @ K @ (plus_plus_nat @ A3 @ B3))))))). % group_cancel.add1
thf(fact_66_group__cancel_Oadd2, axiom,
    ((![B2 : a, K : a, B3 : a, A3 : a]: ((B2 = (plus_plus_a @ K @ B3)) => ((plus_plus_a @ A3 @ B2) = (plus_plus_a @ K @ (plus_plus_a @ A3 @ B3))))))). % group_cancel.add2
thf(fact_67_group__cancel_Oadd2, axiom,
    ((![B2 : nat, K : nat, B3 : nat, A3 : nat]: ((B2 = (plus_plus_nat @ K @ B3)) => ((plus_plus_nat @ A3 @ B2) = (plus_plus_nat @ K @ (plus_plus_nat @ A3 @ B3))))))). % group_cancel.add2
thf(fact_68_add_Oassoc, axiom,
    ((![A3 : a, B3 : a, C : a]: ((plus_plus_a @ (plus_plus_a @ A3 @ B3) @ C) = (plus_plus_a @ A3 @ (plus_plus_a @ B3 @ C)))))). % add.assoc
thf(fact_69_add_Oassoc, axiom,
    ((![A3 : nat, B3 : nat, C : nat]: ((plus_plus_nat @ (plus_plus_nat @ A3 @ B3) @ C) = (plus_plus_nat @ A3 @ (plus_plus_nat @ B3 @ C)))))). % add.assoc
thf(fact_70_add_Ocommute, axiom,
    ((plus_plus_a = (^[A4 : a]: (^[B4 : a]: (plus_plus_a @ B4 @ A4)))))). % add.commute
thf(fact_71_add_Ocommute, axiom,
    ((plus_plus_nat = (^[A4 : nat]: (^[B4 : nat]: (plus_plus_nat @ B4 @ A4)))))). % add.commute
thf(fact_72_add_Oleft__commute, axiom,
    ((![B3 : a, A3 : a, C : a]: ((plus_plus_a @ B3 @ (plus_plus_a @ A3 @ C)) = (plus_plus_a @ A3 @ (plus_plus_a @ B3 @ C)))))). % add.left_commute
thf(fact_73_add_Oleft__commute, axiom,
    ((![B3 : nat, A3 : nat, C : nat]: ((plus_plus_nat @ B3 @ (plus_plus_nat @ A3 @ C)) = (plus_plus_nat @ A3 @ (plus_plus_nat @ B3 @ C)))))). % add.left_commute
thf(fact_74_add__left__imp__eq, axiom,
    ((![A3 : nat, B3 : nat, C : nat]: (((plus_plus_nat @ A3 @ B3) = (plus_plus_nat @ A3 @ C)) => (B3 = C))))). % add_left_imp_eq
thf(fact_75_add__right__imp__eq, axiom,
    ((![B3 : nat, A3 : nat, C : nat]: (((plus_plus_nat @ B3 @ A3) = (plus_plus_nat @ C @ A3)) => (B3 = C))))). % add_right_imp_eq
thf(fact_76_one__reorient, axiom,
    ((![X4 : nat]: ((one_one_nat = X4) = (X4 = one_one_nat))))). % one_reorient
thf(fact_77_comm__monoid__add__class_Oadd__0, axiom,
    ((![A3 : nat]: ((plus_plus_nat @ zero_zero_nat @ A3) = A3)))). % comm_monoid_add_class.add_0
thf(fact_78_comm__monoid__add__class_Oadd__0, axiom,
    ((![A3 : a]: ((plus_plus_a @ zero_zero_a @ A3) = A3)))). % comm_monoid_add_class.add_0
thf(fact_79_add_Ocomm__neutral, axiom,
    ((![A3 : nat]: ((plus_plus_nat @ A3 @ zero_zero_nat) = A3)))). % add.comm_neutral
thf(fact_80_add_Ocomm__neutral, axiom,
    ((![A3 : a]: ((plus_plus_a @ A3 @ zero_zero_a) = A3)))). % add.comm_neutral
thf(fact_81_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_82_Nat_Oadd__0__right, axiom,
    ((![M : nat]: ((plus_plus_nat @ M @ zero_zero_nat) = M)))). % Nat.add_0_right
thf(fact_83_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_84_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_85_plus__nat_Oadd__0, axiom,
    ((![N : nat]: ((plus_plus_nat @ zero_zero_nat @ N) = N)))). % plus_nat.add_0
thf(fact_86_verit__eq__simplify_I9_J, axiom,
    ((![X33 : num, Y3 : num]: (((bit1 @ X33) = (bit1 @ Y3)) = (X33 = Y3))))). % verit_eq_simplify(9)
thf(fact_87_verit__eq__simplify_I8_J, axiom,
    ((![X23 : num, Y22 : num]: (((bit0 @ X23) = (bit0 @ Y22)) = (X23 = Y22))))). % verit_eq_simplify(8)
thf(fact_88_verit__eq__simplify_I12_J, axiom,
    ((![X33 : num]: (~ ((one = (bit1 @ X33))))))). % verit_eq_simplify(12)
thf(fact_89_verit__sum__simplify, axiom,
    ((![A3 : nat]: ((plus_plus_nat @ A3 @ zero_zero_nat) = A3)))). % verit_sum_simplify
thf(fact_90_verit__eq__simplify_I10_J, axiom,
    ((![X23 : num]: (~ ((one = (bit0 @ X23))))))). % verit_eq_simplify(10)
thf(fact_91_verit__eq__simplify_I14_J, axiom,
    ((![X23 : num, X33 : num]: (~ (((bit0 @ X23) = (bit1 @ X33))))))). % verit_eq_simplify(14)
thf(fact_92_Euclid__induct, axiom,
    ((![P : nat > nat > $o, A3 : nat, B3 : nat]: ((![A2 : nat, B : nat]: ((P @ A2 @ B) = (P @ B @ A2))) => ((![A2 : nat]: (P @ A2 @ zero_zero_nat)) => ((![A2 : nat, B : nat]: ((P @ A2 @ B) => (P @ A2 @ (plus_plus_nat @ A2 @ B)))) => (P @ A3 @ B3))))))). % Euclid_induct
thf(fact_93_zero__neq__one, axiom,
    ((~ ((zero_zero_nat = one_one_nat))))). % zero_neq_one
thf(fact_94_add__0__iff, axiom,
    ((![B3 : nat, A3 : nat]: ((B3 = (plus_plus_nat @ B3 @ A3)) = (A3 = zero_zero_nat))))). % add_0_iff
thf(fact_95_bits__1__div__2, axiom,
    (((divide_divide_nat @ one_one_nat @ (numeral_numeral_nat @ (bit0 @ one))) = zero_zero_nat))). % bits_1_div_2
thf(fact_96_div__0, axiom,
    ((![A3 : nat]: ((divide_divide_nat @ zero_zero_nat @ A3) = zero_zero_nat)))). % div_0
thf(fact_97_div__by__0, axiom,
    ((![A3 : nat]: ((divide_divide_nat @ A3 @ zero_zero_nat) = zero_zero_nat)))). % div_by_0
thf(fact_98_bits__div__by__0, axiom,
    ((![A3 : nat]: ((divide_divide_nat @ A3 @ zero_zero_nat) = zero_zero_nat)))). % bits_div_by_0
thf(fact_99_bits__div__0, axiom,
    ((![A3 : nat]: ((divide_divide_nat @ zero_zero_nat @ A3) = zero_zero_nat)))). % bits_div_0
thf(fact_100_div__by__1, axiom,
    ((![A3 : nat]: ((divide_divide_nat @ A3 @ one_one_nat) = A3)))). % div_by_1
thf(fact_101_bits__div__by__1, axiom,
    ((![A3 : nat]: ((divide_divide_nat @ A3 @ one_one_nat) = A3)))). % bits_div_by_1
thf(fact_102_div__self, axiom,
    ((![A3 : nat]: ((~ ((A3 = zero_zero_nat))) => ((divide_divide_nat @ A3 @ A3) = one_one_nat))))). % div_self

% Conjectures (1)
thf(conj_0, conjecture,
    (((groups1145913330_nat_a @ x @ (set_or562006527an_nat @ zero_zero_nat @ (numeral_numeral_nat @ (bit0 @ (bit0 @ one))))) = (plus_plus_a @ (plus_plus_a @ (plus_plus_a @ (x @ zero_zero_nat) @ (x @ one_one_nat)) @ (x @ (numeral_numeral_nat @ (bit0 @ one)))) @ (x @ (numeral_numeral_nat @ (bit1 @ one))))))).
