Current library: Alpha v34, 4,223 checked-use theorems; Stable remains 432. Historical first admissions, original proof editions, and non-admitted aliases are preserved.
Definition in prerequisite notation
PolynomialEquivalent(ab,ac,L,ub,uc,K) ∧ PolynomialEquivalent(bb,bc,M,vb,vc,K)Only definitions earlier in this acyclic notation graph are used here.
Hygienic expanded first-order definition
((forall pfrep_power_working_aligned_definition_left pfrep_left_working_aligned_definition_left pfrep_right_working_aligned_definition_left. ((exists pfrep_position_working_aligned_definition_leftfirst. ((pfrep_position_working_aligned_definition_leftfirst+S (pfrep_power_working_aligned_definition_left)=((L))) /\ ((((exists ff_h_pfp_working_aligned_definition_leftfirstentry. ff_h_pfp_working_aligned_definition_leftfirstentry + S (pfrep_left_working_aligned_definition_left) = S ((S (pfrep_position_working_aligned_definition_leftfirst)) * (ac))) /\ exists ff_q_pfp_working_aligned_definition_leftfirstentry. (ab) = ff_q_pfp_working_aligned_definition_leftfirstentry * S ((S (pfrep_position_working_aligned_definition_leftfirst)) * (ac)) + (pfrep_left_working_aligned_definition_left)))))) \/ (((exists pfrep_gap_working_aligned_definition_leftfirstoutside. pfrep_gap_working_aligned_definition_leftfirstoutside+((L))=(pfrep_power_working_aligned_definition_left)) /\ (((pfrep_left_working_aligned_definition_left)=0))))) -> ((exists pfrep_position_working_aligned_definition_leftsecond. ((pfrep_position_working_aligned_definition_leftsecond+S (pfrep_power_working_aligned_definition_left)=((K))) /\ ((((exists ff_h_pfp_working_aligned_definition_leftsecondentry. ff_h_pfp_working_aligned_definition_leftsecondentry + S (pfrep_right_working_aligned_definition_left) = S ((S (pfrep_position_working_aligned_definition_leftsecond)) * (uc))) /\ exists ff_q_pfp_working_aligned_definition_leftsecondentry. (ub) = ff_q_pfp_working_aligned_definition_leftsecondentry * S ((S (pfrep_position_working_aligned_definition_leftsecond)) * (uc)) + (pfrep_right_working_aligned_definition_left)))))) \/ (((exists pfrep_gap_working_aligned_definition_leftsecondoutside. pfrep_gap_working_aligned_definition_leftsecondoutside+((K))=(pfrep_power_working_aligned_definition_left)) /\ (((pfrep_right_working_aligned_definition_left)=0))))) -> pfrep_left_working_aligned_definition_left=pfrep_right_working_aligned_definition_left) /\ ((forall pfrep_power_working_aligned_definition_right pfrep_left_working_aligned_definition_right pfrep_right_working_aligned_definition_right. ((exists pfrep_position_working_aligned_definition_rightfirst. ((pfrep_position_working_aligned_definition_rightfirst+S (pfrep_power_working_aligned_definition_right)=((M))) /\ ((((exists ff_h_pfp_working_aligned_definition_rightfirstentry. ff_h_pfp_working_aligned_definition_rightfirstentry + S (pfrep_left_working_aligned_definition_right) = S ((S (pfrep_position_working_aligned_definition_rightfirst)) * (bc))) /\ exists ff_q_pfp_working_aligned_definition_rightfirstentry. (bb) = ff_q_pfp_working_aligned_definition_rightfirstentry * S ((S (pfrep_position_working_aligned_definition_rightfirst)) * (bc)) + (pfrep_left_working_aligned_definition_right)))))) \/ (((exists pfrep_gap_working_aligned_definition_rightfirstoutside. pfrep_gap_working_aligned_definition_rightfirstoutside+((M))=(pfrep_power_working_aligned_definition_right)) /\ (((pfrep_left_working_aligned_definition_right)=0))))) -> ((exists pfrep_position_working_aligned_definition_rightsecond. ((pfrep_position_working_aligned_definition_rightsecond+S (pfrep_power_working_aligned_definition_right)=((K))) /\ ((((exists ff_h_pfp_working_aligned_definition_rightsecondentry. ff_h_pfp_working_aligned_definition_rightsecondentry + S (pfrep_right_working_aligned_definition_right) = S ((S (pfrep_position_working_aligned_definition_rightsecond)) * (vc))) /\ exists ff_q_pfp_working_aligned_definition_rightsecondentry. (vb) = ff_q_pfp_working_aligned_definition_rightsecondentry * S ((S (pfrep_position_working_aligned_definition_rightsecond)) * (vc)) + (pfrep_right_working_aligned_definition_right)))))) \/ (((exists pfrep_gap_working_aligned_definition_rightsecondoutside. pfrep_gap_working_aligned_definition_rightsecondoutside+((K))=(pfrep_power_working_aligned_definition_right)) /\ (((pfrep_right_working_aligned_definition_right)=0))))) -> pfrep_left_working_aligned_definition_right=pfrep_right_working_aligned_definition_right)))The unchanged native kernel never receives this surface symbol. Binder-safe expansion produces only its existing first-order syntax.
Direct definition dependencies
Definitions depending on this notation
Checked theorems using this definition
PG0036 · prime_field_polynomial_common_representatives_same_lengthPG0037 · prime_field_polynomial_common_representatives_transportPG0038 · prime_field_polynomial_common_representatives_at_length_existsPG0039 · prime_field_polynomial_common_representatives_existsPG003A · prime_field_polynomial_common_representatives_functionalPG003B · prime_field_polynomial_common_representatives_symmetricPG003C · prime_field_polynomial_aligned_add_from_commonPG0041 · prime_field_polynomial_aligned_add_functionalPG0042 · prime_field_polynomial_aligned_add_existsPG0043 · prime_field_polynomial_aligned_add_realizePG0045 · prime_field_polynomial_aligned_subtract_exists