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 → ZPairAdd(a,b,c) → ZPairAdd(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_add ge_first_rn_output_old_add ge_first_ip_output_old_add ge_first_in_output_old_add ge_second_rp_output_old_add ge_second_rn_output_old_add ge_second_ip_output_old_add ge_second_in_output_old_add. ((exists ge_representation_real_code_output_old_addfirst ge_representation_imaginary_code_output_old_addfirst. (((a) = ((ge_representation_real_code_output_old_addfirst) + (ge_representation_imaginary_code_output_old_addfirst)) * S ((ge_representation_real_code_output_old_addfirst) + (ge_representation_imaginary_code_output_old_addfirst)) + ((ge_representation_imaginary_code_output_old_addfirst) + (ge_representation_imaginary_code_output_old_addfirst))) /\ ((exists ge_balance_positive_output_old_addfirstreal ge_balance_negative_output_old_addfirstreal. (((((ge_representation_real_code_output_old_addfirst) = 2 * (ge_balance_positive_output_old_addfirstreal) /\ (ge_balance_negative_output_old_addfirstreal) = 0) \/ exists ge_signed_half_output_old_addfirstrealdecode. (((ge_representation_real_code_output_old_addfirst) = 2 * ge_signed_half_output_old_addfirstrealdecode + 1 /\ (ge_balance_positive_output_old_addfirstreal) = 0) /\ (ge_balance_negative_output_old_addfirstreal) = S ge_signed_half_output_old_addfirstrealdecode))) /\ ((ge_first_rp_output_old_add) + ge_balance_negative_output_old_addfirstreal = (ge_first_rn_output_old_add) + ge_balance_positive_output_old_addfirstreal))) /\ (exists ge_balance_positive_output_old_addfirstimaginary ge_balance_negative_output_old_addfirstimaginary. (((((ge_representation_imaginary_code_output_old_addfirst) = 2 * (ge_balance_positive_output_old_addfirstimaginary) /\ (ge_balance_negative_output_old_addfirstimaginary) = 0) \/ exists ge_signed_half_output_old_addfirstimaginarydecode. (((ge_representation_imaginary_code_output_old_addfirst) = 2 * ge_signed_half_output_old_addfirstimaginarydecode + 1 /\ (ge_balance_positive_output_old_addfirstimaginary) = 0) /\ (ge_balance_negative_output_old_addfirstimaginary) = S ge_signed_half_output_old_addfirstimaginarydecode))) /\ ((ge_first_ip_output_old_add) + ge_balance_negative_output_old_addfirstimaginary = (ge_first_in_output_old_add) + ge_balance_positive_output_old_addfirstimaginary)))))) /\ ((exists ge_representation_real_code_output_old_addsecond ge_representation_imaginary_code_output_old_addsecond. (((b) = ((ge_representation_real_code_output_old_addsecond) + (ge_representation_imaginary_code_output_old_addsecond)) * S ((ge_representation_real_code_output_old_addsecond) + (ge_representation_imaginary_code_output_old_addsecond)) + ((ge_representation_imaginary_code_output_old_addsecond) + (ge_representation_imaginary_code_output_old_addsecond))) /\ ((exists ge_balance_positive_output_old_addsecondreal ge_balance_negative_output_old_addsecondreal. (((((ge_representation_real_code_output_old_addsecond) = 2 * (ge_balance_positive_output_old_addsecondreal) /\ (ge_balance_negative_output_old_addsecondreal) = 0) \/ exists ge_signed_half_output_old_addsecondrealdecode. (((ge_representation_real_code_output_old_addsecond) = 2 * ge_signed_half_output_old_addsecondrealdecode + 1 /\ (ge_balance_positive_output_old_addsecondreal) = 0) /\ (ge_balance_negative_output_old_addsecondreal) = S ge_signed_half_output_old_addsecondrealdecode))) /\ ((ge_second_rp_output_old_add) + ge_balance_negative_output_old_addsecondreal = (ge_second_rn_output_old_add) + ge_balance_positive_output_old_addsecondreal))) /\ (exists ge_balance_positive_output_old_addsecondimaginary ge_balance_negative_output_old_addsecondimaginary. (((((ge_representation_imaginary_code_output_old_addsecond) = 2 * (ge_balance_positive_output_old_addsecondimaginary) /\ (ge_balance_negative_output_old_addsecondimaginary) = 0) \/ exists ge_signed_half_output_old_addsecondimaginarydecode. (((ge_representation_imaginary_code_output_old_addsecond) = 2 * ge_signed_half_output_old_addsecondimaginarydecode + 1 /\ (ge_balance_positive_output_old_addsecondimaginary) = 0) /\ (ge_balance_negative_output_old_addsecondimaginary) = S ge_signed_half_output_old_addsecondimaginarydecode))) /\ ((ge_second_ip_output_old_add) + ge_balance_negative_output_old_addsecondimaginary = (ge_second_in_output_old_add) + ge_balance_positive_output_old_addsecondimaginary)))))) /\ (exists ge_representation_real_code_output_old_addoutput ge_representation_imaginary_code_output_old_addoutput. (((c) = ((ge_representation_real_code_output_old_addoutput) + (ge_representation_imaginary_code_output_old_addoutput)) * S ((ge_representation_real_code_output_old_addoutput) + (ge_representation_imaginary_code_output_old_addoutput)) + ((ge_representation_imaginary_code_output_old_addoutput) + (ge_representation_imaginary_code_output_old_addoutput))) /\ ((exists ge_balance_positive_output_old_addoutputreal ge_balance_negative_output_old_addoutputreal. (((((ge_representation_real_code_output_old_addoutput) = 2 * (ge_balance_positive_output_old_addoutputreal) /\ (ge_balance_negative_output_old_addoutputreal) = 0) \/ exists ge_signed_half_output_old_addoutputrealdecode. (((ge_representation_real_code_output_old_addoutput) = 2 * ge_signed_half_output_old_addoutputrealdecode + 1 /\ (ge_balance_positive_output_old_addoutputreal) = 0) /\ (ge_balance_negative_output_old_addoutputreal) = S ge_signed_half_output_old_addoutputrealdecode))) /\ ((((ge_first_rp_output_old_add) + (ge_second_rp_output_old_add))) + ge_balance_negative_output_old_addoutputreal = (((ge_first_rn_output_old_add) + (ge_second_rn_output_old_add))) + ge_balance_positive_output_old_addoutputreal))) /\ (exists ge_balance_positive_output_old_addoutputimaginary ge_balance_negative_output_old_addoutputimaginary. (((((ge_representation_imaginary_code_output_old_addoutput) = 2 * (ge_balance_positive_output_old_addoutputimaginary) /\ (ge_balance_negative_output_old_addoutputimaginary) = 0) \/ exists ge_signed_half_output_old_addoutputimaginarydecode. (((ge_representation_imaginary_code_output_old_addoutput) = 2 * ge_signed_half_output_old_addoutputimaginarydecode + 1 /\ (ge_balance_positive_output_old_addoutputimaginary) = 0) /\ (ge_balance_negative_output_old_addoutputimaginary) = S ge_signed_half_output_old_addoutputimaginarydecode))) /\ ((((ge_first_ip_output_old_add) + (ge_second_ip_output_old_add))) + ge_balance_negative_output_old_addoutputimaginary = (((ge_first_in_output_old_add) + (ge_second_in_output_old_add))) + ge_balance_positive_output_old_addoutputimaginary))))))))) -> (exists ge_first_rp_output_new_add ge_first_rn_output_new_add ge_first_ip_output_new_add ge_first_in_output_new_add ge_second_rp_output_new_add ge_second_rn_output_new_add ge_second_ip_output_new_add ge_second_in_output_new_add. ((exists ge_representation_real_code_output_new_addfirst ge_representation_imaginary_code_output_new_addfirst. (((a) = ((ge_representation_real_code_output_new_addfirst) + (ge_representation_imaginary_code_output_new_addfirst)) * S ((ge_representation_real_code_output_new_addfirst) + (ge_representation_imaginary_code_output_new_addfirst)) + ((ge_representation_imaginary_code_output_new_addfirst) + (ge_representation_imaginary_code_output_new_addfirst))) /\ ((exists ge_balance_positive_output_new_addfirstreal ge_balance_negative_output_new_addfirstreal. (((((ge_representation_real_code_output_new_addfirst) = 2 * (ge_balance_positive_output_new_addfirstreal) /\ (ge_balance_negative_output_new_addfirstreal) = 0) \/ exists ge_signed_half_output_new_addfirstrealdecode. (((ge_representation_real_code_output_new_addfirst) = 2 * ge_signed_half_output_new_addfirstrealdecode + 1 /\ (ge_balance_positive_output_new_addfirstreal) = 0) /\ (ge_balance_negative_output_new_addfirstreal) = S ge_signed_half_output_new_addfirstrealdecode))) /\ ((ge_first_rp_output_new_add) + ge_balance_negative_output_new_addfirstreal = (ge_first_rn_output_new_add) + ge_balance_positive_output_new_addfirstreal))) /\ (exists ge_balance_positive_output_new_addfirstimaginary ge_balance_negative_output_new_addfirstimaginary. (((((ge_representation_imaginary_code_output_new_addfirst) = 2 * (ge_balance_positive_output_new_addfirstimaginary) /\ (ge_balance_negative_output_new_addfirstimaginary) = 0) \/ exists ge_signed_half_output_new_addfirstimaginarydecode. (((ge_representation_imaginary_code_output_new_addfirst) = 2 * ge_signed_half_output_new_addfirstimaginarydecode + 1 /\ (ge_balance_positive_output_new_addfirstimaginary) = 0) /\ (ge_balance_negative_output_new_addfirstimaginary) = S ge_signed_half_output_new_addfirstimaginarydecode))) /\ ((ge_first_ip_output_new_add) + ge_balance_negative_output_new_addfirstimaginary = (ge_first_in_output_new_add) + ge_balance_positive_output_new_addfirstimaginary)))))) /\ ((exists ge_representation_real_code_output_new_addsecond ge_representation_imaginary_code_output_new_addsecond. (((b) = ((ge_representation_real_code_output_new_addsecond) + (ge_representation_imaginary_code_output_new_addsecond)) * S ((ge_representation_real_code_output_new_addsecond) + (ge_representation_imaginary_code_output_new_addsecond)) + ((ge_representation_imaginary_code_output_new_addsecond) + (ge_representation_imaginary_code_output_new_addsecond))) /\ ((exists ge_balance_positive_output_new_addsecondreal ge_balance_negative_output_new_addsecondreal. (((((ge_representation_real_code_output_new_addsecond) = 2 * (ge_balance_positive_output_new_addsecondreal) /\ (ge_balance_negative_output_new_addsecondreal) = 0) \/ exists ge_signed_half_output_new_addsecondrealdecode. (((ge_representation_real_code_output_new_addsecond) = 2 * ge_signed_half_output_new_addsecondrealdecode + 1 /\ (ge_balance_positive_output_new_addsecondreal) = 0) /\ (ge_balance_negative_output_new_addsecondreal) = S ge_signed_half_output_new_addsecondrealdecode))) /\ ((ge_second_rp_output_new_add) + ge_balance_negative_output_new_addsecondreal = (ge_second_rn_output_new_add) + ge_balance_positive_output_new_addsecondreal))) /\ (exists ge_balance_positive_output_new_addsecondimaginary ge_balance_negative_output_new_addsecondimaginary. (((((ge_representation_imaginary_code_output_new_addsecond) = 2 * (ge_balance_positive_output_new_addsecondimaginary) /\ (ge_balance_negative_output_new_addsecondimaginary) = 0) \/ exists ge_signed_half_output_new_addsecondimaginarydecode. (((ge_representation_imaginary_code_output_new_addsecond) = 2 * ge_signed_half_output_new_addsecondimaginarydecode + 1 /\ (ge_balance_positive_output_new_addsecondimaginary) = 0) /\ (ge_balance_negative_output_new_addsecondimaginary) = S ge_signed_half_output_new_addsecondimaginarydecode))) /\ ((ge_second_ip_output_new_add) + ge_balance_negative_output_new_addsecondimaginary = (ge_second_in_output_new_add) + ge_balance_positive_output_new_addsecondimaginary)))))) /\ (exists ge_representation_real_code_output_new_addoutput ge_representation_imaginary_code_output_new_addoutput. (((d) = ((ge_representation_real_code_output_new_addoutput) + (ge_representation_imaginary_code_output_new_addoutput)) * S ((ge_representation_real_code_output_new_addoutput) + (ge_representation_imaginary_code_output_new_addoutput)) + ((ge_representation_imaginary_code_output_new_addoutput) + (ge_representation_imaginary_code_output_new_addoutput))) /\ ((exists ge_balance_positive_output_new_addoutputreal ge_balance_negative_output_new_addoutputreal. (((((ge_representation_real_code_output_new_addoutput) = 2 * (ge_balance_positive_output_new_addoutputreal) /\ (ge_balance_negative_output_new_addoutputreal) = 0) \/ exists ge_signed_half_output_new_addoutputrealdecode. (((ge_representation_real_code_output_new_addoutput) = 2 * ge_signed_half_output_new_addoutputrealdecode + 1 /\ (ge_balance_positive_output_new_addoutputreal) = 0) /\ (ge_balance_negative_output_new_addoutputreal) = S ge_signed_half_output_new_addoutputrealdecode))) /\ ((((ge_first_rp_output_new_add) + (ge_second_rp_output_new_add))) + ge_balance_negative_output_new_addoutputreal = (((ge_first_rn_output_new_add) + (ge_second_rn_output_new_add))) + ge_balance_positive_output_new_addoutputreal))) /\ (exists ge_balance_positive_output_new_addoutputimaginary ge_balance_negative_output_new_addoutputimaginary. (((((ge_representation_imaginary_code_output_new_addoutput) = 2 * (ge_balance_positive_output_new_addoutputimaginary) /\ (ge_balance_negative_output_new_addoutputimaginary) = 0) \/ exists ge_signed_half_output_new_addoutputimaginarydecode. (((ge_representation_imaginary_code_output_new_addoutput) = 2 * ge_signed_half_output_new_addoutputimaginarydecode + 1 /\ (ge_balance_positive_output_new_addoutputimaginary) = 0) /\ (ge_balance_negative_output_new_addoutputimaginary) = S ge_signed_half_output_new_addoutputimaginarydecode))) /\ ((((ge_first_ip_output_new_add) + (ge_second_ip_output_new_add))) + ge_balance_negative_output_new_addoutputimaginary = (((ge_first_in_output_new_add) + (ge_second_in_output_new_add))) + ge_balance_positive_output_new_addoutputimaginary)))))))))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