Common actual leading-zero padding preserves the genuine aligned add coefficient operation, including empty prefixes.
Alpha v34 checked-use · first admitted v33 · independently kernel and Lean verified; not Stable
Current library: Alpha v34, 4,223 checked-use theorems; Stable remains 432. Historical first admissions, original proof editions, and non-admitted aliases are preserved.
Coefficients are highest-degree-first. The divisor has a nonzero decoded head; primality supplies its actual inverse. Empty quotients and remainders are included. Functionality compares the constructed execution lengths and decoded coefficients, never arbitrary beta codes. Formal polynomial equivalence compares every coefficient, not evaluations on a finite field. The formal identity and remainder-degree bound are proved separately, not assumed by the execution graph. Arbitrary quotient/remainder-pair uniqueness from a formal identity, multiplication associativity, gcd/Bezout, irreducible-polynomial existence, and the full G091 prime-power-field goal remain open. The seven displayed new names are conservative first-order notation, not new kernel primitives.
forall p ab ac bb bc cb cc L t AB AC BB BC CB CC. (~((p) = 1) /\ forall pfa_factor_left_pad_operation_prime pfa_factor_right_pad_operation_prime. (p) = pfa_factor_left_pad_operation_prime * pfa_factor_right_pad_operation_prime -> pfa_factor_left_pad_operation_prime = 1 \/ pfa_factor_right_pad_operation_prime = 1) -> (forall pfp_index_pad_operation_old. (exists pfa_gap_pad_operation_oldindex. pfa_gap_pad_operation_oldindex + S (pfp_index_pad_operation_old) = (L)) -> exists pfp_left_pad_operation_old pfp_right_pad_operation_old pfp_value_pad_operation_old. ((((exists ff_h_pfp_pad_operation_oldleft. ff_h_pfp_pad_operation_oldleft + S (pfp_left_pad_operation_old) = S ((S (pfp_index_pad_operation_old)) * ac)) /\ exists ff_q_pfp_pad_operation_oldleft. ab = ff_q_pfp_pad_operation_oldleft * S ((S (pfp_index_pad_operation_old)) * ac) + (pfp_left_pad_operation_old))) /\ (((((exists ff_h_pfp_pad_operation_oldright. ff_h_pfp_pad_operation_oldright + S (pfp_right_pad_operation_old) = S ((S (pfp_index_pad_operation_old)) * bc)) /\ exists ff_q_pfp_pad_operation_oldright. bb = ff_q_pfp_pad_operation_oldright * S ((S (pfp_index_pad_operation_old)) * bc) + (pfp_right_pad_operation_old))) /\ (((((exists ff_h_pfp_pad_operation_oldtarget. ff_h_pfp_pad_operation_oldtarget + S (pfp_value_pad_operation_old) = S ((S (pfp_index_pad_operation_old)) * cc)) /\ exists ff_q_pfp_pad_operation_oldtarget. cb = ff_q_pfp_pad_operation_oldtarget * S ((S (pfp_index_pad_operation_old)) * cc) + (pfp_value_pad_operation_old))) /\ ((((exists pfa_gap_pad_operation_oldoperationleft. pfa_gap_pad_operation_oldoperationleft + S (pfp_left_pad_operation_old) = (p)) /\ (((exists pfa_gap_pad_operation_oldoperationright. pfa_gap_pad_operation_oldoperationright + S (pfp_right_pad_operation_old) = (p)) /\ ((((exists pfa_gap_pad_operation_oldoperationresultbound. pfa_gap_pad_operation_oldoperationresultbound + S (pfp_value_pad_operation_old) = (p)) /\ ((exists pfa_offset_left_pad_operation_oldoperationresultcongruence pfa_offset_right_pad_operation_oldoperationresultcongruence. ((pfp_left_pad_operation_old) + (pfp_right_pad_operation_old)) + (p) * pfa_offset_left_pad_operation_oldoperationresultcongruence = (pfp_value_pad_operation_old) + (p) * pfa_offset_right_pad_operation_oldoperationresultcongruence)))))))))))))))) -> (((forall pfp_repeat_index_pad_operation_abzeros. (exists pfa_gap_pad_operation_abzerosindex. pfa_gap_pad_operation_abzerosindex + S (pfp_repeat_index_pad_operation_abzeros) = (t)) -> (((exists ff_h_pfp_pad_operation_abzerosentry. ff_h_pfp_pad_operation_abzerosentry + S (0) = S ((S (pfp_repeat_index_pad_operation_abzeros)) * AC)) /\ exists ff_q_pfp_pad_operation_abzerosentry. AB = ff_q_pfp_pad_operation_abzerosentry * S ((S (pfp_repeat_index_pad_operation_abzeros)) * AC) + (0)))) /\ ((forall pfrep_index_pad_operation_ab pfrep_value_pad_operation_ab. (exists pfa_gap_pad_operation_abbound. pfa_gap_pad_operation_abbound + S (pfrep_index_pad_operation_ab) = (L)) -> (((exists ff_h_pfp_pad_operation_abinput. ff_h_pfp_pad_operation_abinput + S (pfrep_value_pad_operation_ab) = S ((S (pfrep_index_pad_operation_ab)) * ac)) /\ exists ff_q_pfp_pad_operation_abinput. ab = ff_q_pfp_pad_operation_abinput * S ((S (pfrep_index_pad_operation_ab)) * ac) + (pfrep_value_pad_operation_ab))) -> (((exists ff_h_pfp_pad_operation_aboutput. ff_h_pfp_pad_operation_aboutput + S (pfrep_value_pad_operation_ab) = S ((S ((t)+pfrep_index_pad_operation_ab)) * AC)) /\ exists ff_q_pfp_pad_operation_aboutput. AB = ff_q_pfp_pad_operation_aboutput * S ((S ((t)+pfrep_index_pad_operation_ab)) * AC) + (pfrep_value_pad_operation_ab))))))) -> (((forall pfp_repeat_index_pad_operation_bbzeros. (exists pfa_gap_pad_operation_bbzerosindex. pfa_gap_pad_operation_bbzerosindex + S (pfp_repeat_index_pad_operation_bbzeros) = (t)) -> (((exists ff_h_pfp_pad_operation_bbzerosentry. ff_h_pfp_pad_operation_bbzerosentry + S (0) = S ((S (pfp_repeat_index_pad_operation_bbzeros)) * BC)) /\ exists ff_q_pfp_pad_operation_bbzerosentry. BB = ff_q_pfp_pad_operation_bbzerosentry * S ((S (pfp_repeat_index_pad_operation_bbzeros)) * BC) + (0)))) /\ ((forall pfrep_index_pad_operation_bb pfrep_value_pad_operation_bb. (exists pfa_gap_pad_operation_bbbound. pfa_gap_pad_operation_bbbound + S (pfrep_index_pad_operation_bb) = (L)) -> (((exists ff_h_pfp_pad_operation_bbinput. ff_h_pfp_pad_operation_bbinput + S (pfrep_value_pad_operation_bb) = S ((S (pfrep_index_pad_operation_bb)) * bc)) /\ exists ff_q_pfp_pad_operation_bbinput. bb = ff_q_pfp_pad_operation_bbinput * S ((S (pfrep_index_pad_operation_bb)) * bc) + (pfrep_value_pad_operation_bb))) -> (((exists ff_h_pfp_pad_operation_bboutput. ff_h_pfp_pad_operation_bboutput + S (pfrep_value_pad_operation_bb) = S ((S ((t)+pfrep_index_pad_operation_bb)) * BC)) /\ exists ff_q_pfp_pad_operation_bboutput. BB = ff_q_pfp_pad_operation_bboutput * S ((S ((t)+pfrep_index_pad_operation_bb)) * BC) + (pfrep_value_pad_operation_bb))))))) -> (((forall pfp_repeat_index_pad_operation_cbzeros. (exists pfa_gap_pad_operation_cbzerosindex. pfa_gap_pad_operation_cbzerosindex + S (pfp_repeat_index_pad_operation_cbzeros) = (t)) -> (((exists ff_h_pfp_pad_operation_cbzerosentry. ff_h_pfp_pad_operation_cbzerosentry + S (0) = S ((S (pfp_repeat_index_pad_operation_cbzeros)) * CC)) /\ exists ff_q_pfp_pad_operation_cbzerosentry. CB = ff_q_pfp_pad_operation_cbzerosentry * S ((S (pfp_repeat_index_pad_operation_cbzeros)) * CC) + (0)))) /\ ((forall pfrep_index_pad_operation_cb pfrep_value_pad_operation_cb. (exists pfa_gap_pad_operation_cbbound. pfa_gap_pad_operation_cbbound + S (pfrep_index_pad_operation_cb) = (L)) -> (((exists ff_h_pfp_pad_operation_cbinput. ff_h_pfp_pad_operation_cbinput + S (pfrep_value_pad_operation_cb) = S ((S (pfrep_index_pad_operation_cb)) * cc)) /\ exists ff_q_pfp_pad_operation_cbinput. cb = ff_q_pfp_pad_operation_cbinput * S ((S (pfrep_index_pad_operation_cb)) * cc) + (pfrep_value_pad_operation_cb))) -> (((exists ff_h_pfp_pad_operation_cboutput. ff_h_pfp_pad_operation_cboutput + S (pfrep_value_pad_operation_cb) = S ((S ((t)+pfrep_index_pad_operation_cb)) * CC)) /\ exists ff_q_pfp_pad_operation_cboutput. CB = ff_q_pfp_pad_operation_cboutput * S ((S ((t)+pfrep_index_pad_operation_cb)) * CC) + (pfrep_value_pad_operation_cb))))))) -> (forall pfp_index_pad_operation_new. (exists pfa_gap_pad_operation_newindex. pfa_gap_pad_operation_newindex + S (pfp_index_pad_operation_new) = (t+L)) -> exists pfp_left_pad_operation_new pfp_right_pad_operation_new pfp_value_pad_operation_new. ((((exists ff_h_pfp_pad_operation_newleft. ff_h_pfp_pad_operation_newleft + S (pfp_left_pad_operation_new) = S ((S (pfp_index_pad_operation_new)) * AC)) /\ exists ff_q_pfp_pad_operation_newleft. AB = ff_q_pfp_pad_operation_newleft * S ((S (pfp_index_pad_operation_new)) * AC) + (pfp_left_pad_operation_new))) /\ (((((exists ff_h_pfp_pad_operation_newright. ff_h_pfp_pad_operation_newright + S (pfp_right_pad_operation_new) = S ((S (pfp_index_pad_operation_new)) * BC)) /\ exists ff_q_pfp_pad_operation_newright. BB = ff_q_pfp_pad_operation_newright * S ((S (pfp_index_pad_operation_new)) * BC) + (pfp_right_pad_operation_new))) /\ (((((exists ff_h_pfp_pad_operation_newtarget. ff_h_pfp_pad_operation_newtarget + S (pfp_value_pad_operation_new) = S ((S (pfp_index_pad_operation_new)) * CC)) /\ exists ff_q_pfp_pad_operation_newtarget. CB = ff_q_pfp_pad_operation_newtarget * S ((S (pfp_index_pad_operation_new)) * CC) + (pfp_value_pad_operation_new))) /\ ((((exists pfa_gap_pad_operation_newoperationleft. pfa_gap_pad_operation_newoperationleft + S (pfp_left_pad_operation_new) = (p)) /\ (((exists pfa_gap_pad_operation_newoperationright. pfa_gap_pad_operation_newoperationright + S (pfp_right_pad_operation_new) = (p)) /\ ((((exists pfa_gap_pad_operation_newoperationresultbound. pfa_gap_pad_operation_newoperationresultbound + S (pfp_value_pad_operation_new) = (p)) /\ ((exists pfa_offset_left_pad_operation_newoperationresultcongruence pfa_offset_right_pad_operation_newoperationresultcongruence. ((pfp_left_pad_operation_new) + (pfp_right_pad_operation_new)) + (p) * pfa_offset_left_pad_operation_newoperationresultcongruence = (pfp_value_pad_operation_new) + (p) * pfa_offset_right_pad_operation_newoperationresultcongruence))))))))))))))))
Complete tactic proof in conservative notation
All 94 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.
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.
Establish this local claim before using it. It is not an additional assumption. The following proof commands apply prime field polynomial left pad index cases.