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. ∀ b. ∀ c. ∀ d. c = d → GMul(a,b,c) → GMul(a,b,d)Every linked abbreviation expands hygienically to the identical original native formula.
Definition DAG
Actual proof prerequisites
none
Original expanded first-order statement
forall a b c d. c=d -> (exists ge_first_rp_output_old_multiply ge_first_rn_output_old_multiply ge_first_ip_output_old_multiply ge_first_in_output_old_multiply ge_second_rp_output_old_multiply ge_second_rn_output_old_multiply ge_second_ip_output_old_multiply ge_second_in_output_old_multiply. ((exists ge_representation_real_code_output_old_multiplyfirst ge_representation_imaginary_code_output_old_multiplyfirst. (((a) = ((ge_representation_real_code_output_old_multiplyfirst) + (ge_representation_imaginary_code_output_old_multiplyfirst)) * S ((ge_representation_real_code_output_old_multiplyfirst) + (ge_representation_imaginary_code_output_old_multiplyfirst)) + ((ge_representation_imaginary_code_output_old_multiplyfirst) + (ge_representation_imaginary_code_output_old_multiplyfirst))) /\ ((exists ge_balance_positive_output_old_multiplyfirstreal ge_balance_negative_output_old_multiplyfirstreal. (((((ge_representation_real_code_output_old_multiplyfirst) = 2 * (ge_balance_positive_output_old_multiplyfirstreal) /\ (ge_balance_negative_output_old_multiplyfirstreal) = 0) \/ exists ge_signed_half_output_old_multiplyfirstrealdecode. (((ge_representation_real_code_output_old_multiplyfirst) = 2 * ge_signed_half_output_old_multiplyfirstrealdecode + 1 /\ (ge_balance_positive_output_old_multiplyfirstreal) = 0) /\ (ge_balance_negative_output_old_multiplyfirstreal) = S ge_signed_half_output_old_multiplyfirstrealdecode))) /\ ((ge_first_rp_output_old_multiply) + ge_balance_negative_output_old_multiplyfirstreal = (ge_first_rn_output_old_multiply) + ge_balance_positive_output_old_multiplyfirstreal))) /\ (exists ge_balance_positive_output_old_multiplyfirstimaginary ge_balance_negative_output_old_multiplyfirstimaginary. (((((ge_representation_imaginary_code_output_old_multiplyfirst) = 2 * (ge_balance_positive_output_old_multiplyfirstimaginary) /\ (ge_balance_negative_output_old_multiplyfirstimaginary) = 0) \/ exists ge_signed_half_output_old_multiplyfirstimaginarydecode. (((ge_representation_imaginary_code_output_old_multiplyfirst) = 2 * ge_signed_half_output_old_multiplyfirstimaginarydecode + 1 /\ (ge_balance_positive_output_old_multiplyfirstimaginary) = 0) /\ (ge_balance_negative_output_old_multiplyfirstimaginary) = S ge_signed_half_output_old_multiplyfirstimaginarydecode))) /\ ((ge_first_ip_output_old_multiply) + ge_balance_negative_output_old_multiplyfirstimaginary = (ge_first_in_output_old_multiply) + ge_balance_positive_output_old_multiplyfirstimaginary)))))) /\ ((exists ge_representation_real_code_output_old_multiplysecond ge_representation_imaginary_code_output_old_multiplysecond. (((b) = ((ge_representation_real_code_output_old_multiplysecond) + (ge_representation_imaginary_code_output_old_multiplysecond)) * S ((ge_representation_real_code_output_old_multiplysecond) + (ge_representation_imaginary_code_output_old_multiplysecond)) + ((ge_representation_imaginary_code_output_old_multiplysecond) + (ge_representation_imaginary_code_output_old_multiplysecond))) /\ ((exists ge_balance_positive_output_old_multiplysecondreal ge_balance_negative_output_old_multiplysecondreal. (((((ge_representation_real_code_output_old_multiplysecond) = 2 * (ge_balance_positive_output_old_multiplysecondreal) /\ (ge_balance_negative_output_old_multiplysecondreal) = 0) \/ exists ge_signed_half_output_old_multiplysecondrealdecode. (((ge_representation_real_code_output_old_multiplysecond) = 2 * ge_signed_half_output_old_multiplysecondrealdecode + 1 /\ (ge_balance_positive_output_old_multiplysecondreal) = 0) /\ (ge_balance_negative_output_old_multiplysecondreal) = S ge_signed_half_output_old_multiplysecondrealdecode))) /\ ((ge_second_rp_output_old_multiply) + ge_balance_negative_output_old_multiplysecondreal = (ge_second_rn_output_old_multiply) + ge_balance_positive_output_old_multiplysecondreal))) /\ (exists ge_balance_positive_output_old_multiplysecondimaginary ge_balance_negative_output_old_multiplysecondimaginary. (((((ge_representation_imaginary_code_output_old_multiplysecond) = 2 * (ge_balance_positive_output_old_multiplysecondimaginary) /\ (ge_balance_negative_output_old_multiplysecondimaginary) = 0) \/ exists ge_signed_half_output_old_multiplysecondimaginarydecode. (((ge_representation_imaginary_code_output_old_multiplysecond) = 2 * ge_signed_half_output_old_multiplysecondimaginarydecode + 1 /\ (ge_balance_positive_output_old_multiplysecondimaginary) = 0) /\ (ge_balance_negative_output_old_multiplysecondimaginary) = S ge_signed_half_output_old_multiplysecondimaginarydecode))) /\ ((ge_second_ip_output_old_multiply) + ge_balance_negative_output_old_multiplysecondimaginary = (ge_second_in_output_old_multiply) + ge_balance_positive_output_old_multiplysecondimaginary)))))) /\ (exists ge_representation_real_code_output_old_multiplyoutput ge_representation_imaginary_code_output_old_multiplyoutput. (((c) = ((ge_representation_real_code_output_old_multiplyoutput) + (ge_representation_imaginary_code_output_old_multiplyoutput)) * S ((ge_representation_real_code_output_old_multiplyoutput) + (ge_representation_imaginary_code_output_old_multiplyoutput)) + ((ge_representation_imaginary_code_output_old_multiplyoutput) + (ge_representation_imaginary_code_output_old_multiplyoutput))) /\ ((exists ge_balance_positive_output_old_multiplyoutputreal ge_balance_negative_output_old_multiplyoutputreal. (((((ge_representation_real_code_output_old_multiplyoutput) = 2 * (ge_balance_positive_output_old_multiplyoutputreal) /\ (ge_balance_negative_output_old_multiplyoutputreal) = 0) \/ exists ge_signed_half_output_old_multiplyoutputrealdecode. (((ge_representation_real_code_output_old_multiplyoutput) = 2 * ge_signed_half_output_old_multiplyoutputrealdecode + 1 /\ (ge_balance_positive_output_old_multiplyoutputreal) = 0) /\ (ge_balance_negative_output_old_multiplyoutputreal) = S ge_signed_half_output_old_multiplyoutputrealdecode))) /\ ((((((((ge_first_rp_output_old_multiply) * (ge_second_rp_output_old_multiply))) + (((ge_first_rn_output_old_multiply) * (ge_second_rn_output_old_multiply))))) + (((((ge_first_ip_output_old_multiply) * (ge_second_in_output_old_multiply))) + (((ge_first_in_output_old_multiply) * (ge_second_ip_output_old_multiply))))))) + ge_balance_negative_output_old_multiplyoutputreal = (((((((ge_first_rp_output_old_multiply) * (ge_second_rn_output_old_multiply))) + (((ge_first_rn_output_old_multiply) * (ge_second_rp_output_old_multiply))))) + (((((ge_first_ip_output_old_multiply) * (ge_second_ip_output_old_multiply))) + (((ge_first_in_output_old_multiply) * (ge_second_in_output_old_multiply))))))) + ge_balance_positive_output_old_multiplyoutputreal))) /\ (exists ge_balance_positive_output_old_multiplyoutputimaginary ge_balance_negative_output_old_multiplyoutputimaginary. (((((ge_representation_imaginary_code_output_old_multiplyoutput) = 2 * (ge_balance_positive_output_old_multiplyoutputimaginary) /\ (ge_balance_negative_output_old_multiplyoutputimaginary) = 0) \/ exists ge_signed_half_output_old_multiplyoutputimaginarydecode. (((ge_representation_imaginary_code_output_old_multiplyoutput) = 2 * ge_signed_half_output_old_multiplyoutputimaginarydecode + 1 /\ (ge_balance_positive_output_old_multiplyoutputimaginary) = 0) /\ (ge_balance_negative_output_old_multiplyoutputimaginary) = S ge_signed_half_output_old_multiplyoutputimaginarydecode))) /\ ((((((((ge_first_rp_output_old_multiply) * (ge_second_ip_output_old_multiply))) + (((ge_first_rn_output_old_multiply) * (ge_second_in_output_old_multiply))))) + (((((ge_first_ip_output_old_multiply) * (ge_second_rp_output_old_multiply))) + (((ge_first_in_output_old_multiply) * (ge_second_rn_output_old_multiply))))))) + ge_balance_negative_output_old_multiplyoutputimaginary = (((((((ge_first_rp_output_old_multiply) * (ge_second_in_output_old_multiply))) + (((ge_first_rn_output_old_multiply) * (ge_second_ip_output_old_multiply))))) + (((((ge_first_ip_output_old_multiply) * (ge_second_rn_output_old_multiply))) + (((ge_first_in_output_old_multiply) * (ge_second_rp_output_old_multiply))))))) + ge_balance_positive_output_old_multiplyoutputimaginary))))))))) -> (exists ge_first_rp_output_new_multiply ge_first_rn_output_new_multiply ge_first_ip_output_new_multiply ge_first_in_output_new_multiply ge_second_rp_output_new_multiply ge_second_rn_output_new_multiply ge_second_ip_output_new_multiply ge_second_in_output_new_multiply. ((exists ge_representation_real_code_output_new_multiplyfirst ge_representation_imaginary_code_output_new_multiplyfirst. (((a) = ((ge_representation_real_code_output_new_multiplyfirst) + (ge_representation_imaginary_code_output_new_multiplyfirst)) * S ((ge_representation_real_code_output_new_multiplyfirst) + (ge_representation_imaginary_code_output_new_multiplyfirst)) + ((ge_representation_imaginary_code_output_new_multiplyfirst) + (ge_representation_imaginary_code_output_new_multiplyfirst))) /\ ((exists ge_balance_positive_output_new_multiplyfirstreal ge_balance_negative_output_new_multiplyfirstreal. (((((ge_representation_real_code_output_new_multiplyfirst) = 2 * (ge_balance_positive_output_new_multiplyfirstreal) /\ (ge_balance_negative_output_new_multiplyfirstreal) = 0) \/ exists ge_signed_half_output_new_multiplyfirstrealdecode. (((ge_representation_real_code_output_new_multiplyfirst) = 2 * ge_signed_half_output_new_multiplyfirstrealdecode + 1 /\ (ge_balance_positive_output_new_multiplyfirstreal) = 0) /\ (ge_balance_negative_output_new_multiplyfirstreal) = S ge_signed_half_output_new_multiplyfirstrealdecode))) /\ ((ge_first_rp_output_new_multiply) + ge_balance_negative_output_new_multiplyfirstreal = (ge_first_rn_output_new_multiply) + ge_balance_positive_output_new_multiplyfirstreal))) /\ (exists ge_balance_positive_output_new_multiplyfirstimaginary ge_balance_negative_output_new_multiplyfirstimaginary. (((((ge_representation_imaginary_code_output_new_multiplyfirst) = 2 * (ge_balance_positive_output_new_multiplyfirstimaginary) /\ (ge_balance_negative_output_new_multiplyfirstimaginary) = 0) \/ exists ge_signed_half_output_new_multiplyfirstimaginarydecode. (((ge_representation_imaginary_code_output_new_multiplyfirst) = 2 * ge_signed_half_output_new_multiplyfirstimaginarydecode + 1 /\ (ge_balance_positive_output_new_multiplyfirstimaginary) = 0) /\ (ge_balance_negative_output_new_multiplyfirstimaginary) = S ge_signed_half_output_new_multiplyfirstimaginarydecode))) /\ ((ge_first_ip_output_new_multiply) + ge_balance_negative_output_new_multiplyfirstimaginary = (ge_first_in_output_new_multiply) + ge_balance_positive_output_new_multiplyfirstimaginary)))))) /\ ((exists ge_representation_real_code_output_new_multiplysecond ge_representation_imaginary_code_output_new_multiplysecond. (((b) = ((ge_representation_real_code_output_new_multiplysecond) + (ge_representation_imaginary_code_output_new_multiplysecond)) * S ((ge_representation_real_code_output_new_multiplysecond) + (ge_representation_imaginary_code_output_new_multiplysecond)) + ((ge_representation_imaginary_code_output_new_multiplysecond) + (ge_representation_imaginary_code_output_new_multiplysecond))) /\ ((exists ge_balance_positive_output_new_multiplysecondreal ge_balance_negative_output_new_multiplysecondreal. (((((ge_representation_real_code_output_new_multiplysecond) = 2 * (ge_balance_positive_output_new_multiplysecondreal) /\ (ge_balance_negative_output_new_multiplysecondreal) = 0) \/ exists ge_signed_half_output_new_multiplysecondrealdecode. (((ge_representation_real_code_output_new_multiplysecond) = 2 * ge_signed_half_output_new_multiplysecondrealdecode + 1 /\ (ge_balance_positive_output_new_multiplysecondreal) = 0) /\ (ge_balance_negative_output_new_multiplysecondreal) = S ge_signed_half_output_new_multiplysecondrealdecode))) /\ ((ge_second_rp_output_new_multiply) + ge_balance_negative_output_new_multiplysecondreal = (ge_second_rn_output_new_multiply) + ge_balance_positive_output_new_multiplysecondreal))) /\ (exists ge_balance_positive_output_new_multiplysecondimaginary ge_balance_negative_output_new_multiplysecondimaginary. (((((ge_representation_imaginary_code_output_new_multiplysecond) = 2 * (ge_balance_positive_output_new_multiplysecondimaginary) /\ (ge_balance_negative_output_new_multiplysecondimaginary) = 0) \/ exists ge_signed_half_output_new_multiplysecondimaginarydecode. (((ge_representation_imaginary_code_output_new_multiplysecond) = 2 * ge_signed_half_output_new_multiplysecondimaginarydecode + 1 /\ (ge_balance_positive_output_new_multiplysecondimaginary) = 0) /\ (ge_balance_negative_output_new_multiplysecondimaginary) = S ge_signed_half_output_new_multiplysecondimaginarydecode))) /\ ((ge_second_ip_output_new_multiply) + ge_balance_negative_output_new_multiplysecondimaginary = (ge_second_in_output_new_multiply) + ge_balance_positive_output_new_multiplysecondimaginary)))))) /\ (exists ge_representation_real_code_output_new_multiplyoutput ge_representation_imaginary_code_output_new_multiplyoutput. (((d) = ((ge_representation_real_code_output_new_multiplyoutput) + (ge_representation_imaginary_code_output_new_multiplyoutput)) * S ((ge_representation_real_code_output_new_multiplyoutput) + (ge_representation_imaginary_code_output_new_multiplyoutput)) + ((ge_representation_imaginary_code_output_new_multiplyoutput) + (ge_representation_imaginary_code_output_new_multiplyoutput))) /\ ((exists ge_balance_positive_output_new_multiplyoutputreal ge_balance_negative_output_new_multiplyoutputreal. (((((ge_representation_real_code_output_new_multiplyoutput) = 2 * (ge_balance_positive_output_new_multiplyoutputreal) /\ (ge_balance_negative_output_new_multiplyoutputreal) = 0) \/ exists ge_signed_half_output_new_multiplyoutputrealdecode. (((ge_representation_real_code_output_new_multiplyoutput) = 2 * ge_signed_half_output_new_multiplyoutputrealdecode + 1 /\ (ge_balance_positive_output_new_multiplyoutputreal) = 0) /\ (ge_balance_negative_output_new_multiplyoutputreal) = S ge_signed_half_output_new_multiplyoutputrealdecode))) /\ ((((((((ge_first_rp_output_new_multiply) * (ge_second_rp_output_new_multiply))) + (((ge_first_rn_output_new_multiply) * (ge_second_rn_output_new_multiply))))) + (((((ge_first_ip_output_new_multiply) * (ge_second_in_output_new_multiply))) + (((ge_first_in_output_new_multiply) * (ge_second_ip_output_new_multiply))))))) + ge_balance_negative_output_new_multiplyoutputreal = (((((((ge_first_rp_output_new_multiply) * (ge_second_rn_output_new_multiply))) + (((ge_first_rn_output_new_multiply) * (ge_second_rp_output_new_multiply))))) + (((((ge_first_ip_output_new_multiply) * (ge_second_ip_output_new_multiply))) + (((ge_first_in_output_new_multiply) * (ge_second_in_output_new_multiply))))))) + ge_balance_positive_output_new_multiplyoutputreal))) /\ (exists ge_balance_positive_output_new_multiplyoutputimaginary ge_balance_negative_output_new_multiplyoutputimaginary. (((((ge_representation_imaginary_code_output_new_multiplyoutput) = 2 * (ge_balance_positive_output_new_multiplyoutputimaginary) /\ (ge_balance_negative_output_new_multiplyoutputimaginary) = 0) \/ exists ge_signed_half_output_new_multiplyoutputimaginarydecode. (((ge_representation_imaginary_code_output_new_multiplyoutput) = 2 * ge_signed_half_output_new_multiplyoutputimaginarydecode + 1 /\ (ge_balance_positive_output_new_multiplyoutputimaginary) = 0) /\ (ge_balance_negative_output_new_multiplyoutputimaginary) = S ge_signed_half_output_new_multiplyoutputimaginarydecode))) /\ ((((((((ge_first_rp_output_new_multiply) * (ge_second_ip_output_new_multiply))) + (((ge_first_rn_output_new_multiply) * (ge_second_in_output_new_multiply))))) + (((((ge_first_ip_output_new_multiply) * (ge_second_rp_output_new_multiply))) + (((ge_first_in_output_new_multiply) * (ge_second_rn_output_new_multiply))))))) + ge_balance_negative_output_new_multiplyoutputimaginary = (((((((ge_first_rp_output_new_multiply) * (ge_second_in_output_new_multiply))) + (((ge_first_rn_output_new_multiply) * (ge_second_ip_output_new_multiply))))) + (((((ge_first_ip_output_new_multiply) * (ge_second_rn_output_new_multiply))) + (((ge_first_in_output_new_multiply) * (ge_second_rp_output_new_multiply))))))) + ge_balance_positive_output_new_multiplyoutputimaginary)))))))))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 · 3 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.
01Fix variables and assumptionsL1–6
02Calculate and transport equalitiesL7–7
Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.
- L7
rewrite heq at h
03Use earlier factsL8–8
Instantiate or apply named facts and discharge the corresponding proof obligations.
- L8
exact h