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.
Exact expanded first-order arithmetic 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)))))))))Constructive proof overview
Generated structural guide
A proved equal output code preserves the actual Gaussian multiply graph.
The unchanged tactic script uses 0 declared prerequisites and contains 8 exact native proof lines.
Alpha v34 checked-use · first admitted v30 · independently kernel and Lean verified; not Stable
Proof neighborhood
Direct dependencies
none
Direct dependents
GF0030 gaussian_multiply_associative_reverse GF003B gaussian_multiply_cancel_left GF004C gaussian_common_divisor_subtract GF0051 gaussian_division_zero_remainder_divides GF005A gaussian_mutual_divisibility_associate GF0068 gaussian_nonzero_product_divisor_unit_cofactor GF0089 gaussian_product_successor_decomposeFormal native tactic body
Dependencies are introduced as named hypotheses before line 1. Local theorem links identify exact declared prerequisites. This exact body belongs to a complete independently kernel-checked constructive proof bundle and has Alpha checked-use authority; it does not imply Stable membership.
Read the argument
Proof checkpoints
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.
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