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.
Inputs are genuine canonical signed-pair codes, not arbitrary naturals. Products start at the actual Gaussian identity, whose code is six. The factor list uses the proved prime-divisor property; irreducibility alone is not silently renamed primality. Uniqueness supplies equal lengths, a bounded bijection, and an actual unit at each match, including repeated factors. Units have empty factorizations and zero is excluded. Sorted primary representatives, Gaussian prime classification, and Eisenstein factorization are separate targets.
Exact theorem in conservative defined notation
∀ u. ∀ z. GUnit(u) → ZPairValid(z) → GDvd(u,z)Every linked abbreviation expands hygienically to the identical original native formula.
Definition DAG
Actual proof prerequisites
Original expanded first-order statement
forall u z. (exists gr_inverse_unit_divisor. (exists ge_first_rp_unit_divisoridentity ge_first_rn_unit_divisoridentity ge_first_ip_unit_divisoridentity ge_first_in_unit_divisoridentity ge_second_rp_unit_divisoridentity ge_second_rn_unit_divisoridentity ge_second_ip_unit_divisoridentity ge_second_in_unit_divisoridentity. ((exists ge_representation_real_code_unit_divisoridentityfirst ge_representation_imaginary_code_unit_divisoridentityfirst. (((u) = ((ge_representation_real_code_unit_divisoridentityfirst) + (ge_representation_imaginary_code_unit_divisoridentityfirst)) * S ((ge_representation_real_code_unit_divisoridentityfirst) + (ge_representation_imaginary_code_unit_divisoridentityfirst)) + ((ge_representation_imaginary_code_unit_divisoridentityfirst) + (ge_representation_imaginary_code_unit_divisoridentityfirst))) /\ ((exists ge_balance_positive_unit_divisoridentityfirstreal ge_balance_negative_unit_divisoridentityfirstreal. (((((ge_representation_real_code_unit_divisoridentityfirst) = 2 * (ge_balance_positive_unit_divisoridentityfirstreal) /\ (ge_balance_negative_unit_divisoridentityfirstreal) = 0) \/ exists ge_signed_half_unit_divisoridentityfirstrealdecode. (((ge_representation_real_code_unit_divisoridentityfirst) = 2 * ge_signed_half_unit_divisoridentityfirstrealdecode + 1 /\ (ge_balance_positive_unit_divisoridentityfirstreal) = 0) /\ (ge_balance_negative_unit_divisoridentityfirstreal) = S ge_signed_half_unit_divisoridentityfirstrealdecode))) /\ ((ge_first_rp_unit_divisoridentity) + ge_balance_negative_unit_divisoridentityfirstreal = (ge_first_rn_unit_divisoridentity) + ge_balance_positive_unit_divisoridentityfirstreal))) /\ (exists ge_balance_positive_unit_divisoridentityfirstimaginary ge_balance_negative_unit_divisoridentityfirstimaginary. (((((ge_representation_imaginary_code_unit_divisoridentityfirst) = 2 * (ge_balance_positive_unit_divisoridentityfirstimaginary) /\ (ge_balance_negative_unit_divisoridentityfirstimaginary) = 0) \/ exists ge_signed_half_unit_divisoridentityfirstimaginarydecode. (((ge_representation_imaginary_code_unit_divisoridentityfirst) = 2 * ge_signed_half_unit_divisoridentityfirstimaginarydecode + 1 /\ (ge_balance_positive_unit_divisoridentityfirstimaginary) = 0) /\ (ge_balance_negative_unit_divisoridentityfirstimaginary) = S ge_signed_half_unit_divisoridentityfirstimaginarydecode))) /\ ((ge_first_ip_unit_divisoridentity) + ge_balance_negative_unit_divisoridentityfirstimaginary = (ge_first_in_unit_divisoridentity) + ge_balance_positive_unit_divisoridentityfirstimaginary)))))) /\ ((exists ge_representation_real_code_unit_divisoridentitysecond ge_representation_imaginary_code_unit_divisoridentitysecond. (((gr_inverse_unit_divisor) = ((ge_representation_real_code_unit_divisoridentitysecond) + (ge_representation_imaginary_code_unit_divisoridentitysecond)) * S ((ge_representation_real_code_unit_divisoridentitysecond) + (ge_representation_imaginary_code_unit_divisoridentitysecond)) + ((ge_representation_imaginary_code_unit_divisoridentitysecond) + (ge_representation_imaginary_code_unit_divisoridentitysecond))) /\ ((exists ge_balance_positive_unit_divisoridentitysecondreal ge_balance_negative_unit_divisoridentitysecondreal. (((((ge_representation_real_code_unit_divisoridentitysecond) = 2 * (ge_balance_positive_unit_divisoridentitysecondreal) /\ (ge_balance_negative_unit_divisoridentitysecondreal) = 0) \/ exists ge_signed_half_unit_divisoridentitysecondrealdecode. (((ge_representation_real_code_unit_divisoridentitysecond) = 2 * ge_signed_half_unit_divisoridentitysecondrealdecode + 1 /\ (ge_balance_positive_unit_divisoridentitysecondreal) = 0) /\ (ge_balance_negative_unit_divisoridentitysecondreal) = S ge_signed_half_unit_divisoridentitysecondrealdecode))) /\ ((ge_second_rp_unit_divisoridentity) + ge_balance_negative_unit_divisoridentitysecondreal = (ge_second_rn_unit_divisoridentity) + ge_balance_positive_unit_divisoridentitysecondreal))) /\ (exists ge_balance_positive_unit_divisoridentitysecondimaginary ge_balance_negative_unit_divisoridentitysecondimaginary. (((((ge_representation_imaginary_code_unit_divisoridentitysecond) = 2 * (ge_balance_positive_unit_divisoridentitysecondimaginary) /\ (ge_balance_negative_unit_divisoridentitysecondimaginary) = 0) \/ exists ge_signed_half_unit_divisoridentitysecondimaginarydecode. (((ge_representation_imaginary_code_unit_divisoridentitysecond) = 2 * ge_signed_half_unit_divisoridentitysecondimaginarydecode + 1 /\ (ge_balance_positive_unit_divisoridentitysecondimaginary) = 0) /\ (ge_balance_negative_unit_divisoridentitysecondimaginary) = S ge_signed_half_unit_divisoridentitysecondimaginarydecode))) /\ ((ge_second_ip_unit_divisoridentity) + ge_balance_negative_unit_divisoridentitysecondimaginary = (ge_second_in_unit_divisoridentity) + ge_balance_positive_unit_divisoridentitysecondimaginary)))))) /\ (exists ge_representation_real_code_unit_divisoridentityoutput ge_representation_imaginary_code_unit_divisoridentityoutput. (((6) = ((ge_representation_real_code_unit_divisoridentityoutput) + (ge_representation_imaginary_code_unit_divisoridentityoutput)) * S ((ge_representation_real_code_unit_divisoridentityoutput) + (ge_representation_imaginary_code_unit_divisoridentityoutput)) + ((ge_representation_imaginary_code_unit_divisoridentityoutput) + (ge_representation_imaginary_code_unit_divisoridentityoutput))) /\ ((exists ge_balance_positive_unit_divisoridentityoutputreal ge_balance_negative_unit_divisoridentityoutputreal. (((((ge_representation_real_code_unit_divisoridentityoutput) = 2 * (ge_balance_positive_unit_divisoridentityoutputreal) /\ (ge_balance_negative_unit_divisoridentityoutputreal) = 0) \/ exists ge_signed_half_unit_divisoridentityoutputrealdecode. (((ge_representation_real_code_unit_divisoridentityoutput) = 2 * ge_signed_half_unit_divisoridentityoutputrealdecode + 1 /\ (ge_balance_positive_unit_divisoridentityoutputreal) = 0) /\ (ge_balance_negative_unit_divisoridentityoutputreal) = S ge_signed_half_unit_divisoridentityoutputrealdecode))) /\ ((((((((ge_first_rp_unit_divisoridentity) * (ge_second_rp_unit_divisoridentity))) + (((ge_first_rn_unit_divisoridentity) * (ge_second_rn_unit_divisoridentity))))) + (((((ge_first_ip_unit_divisoridentity) * (ge_second_in_unit_divisoridentity))) + (((ge_first_in_unit_divisoridentity) * (ge_second_ip_unit_divisoridentity))))))) + ge_balance_negative_unit_divisoridentityoutputreal = (((((((ge_first_rp_unit_divisoridentity) * (ge_second_rn_unit_divisoridentity))) + (((ge_first_rn_unit_divisoridentity) * (ge_second_rp_unit_divisoridentity))))) + (((((ge_first_ip_unit_divisoridentity) * (ge_second_ip_unit_divisoridentity))) + (((ge_first_in_unit_divisoridentity) * (ge_second_in_unit_divisoridentity))))))) + ge_balance_positive_unit_divisoridentityoutputreal))) /\ (exists ge_balance_positive_unit_divisoridentityoutputimaginary ge_balance_negative_unit_divisoridentityoutputimaginary. (((((ge_representation_imaginary_code_unit_divisoridentityoutput) = 2 * (ge_balance_positive_unit_divisoridentityoutputimaginary) /\ (ge_balance_negative_unit_divisoridentityoutputimaginary) = 0) \/ exists ge_signed_half_unit_divisoridentityoutputimaginarydecode. (((ge_representation_imaginary_code_unit_divisoridentityoutput) = 2 * ge_signed_half_unit_divisoridentityoutputimaginarydecode + 1 /\ (ge_balance_positive_unit_divisoridentityoutputimaginary) = 0) /\ (ge_balance_negative_unit_divisoridentityoutputimaginary) = S ge_signed_half_unit_divisoridentityoutputimaginarydecode))) /\ ((((((((ge_first_rp_unit_divisoridentity) * (ge_second_ip_unit_divisoridentity))) + (((ge_first_rn_unit_divisoridentity) * (ge_second_in_unit_divisoridentity))))) + (((((ge_first_ip_unit_divisoridentity) * (ge_second_rp_unit_divisoridentity))) + (((ge_first_in_unit_divisoridentity) * (ge_second_rn_unit_divisoridentity))))))) + ge_balance_negative_unit_divisoridentityoutputimaginary = (((((((ge_first_rp_unit_divisoridentity) * (ge_second_in_unit_divisoridentity))) + (((ge_first_rn_unit_divisoridentity) * (ge_second_ip_unit_divisoridentity))))) + (((((ge_first_ip_unit_divisoridentity) * (ge_second_rn_unit_divisoridentity))) + (((ge_first_in_unit_divisoridentity) * (ge_second_rp_unit_divisoridentity))))))) + ge_balance_positive_unit_divisoridentityoutputimaginary)))))))))) -> (exists ge_real_positive_unit_multiple_domain ge_real_negative_unit_multiple_domain ge_imaginary_positive_unit_multiple_domain ge_imaginary_negative_unit_multiple_domain. (exists ge_real_code_unit_multiple_domaindecode ge_imaginary_code_unit_multiple_domaindecode. (((z) = ((ge_real_code_unit_multiple_domaindecode) + (ge_imaginary_code_unit_multiple_domaindecode)) * S ((ge_real_code_unit_multiple_domaindecode) + (ge_imaginary_code_unit_multiple_domaindecode)) + ((ge_imaginary_code_unit_multiple_domaindecode) + (ge_imaginary_code_unit_multiple_domaindecode))) /\ (((((ge_real_code_unit_multiple_domaindecode) = 2 * (ge_real_positive_unit_multiple_domain) /\ (ge_real_negative_unit_multiple_domain) = 0) \/ exists ge_signed_half_ge_unit_multiple_domaindecode_real. (((ge_real_code_unit_multiple_domaindecode) = 2 * ge_signed_half_ge_unit_multiple_domaindecode_real + 1 /\ (ge_real_positive_unit_multiple_domain) = 0) /\ (ge_real_negative_unit_multiple_domain) = S ge_signed_half_ge_unit_multiple_domaindecode_real))) /\ ((((ge_imaginary_code_unit_multiple_domaindecode) = 2 * (ge_imaginary_positive_unit_multiple_domain) /\ (ge_imaginary_negative_unit_multiple_domain) = 0) \/ exists ge_signed_half_ge_unit_multiple_domaindecode_imaginary. (((ge_imaginary_code_unit_multiple_domaindecode) = 2 * ge_signed_half_ge_unit_multiple_domaindecode_imaginary + 1 /\ (ge_imaginary_positive_unit_multiple_domain) = 0) /\ (ge_imaginary_negative_unit_multiple_domain) = S ge_signed_half_ge_unit_multiple_domaindecode_imaginary))))))) -> (exists gr_quotient_unit_divides. (exists ge_first_rp_unit_dividesproduct ge_first_rn_unit_dividesproduct ge_first_ip_unit_dividesproduct ge_first_in_unit_dividesproduct ge_second_rp_unit_dividesproduct ge_second_rn_unit_dividesproduct ge_second_ip_unit_dividesproduct ge_second_in_unit_dividesproduct. ((exists ge_representation_real_code_unit_dividesproductfirst ge_representation_imaginary_code_unit_dividesproductfirst. (((u) = ((ge_representation_real_code_unit_dividesproductfirst) + (ge_representation_imaginary_code_unit_dividesproductfirst)) * S ((ge_representation_real_code_unit_dividesproductfirst) + (ge_representation_imaginary_code_unit_dividesproductfirst)) + ((ge_representation_imaginary_code_unit_dividesproductfirst) + (ge_representation_imaginary_code_unit_dividesproductfirst))) /\ ((exists ge_balance_positive_unit_dividesproductfirstreal ge_balance_negative_unit_dividesproductfirstreal. (((((ge_representation_real_code_unit_dividesproductfirst) = 2 * (ge_balance_positive_unit_dividesproductfirstreal) /\ (ge_balance_negative_unit_dividesproductfirstreal) = 0) \/ exists ge_signed_half_unit_dividesproductfirstrealdecode. (((ge_representation_real_code_unit_dividesproductfirst) = 2 * ge_signed_half_unit_dividesproductfirstrealdecode + 1 /\ (ge_balance_positive_unit_dividesproductfirstreal) = 0) /\ (ge_balance_negative_unit_dividesproductfirstreal) = S ge_signed_half_unit_dividesproductfirstrealdecode))) /\ ((ge_first_rp_unit_dividesproduct) + ge_balance_negative_unit_dividesproductfirstreal = (ge_first_rn_unit_dividesproduct) + ge_balance_positive_unit_dividesproductfirstreal))) /\ (exists ge_balance_positive_unit_dividesproductfirstimaginary ge_balance_negative_unit_dividesproductfirstimaginary. (((((ge_representation_imaginary_code_unit_dividesproductfirst) = 2 * (ge_balance_positive_unit_dividesproductfirstimaginary) /\ (ge_balance_negative_unit_dividesproductfirstimaginary) = 0) \/ exists ge_signed_half_unit_dividesproductfirstimaginarydecode. (((ge_representation_imaginary_code_unit_dividesproductfirst) = 2 * ge_signed_half_unit_dividesproductfirstimaginarydecode + 1 /\ (ge_balance_positive_unit_dividesproductfirstimaginary) = 0) /\ (ge_balance_negative_unit_dividesproductfirstimaginary) = S ge_signed_half_unit_dividesproductfirstimaginarydecode))) /\ ((ge_first_ip_unit_dividesproduct) + ge_balance_negative_unit_dividesproductfirstimaginary = (ge_first_in_unit_dividesproduct) + ge_balance_positive_unit_dividesproductfirstimaginary)))))) /\ ((exists ge_representation_real_code_unit_dividesproductsecond ge_representation_imaginary_code_unit_dividesproductsecond. (((gr_quotient_unit_divides) = ((ge_representation_real_code_unit_dividesproductsecond) + (ge_representation_imaginary_code_unit_dividesproductsecond)) * S ((ge_representation_real_code_unit_dividesproductsecond) + (ge_representation_imaginary_code_unit_dividesproductsecond)) + ((ge_representation_imaginary_code_unit_dividesproductsecond) + (ge_representation_imaginary_code_unit_dividesproductsecond))) /\ ((exists ge_balance_positive_unit_dividesproductsecondreal ge_balance_negative_unit_dividesproductsecondreal. (((((ge_representation_real_code_unit_dividesproductsecond) = 2 * (ge_balance_positive_unit_dividesproductsecondreal) /\ (ge_balance_negative_unit_dividesproductsecondreal) = 0) \/ exists ge_signed_half_unit_dividesproductsecondrealdecode. (((ge_representation_real_code_unit_dividesproductsecond) = 2 * ge_signed_half_unit_dividesproductsecondrealdecode + 1 /\ (ge_balance_positive_unit_dividesproductsecondreal) = 0) /\ (ge_balance_negative_unit_dividesproductsecondreal) = S ge_signed_half_unit_dividesproductsecondrealdecode))) /\ ((ge_second_rp_unit_dividesproduct) + ge_balance_negative_unit_dividesproductsecondreal = (ge_second_rn_unit_dividesproduct) + ge_balance_positive_unit_dividesproductsecondreal))) /\ (exists ge_balance_positive_unit_dividesproductsecondimaginary ge_balance_negative_unit_dividesproductsecondimaginary. (((((ge_representation_imaginary_code_unit_dividesproductsecond) = 2 * (ge_balance_positive_unit_dividesproductsecondimaginary) /\ (ge_balance_negative_unit_dividesproductsecondimaginary) = 0) \/ exists ge_signed_half_unit_dividesproductsecondimaginarydecode. (((ge_representation_imaginary_code_unit_dividesproductsecond) = 2 * ge_signed_half_unit_dividesproductsecondimaginarydecode + 1 /\ (ge_balance_positive_unit_dividesproductsecondimaginary) = 0) /\ (ge_balance_negative_unit_dividesproductsecondimaginary) = S ge_signed_half_unit_dividesproductsecondimaginarydecode))) /\ ((ge_second_ip_unit_dividesproduct) + ge_balance_negative_unit_dividesproductsecondimaginary = (ge_second_in_unit_dividesproduct) + ge_balance_positive_unit_dividesproductsecondimaginary)))))) /\ (exists ge_representation_real_code_unit_dividesproductoutput ge_representation_imaginary_code_unit_dividesproductoutput. (((z) = ((ge_representation_real_code_unit_dividesproductoutput) + (ge_representation_imaginary_code_unit_dividesproductoutput)) * S ((ge_representation_real_code_unit_dividesproductoutput) + (ge_representation_imaginary_code_unit_dividesproductoutput)) + ((ge_representation_imaginary_code_unit_dividesproductoutput) + (ge_representation_imaginary_code_unit_dividesproductoutput))) /\ ((exists ge_balance_positive_unit_dividesproductoutputreal ge_balance_negative_unit_dividesproductoutputreal. (((((ge_representation_real_code_unit_dividesproductoutput) = 2 * (ge_balance_positive_unit_dividesproductoutputreal) /\ (ge_balance_negative_unit_dividesproductoutputreal) = 0) \/ exists ge_signed_half_unit_dividesproductoutputrealdecode. (((ge_representation_real_code_unit_dividesproductoutput) = 2 * ge_signed_half_unit_dividesproductoutputrealdecode + 1 /\ (ge_balance_positive_unit_dividesproductoutputreal) = 0) /\ (ge_balance_negative_unit_dividesproductoutputreal) = S ge_signed_half_unit_dividesproductoutputrealdecode))) /\ ((((((((ge_first_rp_unit_dividesproduct) * (ge_second_rp_unit_dividesproduct))) + (((ge_first_rn_unit_dividesproduct) * (ge_second_rn_unit_dividesproduct))))) + (((((ge_first_ip_unit_dividesproduct) * (ge_second_in_unit_dividesproduct))) + (((ge_first_in_unit_dividesproduct) * (ge_second_ip_unit_dividesproduct))))))) + ge_balance_negative_unit_dividesproductoutputreal = (((((((ge_first_rp_unit_dividesproduct) * (ge_second_rn_unit_dividesproduct))) + (((ge_first_rn_unit_dividesproduct) * (ge_second_rp_unit_dividesproduct))))) + (((((ge_first_ip_unit_dividesproduct) * (ge_second_ip_unit_dividesproduct))) + (((ge_first_in_unit_dividesproduct) * (ge_second_in_unit_dividesproduct))))))) + ge_balance_positive_unit_dividesproductoutputreal))) /\ (exists ge_balance_positive_unit_dividesproductoutputimaginary ge_balance_negative_unit_dividesproductoutputimaginary. (((((ge_representation_imaginary_code_unit_dividesproductoutput) = 2 * (ge_balance_positive_unit_dividesproductoutputimaginary) /\ (ge_balance_negative_unit_dividesproductoutputimaginary) = 0) \/ exists ge_signed_half_unit_dividesproductoutputimaginarydecode. (((ge_representation_imaginary_code_unit_dividesproductoutput) = 2 * ge_signed_half_unit_dividesproductoutputimaginarydecode + 1 /\ (ge_balance_positive_unit_dividesproductoutputimaginary) = 0) /\ (ge_balance_negative_unit_dividesproductoutputimaginary) = S ge_signed_half_unit_dividesproductoutputimaginarydecode))) /\ ((((((((ge_first_rp_unit_dividesproduct) * (ge_second_ip_unit_dividesproduct))) + (((ge_first_rn_unit_dividesproduct) * (ge_second_in_unit_dividesproduct))))) + (((((ge_first_ip_unit_dividesproduct) * (ge_second_rp_unit_dividesproduct))) + (((ge_first_in_unit_dividesproduct) * (ge_second_rn_unit_dividesproduct))))))) + ge_balance_negative_unit_dividesproductoutputimaginary = (((((((ge_first_rp_unit_dividesproduct) * (ge_second_in_unit_dividesproduct))) + (((ge_first_rn_unit_dividesproduct) * (ge_second_ip_unit_dividesproduct))))) + (((((ge_first_ip_unit_dividesproduct) * (ge_second_rn_unit_dividesproduct))) + (((ge_first_in_unit_dividesproduct) * (ge_second_rp_unit_dividesproduct))))))) + ge_balance_positive_unit_dividesproductoutputimaginary))))))))))Complete tactic proof in conservative notation
All 12 original proof lines are preserved. Only local proposition formulas are abbreviated; every abbreviation has an exact binder-safe expansion check. The linked exact edition contains the unchanged replay script.
Read the argument
Proof checkpoints
12 script commands · 2 reading checkpoints · 0 local claims
This is a reading aid, not a new proof or a proof-tree certificate. Checkpoint groups are consecutive commands, not inferred branch boundaries. Every step links to the preserved script.
Definition notation is shown below. Open the paired exact edition for the original native formulas. Source pairing is not a new equivalence certificate.
Named ingredients (2)
01Fix variables and assumptionsL1–4
02Use earlier factsL5–12
Instantiate or apply named facts and discharge the corresponding proof obligations.
Original defined command ledger · 12 lines
- 0001
intro u - 0002
intro z - 0003
intro hu - 0004
intro hz - 0005
specialize gaussian_divides_transitive (u) - 0006
specialize gaussian_divides_transitive (6) - 0007
specialize gaussian_divides_transitive (z) - 0008
apply gaussian_divides_transitive - 0009
exact hu - 0010
specialize gaussian_one_divides (z) - 0011
apply gaussian_one_divides - 0012
exact hz