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 b c l m P. l=m -> (exists gr_product_trace_length_transport_product gr_product_scale_length_transport_product. ((((exists ff_h_gprod_length_transport_productstart. ff_h_gprod_length_transport_productstart + S (6) = S ((S (0)) * gr_product_scale_length_transport_product)) /\ exists ff_q_gprod_length_transport_productstart. gr_product_trace_length_transport_product = ff_q_gprod_length_transport_productstart * S ((S (0)) * gr_product_scale_length_transport_product) + (6))) /\ ((((exists ff_h_gprod_length_transport_productend. ff_h_gprod_length_transport_productend + S (P) = S ((S (l)) * gr_product_scale_length_transport_product)) /\ exists ff_q_gprod_length_transport_productend. gr_product_trace_length_transport_product = ff_q_gprod_length_transport_productend * S ((S (l)) * gr_product_scale_length_transport_product) + (P))) /\ (forall gr_product_index_length_transport_productsteps. (exists ge_gap_length_transport_productstepsindex_bound. ge_gap_length_transport_productstepsindex_bound + S (gr_product_index_length_transport_productsteps) = (l)) -> exists gr_product_factor_length_transport_productsteps gr_product_before_length_transport_productsteps gr_product_after_length_transport_productsteps. ((((exists ff_h_gprod_length_transport_productstepsfactor. ff_h_gprod_length_transport_productstepsfactor + S (gr_product_factor_length_transport_productsteps) = S ((S (gr_product_index_length_transport_productsteps)) * c)) /\ exists ff_q_gprod_length_transport_productstepsfactor. b = ff_q_gprod_length_transport_productstepsfactor * S ((S (gr_product_index_length_transport_productsteps)) * c) + (gr_product_factor_length_transport_productsteps))) /\ ((((exists ff_h_gprod_length_transport_productstepsbefore. ff_h_gprod_length_transport_productstepsbefore + S (gr_product_before_length_transport_productsteps) = S ((S (gr_product_index_length_transport_productsteps)) * gr_product_scale_length_transport_product)) /\ exists ff_q_gprod_length_transport_productstepsbefore. gr_product_trace_length_transport_product = ff_q_gprod_length_transport_productstepsbefore * S ((S (gr_product_index_length_transport_productsteps)) * gr_product_scale_length_transport_product) + (gr_product_before_length_transport_productsteps))) /\ ((((exists ff_h_gprod_length_transport_productstepsafter. ff_h_gprod_length_transport_productstepsafter + S (gr_product_after_length_transport_productsteps) = S ((S (S (gr_product_index_length_transport_productsteps))) * gr_product_scale_length_transport_product)) /\ exists ff_q_gprod_length_transport_productstepsafter. gr_product_trace_length_transport_product = ff_q_gprod_length_transport_productstepsafter * S ((S (S (gr_product_index_length_transport_productsteps))) * gr_product_scale_length_transport_product) + (gr_product_after_length_transport_productsteps))) /\ (exists ge_first_rp_length_transport_productstepsmultiply ge_first_rn_length_transport_productstepsmultiply ge_first_ip_length_transport_productstepsmultiply ge_first_in_length_transport_productstepsmultiply ge_second_rp_length_transport_productstepsmultiply ge_second_rn_length_transport_productstepsmultiply ge_second_ip_length_transport_productstepsmultiply ge_second_in_length_transport_productstepsmultiply. ((exists ge_representation_real_code_length_transport_productstepsmultiplyfirst ge_representation_imaginary_code_length_transport_productstepsmultiplyfirst. (((gr_product_before_length_transport_productsteps) = ((ge_representation_real_code_length_transport_productstepsmultiplyfirst) + (ge_representation_imaginary_code_length_transport_productstepsmultiplyfirst)) * S ((ge_representation_real_code_length_transport_productstepsmultiplyfirst) + (ge_representation_imaginary_code_length_transport_productstepsmultiplyfirst)) + ((ge_representation_imaginary_code_length_transport_productstepsmultiplyfirst) + (ge_representation_imaginary_code_length_transport_productstepsmultiplyfirst))) /\ ((exists ge_balance_positive_length_transport_productstepsmultiplyfirstreal ge_balance_negative_length_transport_productstepsmultiplyfirstreal. (((((ge_representation_real_code_length_transport_productstepsmultiplyfirst) = 2 * (ge_balance_positive_length_transport_productstepsmultiplyfirstreal) /\ (ge_balance_negative_length_transport_productstepsmultiplyfirstreal) = 0) \/ exists ge_signed_half_length_transport_productstepsmultiplyfirstrealdecode. (((ge_representation_real_code_length_transport_productstepsmultiplyfirst) = 2 * ge_signed_half_length_transport_productstepsmultiplyfirstrealdecode + 1 /\ (ge_balance_positive_length_transport_productstepsmultiplyfirstreal) = 0) /\ (ge_balance_negative_length_transport_productstepsmultiplyfirstreal) = S ge_signed_half_length_transport_productstepsmultiplyfirstrealdecode))) /\ ((ge_first_rp_length_transport_productstepsmultiply) + ge_balance_negative_length_transport_productstepsmultiplyfirstreal = (ge_first_rn_length_transport_productstepsmultiply) + ge_balance_positive_length_transport_productstepsmultiplyfirstreal))) /\ (exists ge_balance_positive_length_transport_productstepsmultiplyfirstimaginary ge_balance_negative_length_transport_productstepsmultiplyfirstimaginary. (((((ge_representation_imaginary_code_length_transport_productstepsmultiplyfirst) = 2 * (ge_balance_positive_length_transport_productstepsmultiplyfirstimaginary) /\ (ge_balance_negative_length_transport_productstepsmultiplyfirstimaginary) = 0) \/ exists ge_signed_half_length_transport_productstepsmultiplyfirstimaginarydecode. (((ge_representation_imaginary_code_length_transport_productstepsmultiplyfirst) = 2 * ge_signed_half_length_transport_productstepsmultiplyfirstimaginarydecode + 1 /\ (ge_balance_positive_length_transport_productstepsmultiplyfirstimaginary) = 0) /\ (ge_balance_negative_length_transport_productstepsmultiplyfirstimaginary) = S ge_signed_half_length_transport_productstepsmultiplyfirstimaginarydecode))) /\ ((ge_first_ip_length_transport_productstepsmultiply) + ge_balance_negative_length_transport_productstepsmultiplyfirstimaginary = (ge_first_in_length_transport_productstepsmultiply) + ge_balance_positive_length_transport_productstepsmultiplyfirstimaginary)))))) /\ ((exists ge_representation_real_code_length_transport_productstepsmultiplysecond ge_representation_imaginary_code_length_transport_productstepsmultiplysecond. (((gr_product_factor_length_transport_productsteps) = ((ge_representation_real_code_length_transport_productstepsmultiplysecond) + (ge_representation_imaginary_code_length_transport_productstepsmultiplysecond)) * S ((ge_representation_real_code_length_transport_productstepsmultiplysecond) + (ge_representation_imaginary_code_length_transport_productstepsmultiplysecond)) + ((ge_representation_imaginary_code_length_transport_productstepsmultiplysecond) + (ge_representation_imaginary_code_length_transport_productstepsmultiplysecond))) /\ ((exists ge_balance_positive_length_transport_productstepsmultiplysecondreal ge_balance_negative_length_transport_productstepsmultiplysecondreal. (((((ge_representation_real_code_length_transport_productstepsmultiplysecond) = 2 * (ge_balance_positive_length_transport_productstepsmultiplysecondreal) /\ (ge_balance_negative_length_transport_productstepsmultiplysecondreal) = 0) \/ exists ge_signed_half_length_transport_productstepsmultiplysecondrealdecode. (((ge_representation_real_code_length_transport_productstepsmultiplysecond) = 2 * ge_signed_half_length_transport_productstepsmultiplysecondrealdecode + 1 /\ (ge_balance_positive_length_transport_productstepsmultiplysecondreal) = 0) /\ (ge_balance_negative_length_transport_productstepsmultiplysecondreal) = S ge_signed_half_length_transport_productstepsmultiplysecondrealdecode))) /\ ((ge_second_rp_length_transport_productstepsmultiply) + ge_balance_negative_length_transport_productstepsmultiplysecondreal = (ge_second_rn_length_transport_productstepsmultiply) + ge_balance_positive_length_transport_productstepsmultiplysecondreal))) /\ (exists ge_balance_positive_length_transport_productstepsmultiplysecondimaginary ge_balance_negative_length_transport_productstepsmultiplysecondimaginary. (((((ge_representation_imaginary_code_length_transport_productstepsmultiplysecond) = 2 * (ge_balance_positive_length_transport_productstepsmultiplysecondimaginary) /\ (ge_balance_negative_length_transport_productstepsmultiplysecondimaginary) = 0) \/ exists ge_signed_half_length_transport_productstepsmultiplysecondimaginarydecode. (((ge_representation_imaginary_code_length_transport_productstepsmultiplysecond) = 2 * ge_signed_half_length_transport_productstepsmultiplysecondimaginarydecode + 1 /\ (ge_balance_positive_length_transport_productstepsmultiplysecondimaginary) = 0) /\ (ge_balance_negative_length_transport_productstepsmultiplysecondimaginary) = S ge_signed_half_length_transport_productstepsmultiplysecondimaginarydecode))) /\ ((ge_second_ip_length_transport_productstepsmultiply) + ge_balance_negative_length_transport_productstepsmultiplysecondimaginary = (ge_second_in_length_transport_productstepsmultiply) + ge_balance_positive_length_transport_productstepsmultiplysecondimaginary)))))) /\ (exists ge_representation_real_code_length_transport_productstepsmultiplyoutput ge_representation_imaginary_code_length_transport_productstepsmultiplyoutput. (((gr_product_after_length_transport_productsteps) = ((ge_representation_real_code_length_transport_productstepsmultiplyoutput) + (ge_representation_imaginary_code_length_transport_productstepsmultiplyoutput)) * S ((ge_representation_real_code_length_transport_productstepsmultiplyoutput) + (ge_representation_imaginary_code_length_transport_productstepsmultiplyoutput)) + ((ge_representation_imaginary_code_length_transport_productstepsmultiplyoutput) + (ge_representation_imaginary_code_length_transport_productstepsmultiplyoutput))) /\ ((exists ge_balance_positive_length_transport_productstepsmultiplyoutputreal ge_balance_negative_length_transport_productstepsmultiplyoutputreal. (((((ge_representation_real_code_length_transport_productstepsmultiplyoutput) = 2 * (ge_balance_positive_length_transport_productstepsmultiplyoutputreal) /\ (ge_balance_negative_length_transport_productstepsmultiplyoutputreal) = 0) \/ exists ge_signed_half_length_transport_productstepsmultiplyoutputrealdecode. (((ge_representation_real_code_length_transport_productstepsmultiplyoutput) = 2 * ge_signed_half_length_transport_productstepsmultiplyoutputrealdecode + 1 /\ (ge_balance_positive_length_transport_productstepsmultiplyoutputreal) = 0) /\ (ge_balance_negative_length_transport_productstepsmultiplyoutputreal) = S ge_signed_half_length_transport_productstepsmultiplyoutputrealdecode))) /\ ((((((((ge_first_rp_length_transport_productstepsmultiply) * (ge_second_rp_length_transport_productstepsmultiply))) + (((ge_first_rn_length_transport_productstepsmultiply) * (ge_second_rn_length_transport_productstepsmultiply))))) + (((((ge_first_ip_length_transport_productstepsmultiply) * (ge_second_in_length_transport_productstepsmultiply))) + (((ge_first_in_length_transport_productstepsmultiply) * (ge_second_ip_length_transport_productstepsmultiply))))))) + ge_balance_negative_length_transport_productstepsmultiplyoutputreal = (((((((ge_first_rp_length_transport_productstepsmultiply) * (ge_second_rn_length_transport_productstepsmultiply))) + (((ge_first_rn_length_transport_productstepsmultiply) * (ge_second_rp_length_transport_productstepsmultiply))))) + (((((ge_first_ip_length_transport_productstepsmultiply) * (ge_second_ip_length_transport_productstepsmultiply))) + (((ge_first_in_length_transport_productstepsmultiply) * (ge_second_in_length_transport_productstepsmultiply))))))) + ge_balance_positive_length_transport_productstepsmultiplyoutputreal))) /\ (exists ge_balance_positive_length_transport_productstepsmultiplyoutputimaginary ge_balance_negative_length_transport_productstepsmultiplyoutputimaginary. (((((ge_representation_imaginary_code_length_transport_productstepsmultiplyoutput) = 2 * (ge_balance_positive_length_transport_productstepsmultiplyoutputimaginary) /\ (ge_balance_negative_length_transport_productstepsmultiplyoutputimaginary) = 0) \/ exists ge_signed_half_length_transport_productstepsmultiplyoutputimaginarydecode. (((ge_representation_imaginary_code_length_transport_productstepsmultiplyoutput) = 2 * ge_signed_half_length_transport_productstepsmultiplyoutputimaginarydecode + 1 /\ (ge_balance_positive_length_transport_productstepsmultiplyoutputimaginary) = 0) /\ (ge_balance_negative_length_transport_productstepsmultiplyoutputimaginary) = S ge_signed_half_length_transport_productstepsmultiplyoutputimaginarydecode))) /\ ((((((((ge_first_rp_length_transport_productstepsmultiply) * (ge_second_ip_length_transport_productstepsmultiply))) + (((ge_first_rn_length_transport_productstepsmultiply) * (ge_second_in_length_transport_productstepsmultiply))))) + (((((ge_first_ip_length_transport_productstepsmultiply) * (ge_second_rp_length_transport_productstepsmultiply))) + (((ge_first_in_length_transport_productstepsmultiply) * (ge_second_rn_length_transport_productstepsmultiply))))))) + ge_balance_negative_length_transport_productstepsmultiplyoutputimaginary = (((((((ge_first_rp_length_transport_productstepsmultiply) * (ge_second_in_length_transport_productstepsmultiply))) + (((ge_first_rn_length_transport_productstepsmultiply) * (ge_second_ip_length_transport_productstepsmultiply))))) + (((((ge_first_ip_length_transport_productstepsmultiply) * (ge_second_rn_length_transport_productstepsmultiply))) + (((ge_first_in_length_transport_productstepsmultiply) * (ge_second_rp_length_transport_productstepsmultiply))))))) + ge_balance_positive_length_transport_productstepsmultiplyoutputimaginary)))))))))))))))) -> (exists gr_product_trace_length_transport_result gr_product_scale_length_transport_result. ((((exists ff_h_gprod_length_transport_resultstart. ff_h_gprod_length_transport_resultstart + S (6) = S ((S (0)) * gr_product_scale_length_transport_result)) /\ exists ff_q_gprod_length_transport_resultstart. gr_product_trace_length_transport_result = ff_q_gprod_length_transport_resultstart * S ((S (0)) * gr_product_scale_length_transport_result) + (6))) /\ ((((exists ff_h_gprod_length_transport_resultend. ff_h_gprod_length_transport_resultend + S (P) = S ((S (m)) * gr_product_scale_length_transport_result)) /\ exists ff_q_gprod_length_transport_resultend. gr_product_trace_length_transport_result = ff_q_gprod_length_transport_resultend * S ((S (m)) * gr_product_scale_length_transport_result) + (P))) /\ (forall gr_product_index_length_transport_resultsteps. (exists ge_gap_length_transport_resultstepsindex_bound. ge_gap_length_transport_resultstepsindex_bound + S (gr_product_index_length_transport_resultsteps) = (m)) -> exists gr_product_factor_length_transport_resultsteps gr_product_before_length_transport_resultsteps gr_product_after_length_transport_resultsteps. ((((exists ff_h_gprod_length_transport_resultstepsfactor. ff_h_gprod_length_transport_resultstepsfactor + S (gr_product_factor_length_transport_resultsteps) = S ((S (gr_product_index_length_transport_resultsteps)) * c)) /\ exists ff_q_gprod_length_transport_resultstepsfactor. b = ff_q_gprod_length_transport_resultstepsfactor * S ((S (gr_product_index_length_transport_resultsteps)) * c) + (gr_product_factor_length_transport_resultsteps))) /\ ((((exists ff_h_gprod_length_transport_resultstepsbefore. ff_h_gprod_length_transport_resultstepsbefore + S (gr_product_before_length_transport_resultsteps) = S ((S (gr_product_index_length_transport_resultsteps)) * gr_product_scale_length_transport_result)) /\ exists ff_q_gprod_length_transport_resultstepsbefore. gr_product_trace_length_transport_result = ff_q_gprod_length_transport_resultstepsbefore * S ((S (gr_product_index_length_transport_resultsteps)) * gr_product_scale_length_transport_result) + (gr_product_before_length_transport_resultsteps))) /\ ((((exists ff_h_gprod_length_transport_resultstepsafter. ff_h_gprod_length_transport_resultstepsafter + S (gr_product_after_length_transport_resultsteps) = S ((S (S (gr_product_index_length_transport_resultsteps))) * gr_product_scale_length_transport_result)) /\ exists ff_q_gprod_length_transport_resultstepsafter. gr_product_trace_length_transport_result = ff_q_gprod_length_transport_resultstepsafter * S ((S (S (gr_product_index_length_transport_resultsteps))) * gr_product_scale_length_transport_result) + (gr_product_after_length_transport_resultsteps))) /\ (exists ge_first_rp_length_transport_resultstepsmultiply ge_first_rn_length_transport_resultstepsmultiply ge_first_ip_length_transport_resultstepsmultiply ge_first_in_length_transport_resultstepsmultiply ge_second_rp_length_transport_resultstepsmultiply ge_second_rn_length_transport_resultstepsmultiply ge_second_ip_length_transport_resultstepsmultiply ge_second_in_length_transport_resultstepsmultiply. ((exists ge_representation_real_code_length_transport_resultstepsmultiplyfirst ge_representation_imaginary_code_length_transport_resultstepsmultiplyfirst. (((gr_product_before_length_transport_resultsteps) = ((ge_representation_real_code_length_transport_resultstepsmultiplyfirst) + (ge_representation_imaginary_code_length_transport_resultstepsmultiplyfirst)) * S ((ge_representation_real_code_length_transport_resultstepsmultiplyfirst) + (ge_representation_imaginary_code_length_transport_resultstepsmultiplyfirst)) + ((ge_representation_imaginary_code_length_transport_resultstepsmultiplyfirst) + (ge_representation_imaginary_code_length_transport_resultstepsmultiplyfirst))) /\ ((exists ge_balance_positive_length_transport_resultstepsmultiplyfirstreal ge_balance_negative_length_transport_resultstepsmultiplyfirstreal. (((((ge_representation_real_code_length_transport_resultstepsmultiplyfirst) = 2 * (ge_balance_positive_length_transport_resultstepsmultiplyfirstreal) /\ (ge_balance_negative_length_transport_resultstepsmultiplyfirstreal) = 0) \/ exists ge_signed_half_length_transport_resultstepsmultiplyfirstrealdecode. (((ge_representation_real_code_length_transport_resultstepsmultiplyfirst) = 2 * ge_signed_half_length_transport_resultstepsmultiplyfirstrealdecode + 1 /\ (ge_balance_positive_length_transport_resultstepsmultiplyfirstreal) = 0) /\ (ge_balance_negative_length_transport_resultstepsmultiplyfirstreal) = S ge_signed_half_length_transport_resultstepsmultiplyfirstrealdecode))) /\ ((ge_first_rp_length_transport_resultstepsmultiply) + ge_balance_negative_length_transport_resultstepsmultiplyfirstreal = (ge_first_rn_length_transport_resultstepsmultiply) + ge_balance_positive_length_transport_resultstepsmultiplyfirstreal))) /\ (exists ge_balance_positive_length_transport_resultstepsmultiplyfirstimaginary ge_balance_negative_length_transport_resultstepsmultiplyfirstimaginary. (((((ge_representation_imaginary_code_length_transport_resultstepsmultiplyfirst) = 2 * (ge_balance_positive_length_transport_resultstepsmultiplyfirstimaginary) /\ (ge_balance_negative_length_transport_resultstepsmultiplyfirstimaginary) = 0) \/ exists ge_signed_half_length_transport_resultstepsmultiplyfirstimaginarydecode. (((ge_representation_imaginary_code_length_transport_resultstepsmultiplyfirst) = 2 * ge_signed_half_length_transport_resultstepsmultiplyfirstimaginarydecode + 1 /\ (ge_balance_positive_length_transport_resultstepsmultiplyfirstimaginary) = 0) /\ (ge_balance_negative_length_transport_resultstepsmultiplyfirstimaginary) = S ge_signed_half_length_transport_resultstepsmultiplyfirstimaginarydecode))) /\ ((ge_first_ip_length_transport_resultstepsmultiply) + ge_balance_negative_length_transport_resultstepsmultiplyfirstimaginary = (ge_first_in_length_transport_resultstepsmultiply) + ge_balance_positive_length_transport_resultstepsmultiplyfirstimaginary)))))) /\ ((exists ge_representation_real_code_length_transport_resultstepsmultiplysecond ge_representation_imaginary_code_length_transport_resultstepsmultiplysecond. (((gr_product_factor_length_transport_resultsteps) = ((ge_representation_real_code_length_transport_resultstepsmultiplysecond) + (ge_representation_imaginary_code_length_transport_resultstepsmultiplysecond)) * S ((ge_representation_real_code_length_transport_resultstepsmultiplysecond) + (ge_representation_imaginary_code_length_transport_resultstepsmultiplysecond)) + ((ge_representation_imaginary_code_length_transport_resultstepsmultiplysecond) + (ge_representation_imaginary_code_length_transport_resultstepsmultiplysecond))) /\ ((exists ge_balance_positive_length_transport_resultstepsmultiplysecondreal ge_balance_negative_length_transport_resultstepsmultiplysecondreal. (((((ge_representation_real_code_length_transport_resultstepsmultiplysecond) = 2 * (ge_balance_positive_length_transport_resultstepsmultiplysecondreal) /\ (ge_balance_negative_length_transport_resultstepsmultiplysecondreal) = 0) \/ exists ge_signed_half_length_transport_resultstepsmultiplysecondrealdecode. (((ge_representation_real_code_length_transport_resultstepsmultiplysecond) = 2 * ge_signed_half_length_transport_resultstepsmultiplysecondrealdecode + 1 /\ (ge_balance_positive_length_transport_resultstepsmultiplysecondreal) = 0) /\ (ge_balance_negative_length_transport_resultstepsmultiplysecondreal) = S ge_signed_half_length_transport_resultstepsmultiplysecondrealdecode))) /\ ((ge_second_rp_length_transport_resultstepsmultiply) + ge_balance_negative_length_transport_resultstepsmultiplysecondreal = (ge_second_rn_length_transport_resultstepsmultiply) + ge_balance_positive_length_transport_resultstepsmultiplysecondreal))) /\ (exists ge_balance_positive_length_transport_resultstepsmultiplysecondimaginary ge_balance_negative_length_transport_resultstepsmultiplysecondimaginary. (((((ge_representation_imaginary_code_length_transport_resultstepsmultiplysecond) = 2 * (ge_balance_positive_length_transport_resultstepsmultiplysecondimaginary) /\ (ge_balance_negative_length_transport_resultstepsmultiplysecondimaginary) = 0) \/ exists ge_signed_half_length_transport_resultstepsmultiplysecondimaginarydecode. (((ge_representation_imaginary_code_length_transport_resultstepsmultiplysecond) = 2 * ge_signed_half_length_transport_resultstepsmultiplysecondimaginarydecode + 1 /\ (ge_balance_positive_length_transport_resultstepsmultiplysecondimaginary) = 0) /\ (ge_balance_negative_length_transport_resultstepsmultiplysecondimaginary) = S ge_signed_half_length_transport_resultstepsmultiplysecondimaginarydecode))) /\ ((ge_second_ip_length_transport_resultstepsmultiply) + ge_balance_negative_length_transport_resultstepsmultiplysecondimaginary = (ge_second_in_length_transport_resultstepsmultiply) + ge_balance_positive_length_transport_resultstepsmultiplysecondimaginary)))))) /\ (exists ge_representation_real_code_length_transport_resultstepsmultiplyoutput ge_representation_imaginary_code_length_transport_resultstepsmultiplyoutput. (((gr_product_after_length_transport_resultsteps) = ((ge_representation_real_code_length_transport_resultstepsmultiplyoutput) + (ge_representation_imaginary_code_length_transport_resultstepsmultiplyoutput)) * S ((ge_representation_real_code_length_transport_resultstepsmultiplyoutput) + (ge_representation_imaginary_code_length_transport_resultstepsmultiplyoutput)) + ((ge_representation_imaginary_code_length_transport_resultstepsmultiplyoutput) + (ge_representation_imaginary_code_length_transport_resultstepsmultiplyoutput))) /\ ((exists ge_balance_positive_length_transport_resultstepsmultiplyoutputreal ge_balance_negative_length_transport_resultstepsmultiplyoutputreal. (((((ge_representation_real_code_length_transport_resultstepsmultiplyoutput) = 2 * (ge_balance_positive_length_transport_resultstepsmultiplyoutputreal) /\ (ge_balance_negative_length_transport_resultstepsmultiplyoutputreal) = 0) \/ exists ge_signed_half_length_transport_resultstepsmultiplyoutputrealdecode. (((ge_representation_real_code_length_transport_resultstepsmultiplyoutput) = 2 * ge_signed_half_length_transport_resultstepsmultiplyoutputrealdecode + 1 /\ (ge_balance_positive_length_transport_resultstepsmultiplyoutputreal) = 0) /\ (ge_balance_negative_length_transport_resultstepsmultiplyoutputreal) = S ge_signed_half_length_transport_resultstepsmultiplyoutputrealdecode))) /\ ((((((((ge_first_rp_length_transport_resultstepsmultiply) * (ge_second_rp_length_transport_resultstepsmultiply))) + (((ge_first_rn_length_transport_resultstepsmultiply) * (ge_second_rn_length_transport_resultstepsmultiply))))) + (((((ge_first_ip_length_transport_resultstepsmultiply) * (ge_second_in_length_transport_resultstepsmultiply))) + (((ge_first_in_length_transport_resultstepsmultiply) * (ge_second_ip_length_transport_resultstepsmultiply))))))) + ge_balance_negative_length_transport_resultstepsmultiplyoutputreal = (((((((ge_first_rp_length_transport_resultstepsmultiply) * (ge_second_rn_length_transport_resultstepsmultiply))) + (((ge_first_rn_length_transport_resultstepsmultiply) * (ge_second_rp_length_transport_resultstepsmultiply))))) + (((((ge_first_ip_length_transport_resultstepsmultiply) * (ge_second_ip_length_transport_resultstepsmultiply))) + (((ge_first_in_length_transport_resultstepsmultiply) * (ge_second_in_length_transport_resultstepsmultiply))))))) + ge_balance_positive_length_transport_resultstepsmultiplyoutputreal))) /\ (exists ge_balance_positive_length_transport_resultstepsmultiplyoutputimaginary ge_balance_negative_length_transport_resultstepsmultiplyoutputimaginary. (((((ge_representation_imaginary_code_length_transport_resultstepsmultiplyoutput) = 2 * (ge_balance_positive_length_transport_resultstepsmultiplyoutputimaginary) /\ (ge_balance_negative_length_transport_resultstepsmultiplyoutputimaginary) = 0) \/ exists ge_signed_half_length_transport_resultstepsmultiplyoutputimaginarydecode. (((ge_representation_imaginary_code_length_transport_resultstepsmultiplyoutput) = 2 * ge_signed_half_length_transport_resultstepsmultiplyoutputimaginarydecode + 1 /\ (ge_balance_positive_length_transport_resultstepsmultiplyoutputimaginary) = 0) /\ (ge_balance_negative_length_transport_resultstepsmultiplyoutputimaginary) = S ge_signed_half_length_transport_resultstepsmultiplyoutputimaginarydecode))) /\ ((((((((ge_first_rp_length_transport_resultstepsmultiply) * (ge_second_ip_length_transport_resultstepsmultiply))) + (((ge_first_rn_length_transport_resultstepsmultiply) * (ge_second_in_length_transport_resultstepsmultiply))))) + (((((ge_first_ip_length_transport_resultstepsmultiply) * (ge_second_rp_length_transport_resultstepsmultiply))) + (((ge_first_in_length_transport_resultstepsmultiply) * (ge_second_rn_length_transport_resultstepsmultiply))))))) + ge_balance_negative_length_transport_resultstepsmultiplyoutputimaginary = (((((((ge_first_rp_length_transport_resultstepsmultiply) * (ge_second_in_length_transport_resultstepsmultiply))) + (((ge_first_rn_length_transport_resultstepsmultiply) * (ge_second_ip_length_transport_resultstepsmultiply))))) + (((((ge_first_ip_length_transport_resultstepsmultiply) * (ge_second_rn_length_transport_resultstepsmultiply))) + (((ge_first_in_length_transport_resultstepsmultiply) * (ge_second_rp_length_transport_resultstepsmultiply))))))) + ge_balance_positive_length_transport_resultstepsmultiplyoutputimaginary))))))))))))))))Constructive proof overview
Generated structural guide
Equality of natural lengths transports the exact endpoint and bound of an actual Gaussian multiplication trace.
The unchanged tactic script uses 0 declared prerequisites and contains 11 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
Formal 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 assumptionsL1–7
02Calculate and transport equalitiesL8–10
03Use earlier factsL11–11
Instantiate or apply named facts and discharge the corresponding proof obligations.
- L11
exact h