Current library: Alpha v34, 4,223 checked-use theorems; Stable remains 432. Historical first admissions, original proof editions, and non-admitted aliases are preserved. Exact original first-admission records.
Definition in prerequisite notation
∃ ge_real_positive_lowerlayer. ∃ ge_real_negative_lowerlayer. ∃ ge_imaginary_positive_lowerlayer. ∃ ge_imaginary_negative_lowerlayer. ZPairDecode(z,ge_real_positive_lowerlayer,ge_real_negative_lowerlayer,ge_imaginary_positive_lowerlayer,ge_imaginary_negative_lowerlayer)Only definitions earlier in this acyclic notation graph are used here.
Hygienic expanded first-order definition
exists ge_real_positive_lowerlayer ge_real_negative_lowerlayer ge_imaginary_positive_lowerlayer ge_imaginary_negative_lowerlayer. (exists ge_real_code_lowerlayerdecode ge_imaginary_code_lowerlayerdecode. (((z) = ((ge_real_code_lowerlayerdecode) + (ge_imaginary_code_lowerlayerdecode)) * S ((ge_real_code_lowerlayerdecode) + (ge_imaginary_code_lowerlayerdecode)) + ((ge_imaginary_code_lowerlayerdecode) + (ge_imaginary_code_lowerlayerdecode))) /\ (((((ge_real_code_lowerlayerdecode) = 2 * (ge_real_positive_lowerlayer) /\ (ge_real_negative_lowerlayer) = 0) \/ exists ge_signed_half_ge_lowerlayerdecode_real. (((ge_real_code_lowerlayerdecode) = 2 * ge_signed_half_ge_lowerlayerdecode_real + 1 /\ (ge_real_positive_lowerlayer) = 0) /\ (ge_real_negative_lowerlayer) = S ge_signed_half_ge_lowerlayerdecode_real))) /\ ((((ge_imaginary_code_lowerlayerdecode) = 2 * (ge_imaginary_positive_lowerlayer) /\ (ge_imaginary_negative_lowerlayer) = 0) \/ exists ge_signed_half_ge_lowerlayerdecode_imaginary. (((ge_imaginary_code_lowerlayerdecode) = 2 * ge_signed_half_ge_lowerlayerdecode_imaginary + 1 /\ (ge_imaginary_positive_lowerlayer) = 0) /\ (ge_imaginary_negative_lowerlayer) = S ge_signed_half_ge_lowerlayerdecode_imaginary))))))The unchanged native kernel never receives this surface symbol. Binder-safe expansion produces only its existing first-order syntax.
Direct definition dependencies
Definitions depending on this notation
Checked theorems using this definition
GF0001 · gaussian_valid_has_representationGF0003 · gaussian_norm_input_validGF0004 · gaussian_add_input_left_validGF0005 · gaussian_add_input_right_validGF0006 · gaussian_add_output_validGF0007 · gaussian_multiply_input_left_validGF0008 · gaussian_multiply_input_right_validGF0009 · gaussian_multiply_output_validGF000C · gaussian_zero_validGF000F · gaussian_one_validGF001C · gaussian_unit_decidableGF001E · gaussian_unit_validGF0026 · gaussian_add_zero_rightGF0027 · gaussian_add_zero_leftGF0028 · gaussian_multiply_one_rightGF0029 · gaussian_multiply_one_leftGF002A · gaussian_multiply_zero_rightGF002B · gaussian_multiply_zero_leftGF002C · gaussian_subtract_existsGF0042 · gaussian_divides_input_validGF0043 · gaussian_divides_value_validGF0044 · gaussian_divides_reflexiveGF0045 · gaussian_divides_zeroGF0046 · gaussian_one_dividesGF004D · gaussian_unit_dividesGF0053 · gaussian_divides_decidableGF0054 · gaussian_associate_reflexiveGF005F · gaussian_gcd_bezout_zero_rightGF0060 · gaussian_gcd_bezout_zero_caseGF0064 · gaussian_gcd_bezout_bounded_existsGF0065 · gaussian_gcd_bezout_existsGF006F · gaussian_search_pair_validGF0070 · gaussian_norm_bounded_coordinatesGF0074 · gaussian_proper_norm_divisor_decidableGF0075 · gaussian_factor_search_coordinate_rowGF0076 · gaussian_factor_search_coordinate_rectangleGF007F · gaussian_irreducible_decidableGF0081 · gaussian_irreducible_divisor_existsGF008E · gaussian_product_result_validGF0095 · gaussian_irreducible_factorization_existsGF0098 · gaussian_prime_factorization_existsGF009D · gaussian_factorization_value_validGF00B2 · gaussian_unique_prime_factorization