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
∀ a. ZPairValid(a) → GAssociate(a,a)Every linked abbreviation expands hygienically to the identical original native formula.
Definition DAG
Actual proof prerequisites
Original expanded first-order statement
forall a. (exists ge_real_positive_associate_refl_domain ge_real_negative_associate_refl_domain ge_imaginary_positive_associate_refl_domain ge_imaginary_negative_associate_refl_domain. (exists ge_real_code_associate_refl_domaindecode ge_imaginary_code_associate_refl_domaindecode. (((a) = ((ge_real_code_associate_refl_domaindecode) + (ge_imaginary_code_associate_refl_domaindecode)) * S ((ge_real_code_associate_refl_domaindecode) + (ge_imaginary_code_associate_refl_domaindecode)) + ((ge_imaginary_code_associate_refl_domaindecode) + (ge_imaginary_code_associate_refl_domaindecode))) /\ (((((ge_real_code_associate_refl_domaindecode) = 2 * (ge_real_positive_associate_refl_domain) /\ (ge_real_negative_associate_refl_domain) = 0) \/ exists ge_signed_half_ge_associate_refl_domaindecode_real. (((ge_real_code_associate_refl_domaindecode) = 2 * ge_signed_half_ge_associate_refl_domaindecode_real + 1 /\ (ge_real_positive_associate_refl_domain) = 0) /\ (ge_real_negative_associate_refl_domain) = S ge_signed_half_ge_associate_refl_domaindecode_real))) /\ ((((ge_imaginary_code_associate_refl_domaindecode) = 2 * (ge_imaginary_positive_associate_refl_domain) /\ (ge_imaginary_negative_associate_refl_domain) = 0) \/ exists ge_signed_half_ge_associate_refl_domaindecode_imaginary. (((ge_imaginary_code_associate_refl_domaindecode) = 2 * ge_signed_half_ge_associate_refl_domaindecode_imaginary + 1 /\ (ge_imaginary_positive_associate_refl_domain) = 0) /\ (ge_imaginary_negative_associate_refl_domain) = S ge_signed_half_ge_associate_refl_domaindecode_imaginary))))))) -> (exists gr_unit_associate_refl. ((exists gr_inverse_associate_reflunit. (exists ge_first_rp_associate_reflunitidentity ge_first_rn_associate_reflunitidentity ge_first_ip_associate_reflunitidentity ge_first_in_associate_reflunitidentity ge_second_rp_associate_reflunitidentity ge_second_rn_associate_reflunitidentity ge_second_ip_associate_reflunitidentity ge_second_in_associate_reflunitidentity. ((exists ge_representation_real_code_associate_reflunitidentityfirst ge_representation_imaginary_code_associate_reflunitidentityfirst. (((gr_unit_associate_refl) = ((ge_representation_real_code_associate_reflunitidentityfirst) + (ge_representation_imaginary_code_associate_reflunitidentityfirst)) * S ((ge_representation_real_code_associate_reflunitidentityfirst) + (ge_representation_imaginary_code_associate_reflunitidentityfirst)) + ((ge_representation_imaginary_code_associate_reflunitidentityfirst) + (ge_representation_imaginary_code_associate_reflunitidentityfirst))) /\ ((exists ge_balance_positive_associate_reflunitidentityfirstreal ge_balance_negative_associate_reflunitidentityfirstreal. (((((ge_representation_real_code_associate_reflunitidentityfirst) = 2 * (ge_balance_positive_associate_reflunitidentityfirstreal) /\ (ge_balance_negative_associate_reflunitidentityfirstreal) = 0) \/ exists ge_signed_half_associate_reflunitidentityfirstrealdecode. (((ge_representation_real_code_associate_reflunitidentityfirst) = 2 * ge_signed_half_associate_reflunitidentityfirstrealdecode + 1 /\ (ge_balance_positive_associate_reflunitidentityfirstreal) = 0) /\ (ge_balance_negative_associate_reflunitidentityfirstreal) = S ge_signed_half_associate_reflunitidentityfirstrealdecode))) /\ ((ge_first_rp_associate_reflunitidentity) + ge_balance_negative_associate_reflunitidentityfirstreal = (ge_first_rn_associate_reflunitidentity) + ge_balance_positive_associate_reflunitidentityfirstreal))) /\ (exists ge_balance_positive_associate_reflunitidentityfirstimaginary ge_balance_negative_associate_reflunitidentityfirstimaginary. (((((ge_representation_imaginary_code_associate_reflunitidentityfirst) = 2 * (ge_balance_positive_associate_reflunitidentityfirstimaginary) /\ (ge_balance_negative_associate_reflunitidentityfirstimaginary) = 0) \/ exists ge_signed_half_associate_reflunitidentityfirstimaginarydecode. (((ge_representation_imaginary_code_associate_reflunitidentityfirst) = 2 * ge_signed_half_associate_reflunitidentityfirstimaginarydecode + 1 /\ (ge_balance_positive_associate_reflunitidentityfirstimaginary) = 0) /\ (ge_balance_negative_associate_reflunitidentityfirstimaginary) = S ge_signed_half_associate_reflunitidentityfirstimaginarydecode))) /\ ((ge_first_ip_associate_reflunitidentity) + ge_balance_negative_associate_reflunitidentityfirstimaginary = (ge_first_in_associate_reflunitidentity) + ge_balance_positive_associate_reflunitidentityfirstimaginary)))))) /\ ((exists ge_representation_real_code_associate_reflunitidentitysecond ge_representation_imaginary_code_associate_reflunitidentitysecond. (((gr_inverse_associate_reflunit) = ((ge_representation_real_code_associate_reflunitidentitysecond) + (ge_representation_imaginary_code_associate_reflunitidentitysecond)) * S ((ge_representation_real_code_associate_reflunitidentitysecond) + (ge_representation_imaginary_code_associate_reflunitidentitysecond)) + ((ge_representation_imaginary_code_associate_reflunitidentitysecond) + (ge_representation_imaginary_code_associate_reflunitidentitysecond))) /\ ((exists ge_balance_positive_associate_reflunitidentitysecondreal ge_balance_negative_associate_reflunitidentitysecondreal. (((((ge_representation_real_code_associate_reflunitidentitysecond) = 2 * (ge_balance_positive_associate_reflunitidentitysecondreal) /\ (ge_balance_negative_associate_reflunitidentitysecondreal) = 0) \/ exists ge_signed_half_associate_reflunitidentitysecondrealdecode. (((ge_representation_real_code_associate_reflunitidentitysecond) = 2 * ge_signed_half_associate_reflunitidentitysecondrealdecode + 1 /\ (ge_balance_positive_associate_reflunitidentitysecondreal) = 0) /\ (ge_balance_negative_associate_reflunitidentitysecondreal) = S ge_signed_half_associate_reflunitidentitysecondrealdecode))) /\ ((ge_second_rp_associate_reflunitidentity) + ge_balance_negative_associate_reflunitidentitysecondreal = (ge_second_rn_associate_reflunitidentity) + ge_balance_positive_associate_reflunitidentitysecondreal))) /\ (exists ge_balance_positive_associate_reflunitidentitysecondimaginary ge_balance_negative_associate_reflunitidentitysecondimaginary. (((((ge_representation_imaginary_code_associate_reflunitidentitysecond) = 2 * (ge_balance_positive_associate_reflunitidentitysecondimaginary) /\ (ge_balance_negative_associate_reflunitidentitysecondimaginary) = 0) \/ exists ge_signed_half_associate_reflunitidentitysecondimaginarydecode. (((ge_representation_imaginary_code_associate_reflunitidentitysecond) = 2 * ge_signed_half_associate_reflunitidentitysecondimaginarydecode + 1 /\ (ge_balance_positive_associate_reflunitidentitysecondimaginary) = 0) /\ (ge_balance_negative_associate_reflunitidentitysecondimaginary) = S ge_signed_half_associate_reflunitidentitysecondimaginarydecode))) /\ ((ge_second_ip_associate_reflunitidentity) + ge_balance_negative_associate_reflunitidentitysecondimaginary = (ge_second_in_associate_reflunitidentity) + ge_balance_positive_associate_reflunitidentitysecondimaginary)))))) /\ (exists ge_representation_real_code_associate_reflunitidentityoutput ge_representation_imaginary_code_associate_reflunitidentityoutput. (((6) = ((ge_representation_real_code_associate_reflunitidentityoutput) + (ge_representation_imaginary_code_associate_reflunitidentityoutput)) * S ((ge_representation_real_code_associate_reflunitidentityoutput) + (ge_representation_imaginary_code_associate_reflunitidentityoutput)) + ((ge_representation_imaginary_code_associate_reflunitidentityoutput) + (ge_representation_imaginary_code_associate_reflunitidentityoutput))) /\ ((exists ge_balance_positive_associate_reflunitidentityoutputreal ge_balance_negative_associate_reflunitidentityoutputreal. (((((ge_representation_real_code_associate_reflunitidentityoutput) = 2 * (ge_balance_positive_associate_reflunitidentityoutputreal) /\ (ge_balance_negative_associate_reflunitidentityoutputreal) = 0) \/ exists ge_signed_half_associate_reflunitidentityoutputrealdecode. (((ge_representation_real_code_associate_reflunitidentityoutput) = 2 * ge_signed_half_associate_reflunitidentityoutputrealdecode + 1 /\ (ge_balance_positive_associate_reflunitidentityoutputreal) = 0) /\ (ge_balance_negative_associate_reflunitidentityoutputreal) = S ge_signed_half_associate_reflunitidentityoutputrealdecode))) /\ ((((((((ge_first_rp_associate_reflunitidentity) * (ge_second_rp_associate_reflunitidentity))) + (((ge_first_rn_associate_reflunitidentity) * (ge_second_rn_associate_reflunitidentity))))) + (((((ge_first_ip_associate_reflunitidentity) * (ge_second_in_associate_reflunitidentity))) + (((ge_first_in_associate_reflunitidentity) * (ge_second_ip_associate_reflunitidentity))))))) + ge_balance_negative_associate_reflunitidentityoutputreal = (((((((ge_first_rp_associate_reflunitidentity) * (ge_second_rn_associate_reflunitidentity))) + (((ge_first_rn_associate_reflunitidentity) * (ge_second_rp_associate_reflunitidentity))))) + (((((ge_first_ip_associate_reflunitidentity) * (ge_second_ip_associate_reflunitidentity))) + (((ge_first_in_associate_reflunitidentity) * (ge_second_in_associate_reflunitidentity))))))) + ge_balance_positive_associate_reflunitidentityoutputreal))) /\ (exists ge_balance_positive_associate_reflunitidentityoutputimaginary ge_balance_negative_associate_reflunitidentityoutputimaginary. (((((ge_representation_imaginary_code_associate_reflunitidentityoutput) = 2 * (ge_balance_positive_associate_reflunitidentityoutputimaginary) /\ (ge_balance_negative_associate_reflunitidentityoutputimaginary) = 0) \/ exists ge_signed_half_associate_reflunitidentityoutputimaginarydecode. (((ge_representation_imaginary_code_associate_reflunitidentityoutput) = 2 * ge_signed_half_associate_reflunitidentityoutputimaginarydecode + 1 /\ (ge_balance_positive_associate_reflunitidentityoutputimaginary) = 0) /\ (ge_balance_negative_associate_reflunitidentityoutputimaginary) = S ge_signed_half_associate_reflunitidentityoutputimaginarydecode))) /\ ((((((((ge_first_rp_associate_reflunitidentity) * (ge_second_ip_associate_reflunitidentity))) + (((ge_first_rn_associate_reflunitidentity) * (ge_second_in_associate_reflunitidentity))))) + (((((ge_first_ip_associate_reflunitidentity) * (ge_second_rp_associate_reflunitidentity))) + (((ge_first_in_associate_reflunitidentity) * (ge_second_rn_associate_reflunitidentity))))))) + ge_balance_negative_associate_reflunitidentityoutputimaginary = (((((((ge_first_rp_associate_reflunitidentity) * (ge_second_in_associate_reflunitidentity))) + (((ge_first_rn_associate_reflunitidentity) * (ge_second_ip_associate_reflunitidentity))))) + (((((ge_first_ip_associate_reflunitidentity) * (ge_second_rn_associate_reflunitidentity))) + (((ge_first_in_associate_reflunitidentity) * (ge_second_rp_associate_reflunitidentity))))))) + ge_balance_positive_associate_reflunitidentityoutputimaginary)))))))))) /\ (exists ge_first_rp_associate_refltransport ge_first_rn_associate_refltransport ge_first_ip_associate_refltransport ge_first_in_associate_refltransport ge_second_rp_associate_refltransport ge_second_rn_associate_refltransport ge_second_ip_associate_refltransport ge_second_in_associate_refltransport. ((exists ge_representation_real_code_associate_refltransportfirst ge_representation_imaginary_code_associate_refltransportfirst. (((gr_unit_associate_refl) = ((ge_representation_real_code_associate_refltransportfirst) + (ge_representation_imaginary_code_associate_refltransportfirst)) * S ((ge_representation_real_code_associate_refltransportfirst) + (ge_representation_imaginary_code_associate_refltransportfirst)) + ((ge_representation_imaginary_code_associate_refltransportfirst) + (ge_representation_imaginary_code_associate_refltransportfirst))) /\ ((exists ge_balance_positive_associate_refltransportfirstreal ge_balance_negative_associate_refltransportfirstreal. (((((ge_representation_real_code_associate_refltransportfirst) = 2 * (ge_balance_positive_associate_refltransportfirstreal) /\ (ge_balance_negative_associate_refltransportfirstreal) = 0) \/ exists ge_signed_half_associate_refltransportfirstrealdecode. (((ge_representation_real_code_associate_refltransportfirst) = 2 * ge_signed_half_associate_refltransportfirstrealdecode + 1 /\ (ge_balance_positive_associate_refltransportfirstreal) = 0) /\ (ge_balance_negative_associate_refltransportfirstreal) = S ge_signed_half_associate_refltransportfirstrealdecode))) /\ ((ge_first_rp_associate_refltransport) + ge_balance_negative_associate_refltransportfirstreal = (ge_first_rn_associate_refltransport) + ge_balance_positive_associate_refltransportfirstreal))) /\ (exists ge_balance_positive_associate_refltransportfirstimaginary ge_balance_negative_associate_refltransportfirstimaginary. (((((ge_representation_imaginary_code_associate_refltransportfirst) = 2 * (ge_balance_positive_associate_refltransportfirstimaginary) /\ (ge_balance_negative_associate_refltransportfirstimaginary) = 0) \/ exists ge_signed_half_associate_refltransportfirstimaginarydecode. (((ge_representation_imaginary_code_associate_refltransportfirst) = 2 * ge_signed_half_associate_refltransportfirstimaginarydecode + 1 /\ (ge_balance_positive_associate_refltransportfirstimaginary) = 0) /\ (ge_balance_negative_associate_refltransportfirstimaginary) = S ge_signed_half_associate_refltransportfirstimaginarydecode))) /\ ((ge_first_ip_associate_refltransport) + ge_balance_negative_associate_refltransportfirstimaginary = (ge_first_in_associate_refltransport) + ge_balance_positive_associate_refltransportfirstimaginary)))))) /\ ((exists ge_representation_real_code_associate_refltransportsecond ge_representation_imaginary_code_associate_refltransportsecond. (((a) = ((ge_representation_real_code_associate_refltransportsecond) + (ge_representation_imaginary_code_associate_refltransportsecond)) * S ((ge_representation_real_code_associate_refltransportsecond) + (ge_representation_imaginary_code_associate_refltransportsecond)) + ((ge_representation_imaginary_code_associate_refltransportsecond) + (ge_representation_imaginary_code_associate_refltransportsecond))) /\ ((exists ge_balance_positive_associate_refltransportsecondreal ge_balance_negative_associate_refltransportsecondreal. (((((ge_representation_real_code_associate_refltransportsecond) = 2 * (ge_balance_positive_associate_refltransportsecondreal) /\ (ge_balance_negative_associate_refltransportsecondreal) = 0) \/ exists ge_signed_half_associate_refltransportsecondrealdecode. (((ge_representation_real_code_associate_refltransportsecond) = 2 * ge_signed_half_associate_refltransportsecondrealdecode + 1 /\ (ge_balance_positive_associate_refltransportsecondreal) = 0) /\ (ge_balance_negative_associate_refltransportsecondreal) = S ge_signed_half_associate_refltransportsecondrealdecode))) /\ ((ge_second_rp_associate_refltransport) + ge_balance_negative_associate_refltransportsecondreal = (ge_second_rn_associate_refltransport) + ge_balance_positive_associate_refltransportsecondreal))) /\ (exists ge_balance_positive_associate_refltransportsecondimaginary ge_balance_negative_associate_refltransportsecondimaginary. (((((ge_representation_imaginary_code_associate_refltransportsecond) = 2 * (ge_balance_positive_associate_refltransportsecondimaginary) /\ (ge_balance_negative_associate_refltransportsecondimaginary) = 0) \/ exists ge_signed_half_associate_refltransportsecondimaginarydecode. (((ge_representation_imaginary_code_associate_refltransportsecond) = 2 * ge_signed_half_associate_refltransportsecondimaginarydecode + 1 /\ (ge_balance_positive_associate_refltransportsecondimaginary) = 0) /\ (ge_balance_negative_associate_refltransportsecondimaginary) = S ge_signed_half_associate_refltransportsecondimaginarydecode))) /\ ((ge_second_ip_associate_refltransport) + ge_balance_negative_associate_refltransportsecondimaginary = (ge_second_in_associate_refltransport) + ge_balance_positive_associate_refltransportsecondimaginary)))))) /\ (exists ge_representation_real_code_associate_refltransportoutput ge_representation_imaginary_code_associate_refltransportoutput. (((a) = ((ge_representation_real_code_associate_refltransportoutput) + (ge_representation_imaginary_code_associate_refltransportoutput)) * S ((ge_representation_real_code_associate_refltransportoutput) + (ge_representation_imaginary_code_associate_refltransportoutput)) + ((ge_representation_imaginary_code_associate_refltransportoutput) + (ge_representation_imaginary_code_associate_refltransportoutput))) /\ ((exists ge_balance_positive_associate_refltransportoutputreal ge_balance_negative_associate_refltransportoutputreal. (((((ge_representation_real_code_associate_refltransportoutput) = 2 * (ge_balance_positive_associate_refltransportoutputreal) /\ (ge_balance_negative_associate_refltransportoutputreal) = 0) \/ exists ge_signed_half_associate_refltransportoutputrealdecode. (((ge_representation_real_code_associate_refltransportoutput) = 2 * ge_signed_half_associate_refltransportoutputrealdecode + 1 /\ (ge_balance_positive_associate_refltransportoutputreal) = 0) /\ (ge_balance_negative_associate_refltransportoutputreal) = S ge_signed_half_associate_refltransportoutputrealdecode))) /\ ((((((((ge_first_rp_associate_refltransport) * (ge_second_rp_associate_refltransport))) + (((ge_first_rn_associate_refltransport) * (ge_second_rn_associate_refltransport))))) + (((((ge_first_ip_associate_refltransport) * (ge_second_in_associate_refltransport))) + (((ge_first_in_associate_refltransport) * (ge_second_ip_associate_refltransport))))))) + ge_balance_negative_associate_refltransportoutputreal = (((((((ge_first_rp_associate_refltransport) * (ge_second_rn_associate_refltransport))) + (((ge_first_rn_associate_refltransport) * (ge_second_rp_associate_refltransport))))) + (((((ge_first_ip_associate_refltransport) * (ge_second_ip_associate_refltransport))) + (((ge_first_in_associate_refltransport) * (ge_second_in_associate_refltransport))))))) + ge_balance_positive_associate_refltransportoutputreal))) /\ (exists ge_balance_positive_associate_refltransportoutputimaginary ge_balance_negative_associate_refltransportoutputimaginary. (((((ge_representation_imaginary_code_associate_refltransportoutput) = 2 * (ge_balance_positive_associate_refltransportoutputimaginary) /\ (ge_balance_negative_associate_refltransportoutputimaginary) = 0) \/ exists ge_signed_half_associate_refltransportoutputimaginarydecode. (((ge_representation_imaginary_code_associate_refltransportoutput) = 2 * ge_signed_half_associate_refltransportoutputimaginarydecode + 1 /\ (ge_balance_positive_associate_refltransportoutputimaginary) = 0) /\ (ge_balance_negative_associate_refltransportoutputimaginary) = S ge_signed_half_associate_refltransportoutputimaginarydecode))) /\ ((((((((ge_first_rp_associate_refltransport) * (ge_second_ip_associate_refltransport))) + (((ge_first_rn_associate_refltransport) * (ge_second_in_associate_refltransport))))) + (((((ge_first_ip_associate_refltransport) * (ge_second_rp_associate_refltransport))) + (((ge_first_in_associate_refltransport) * (ge_second_rn_associate_refltransport))))))) + ge_balance_negative_associate_refltransportoutputimaginary = (((((((ge_first_rp_associate_refltransport) * (ge_second_in_associate_refltransport))) + (((ge_first_rn_associate_refltransport) * (ge_second_ip_associate_refltransport))))) + (((((ge_first_ip_associate_refltransport) * (ge_second_rn_associate_refltransport))) + (((ge_first_in_associate_refltransport) * (ge_second_rp_associate_refltransport))))))) + ge_balance_positive_associate_refltransportoutputimaginary)))))))))))Complete tactic proof in conservative notation
All 8 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
8 script commands · 4 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)
Original defined command ledger · 8 lines
- 0001
intro a - 0002
intro h - 0003
exists (6) - 0004
split - 0005
exact gaussian_one_unit - 0006
specialize gaussian_multiply_one_left (a) - 0007
apply gaussian_multiply_one_left - 0008
exact h