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_representation_real_code_lowerlayer. ∃ ge_representation_imaginary_code_lowerlayer. z = (ge_representation_real_code_lowerlayer + ge_representation_imaginary_code_lowerlayer) · S (ge_representation_real_code_lowerlayer + ge_representation_imaginary_code_lowerlayer) + (ge_representation_imaginary_code_lowerlayer + ge_representation_imaginary_code_lowerlayer) ∧ (SignedBalance(ge_representation_real_code_lowerlayer,ap,an) ∧ SignedBalance(ge_representation_imaginary_code_lowerlayer,bp,bn))Only definitions earlier in this acyclic notation graph are used here.
Hygienic expanded first-order definition
exists ge_representation_real_code_lowerlayer ge_representation_imaginary_code_lowerlayer. (((z) = ((ge_representation_real_code_lowerlayer) + (ge_representation_imaginary_code_lowerlayer)) * S ((ge_representation_real_code_lowerlayer) + (ge_representation_imaginary_code_lowerlayer)) + ((ge_representation_imaginary_code_lowerlayer) + (ge_representation_imaginary_code_lowerlayer))) /\ ((exists ge_balance_positive_lowerlayerreal ge_balance_negative_lowerlayerreal. (((((ge_representation_real_code_lowerlayer) = 2 * (ge_balance_positive_lowerlayerreal) /\ (ge_balance_negative_lowerlayerreal) = 0) \/ exists ge_signed_half_lowerlayerrealdecode. (((ge_representation_real_code_lowerlayer) = 2 * ge_signed_half_lowerlayerrealdecode + 1 /\ (ge_balance_positive_lowerlayerreal) = 0) /\ (ge_balance_negative_lowerlayerreal) = S ge_signed_half_lowerlayerrealdecode))) /\ ((ap) + ge_balance_negative_lowerlayerreal = (an) + ge_balance_positive_lowerlayerreal))) /\ (exists ge_balance_positive_lowerlayerimaginary ge_balance_negative_lowerlayerimaginary. (((((ge_representation_imaginary_code_lowerlayer) = 2 * (ge_balance_positive_lowerlayerimaginary) /\ (ge_balance_negative_lowerlayerimaginary) = 0) \/ exists ge_signed_half_lowerlayerimaginarydecode. (((ge_representation_imaginary_code_lowerlayer) = 2 * ge_signed_half_lowerlayerimaginarydecode + 1 /\ (ge_balance_positive_lowerlayerimaginary) = 0) /\ (ge_balance_negative_lowerlayerimaginary) = S ge_signed_half_lowerlayerimaginarydecode))) /\ ((bp) + ge_balance_negative_lowerlayerimaginary = (bn) + ge_balance_positive_lowerlayerimaginary)))))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
EI0034 · eisenstein_norm_of_representationEI0035 · eisenstein_norm_for_representationEI003B · eisenstein_multiply_of_representationsEI003C · eisenstein_multiply_for_representationsEI003D · eisenstein_multiply_existsEI0040 · eisenstein_division_remainder_of_representationsEI0041 · eisenstein_euclidean_division_exists