PQ0019 — prime_field_polynomial_subtract_add_cancel

Current library: Alpha v34, 4,223 checked-use theorems; Stable remains 432. Historical first admissions, original proof editions, and non-admitted aliases are preserved.

All coefficients retain the established highest-degree-first order. Trimming handles empty and all-zero prefixes; monic normalization requires a nonzero leading coefficient. Synthetic division has a nonempty input of length S n and a quotient of length n, unique in decoded values. Its coefficient recurrence, actual evaluation remainder and positive-degree drop are checked. General polynomial Euclidean division, gcd/Bezout, an arbitrary-convolution factor theorem, irreducible-polynomial existence and the full G091 prime-power-field endpoint remain open. These exact theorems are first admitted to Alpha v32; Stable remains unchanged.

Exact theorem in conservative defined notation

∀ p. ∀ ab. ∀ ac. ∀ bb. ∀ bc. ∀ cb. ∀ cc. ∀ rb. ∀ rc. ∀ l. FpPolyAdd(p,ab,ac,bb,bc,cb,cc,l) → FpCoefficientSubtraction(p,cb,cc,ab,ac,rb,rc,l) → ∀ x. ∀ y. Lt(x,l) → BetaAt(bb,bc,x,y) → BetaAt(rb,rc,x,y)

Every linked abbreviation expands hygienically to the identical original native formula.

Definition DAG

FpPolyAdd(p,ab,ac,bb,bc,cb,cc,l) · 1 FpCoefficientSubtraction(p,ab,ac,bb,bc,rb,rc,L) · 1 Lt(a,b) · 1 BetaAt(b,c,i,x) · 2

Actual proof prerequisites

prime_field_polynomial_subtract_functional prime_field_polynomial_subtract_from_add

Original expanded first-order statement

forall p ab ac bb bc cb cc rb rc l. (forall pfp_index_sub_cancel_sum. (exists pfa_gap_sub_cancel_sumindex. pfa_gap_sub_cancel_sumindex + S (pfp_index_sub_cancel_sum) = (l)) -> exists pfp_left_sub_cancel_sum pfp_right_sub_cancel_sum pfp_value_sub_cancel_sum. ((((exists ff_h_pfp_sub_cancel_sumleft. ff_h_pfp_sub_cancel_sumleft + S (pfp_left_sub_cancel_sum) = S ((S (pfp_index_sub_cancel_sum)) * ac)) /\ exists ff_q_pfp_sub_cancel_sumleft. ab = ff_q_pfp_sub_cancel_sumleft * S ((S (pfp_index_sub_cancel_sum)) * ac) + (pfp_left_sub_cancel_sum))) /\ (((((exists ff_h_pfp_sub_cancel_sumright. ff_h_pfp_sub_cancel_sumright + S (pfp_right_sub_cancel_sum) = S ((S (pfp_index_sub_cancel_sum)) * bc)) /\ exists ff_q_pfp_sub_cancel_sumright. bb = ff_q_pfp_sub_cancel_sumright * S ((S (pfp_index_sub_cancel_sum)) * bc) + (pfp_right_sub_cancel_sum))) /\ (((((exists ff_h_pfp_sub_cancel_sumtarget. ff_h_pfp_sub_cancel_sumtarget + S (pfp_value_sub_cancel_sum) = S ((S (pfp_index_sub_cancel_sum)) * cc)) /\ exists ff_q_pfp_sub_cancel_sumtarget. cb = ff_q_pfp_sub_cancel_sumtarget * S ((S (pfp_index_sub_cancel_sum)) * cc) + (pfp_value_sub_cancel_sum))) /\ ((((exists pfa_gap_sub_cancel_sumoperationleft. pfa_gap_sub_cancel_sumoperationleft + S (pfp_left_sub_cancel_sum) = (p)) /\ (((exists pfa_gap_sub_cancel_sumoperationright. pfa_gap_sub_cancel_sumoperationright + S (pfp_right_sub_cancel_sum) = (p)) /\ ((((exists pfa_gap_sub_cancel_sumoperationresultbound. pfa_gap_sub_cancel_sumoperationresultbound + S (pfp_value_sub_cancel_sum) = (p)) /\ ((exists pfa_offset_left_sub_cancel_sumoperationresultcongruence pfa_offset_right_sub_cancel_sumoperationresultcongruence. ((pfp_left_sub_cancel_sum) + (pfp_right_sub_cancel_sum)) + (p) * pfa_offset_left_sub_cancel_sumoperationresultcongruence = (pfp_value_sub_cancel_sum) + (p) * pfa_offset_right_sub_cancel_sumoperationresultcongruence)))))))))))))))) -> (forall pfs_index_sub_cancel_difference. (exists pfa_gap_sub_cancel_differenceindex. pfa_gap_sub_cancel_differenceindex + S (pfs_index_sub_cancel_difference) = (l)) -> exists pfs_left_sub_cancel_difference pfs_right_sub_cancel_difference pfs_result_sub_cancel_difference. ((((exists ff_h_pfp_sub_cancel_differenceleft. ff_h_pfp_sub_cancel_differenceleft + S (pfs_left_sub_cancel_difference) = S ((S (pfs_index_sub_cancel_difference)) * cc)) /\ exists ff_q_pfp_sub_cancel_differenceleft. cb = ff_q_pfp_sub_cancel_differenceleft * S ((S (pfs_index_sub_cancel_difference)) * cc) + (pfs_left_sub_cancel_difference))) /\ (((((exists ff_h_pfp_sub_cancel_differenceright. ff_h_pfp_sub_cancel_differenceright + S (pfs_right_sub_cancel_difference) = S ((S (pfs_index_sub_cancel_difference)) * ac)) /\ exists ff_q_pfp_sub_cancel_differenceright. ab = ff_q_pfp_sub_cancel_differenceright * S ((S (pfs_index_sub_cancel_difference)) * ac) + (pfs_right_sub_cancel_difference))) /\ (((((exists ff_h_pfp_sub_cancel_differenceresult. ff_h_pfp_sub_cancel_differenceresult + S (pfs_result_sub_cancel_difference) = S ((S (pfs_index_sub_cancel_difference)) * rc)) /\ exists ff_q_pfp_sub_cancel_differenceresult. rb = ff_q_pfp_sub_cancel_differenceresult * S ((S (pfs_index_sub_cancel_difference)) * rc) + (pfs_result_sub_cancel_difference))) /\ ((((exists pfa_gap_sub_cancel_differenceoperationleft. pfa_gap_sub_cancel_differenceoperationleft + S (pfs_right_sub_cancel_difference) = (p)) /\ (((exists pfa_gap_sub_cancel_differenceoperationright. pfa_gap_sub_cancel_differenceoperationright + S (pfs_result_sub_cancel_difference) = (p)) /\ ((((exists pfa_gap_sub_cancel_differenceoperationresultbound. pfa_gap_sub_cancel_differenceoperationresultbound + S (pfs_left_sub_cancel_difference) = (p)) /\ ((exists pfa_offset_left_sub_cancel_differenceoperationresultcongruence pfa_offset_right_sub_cancel_differenceoperationresultcongruence. ((pfs_right_sub_cancel_difference) + (pfs_result_sub_cancel_difference)) + (p) * pfa_offset_left_sub_cancel_differenceoperationresultcongruence = (pfs_left_sub_cancel_difference) + (p) * pfa_offset_right_sub_cancel_differenceoperationresultcongruence)))))))))))))))) -> (forall mdr_i_pfp_sub_cancel_result mdr_a_pfp_sub_cancel_result. (exists mdr_gap_pfp_sub_cancel_resultb. mdr_gap_pfp_sub_cancel_resultb + S (mdr_i_pfp_sub_cancel_result) = (l)) -> (((exists ff_h_mdr_pfp_sub_cancel_resulto. ff_h_mdr_pfp_sub_cancel_resulto + S (mdr_a_pfp_sub_cancel_result) = S ((S (mdr_i_pfp_sub_cancel_result)) * bc)) /\ exists ff_q_mdr_pfp_sub_cancel_resulto. bb = ff_q_mdr_pfp_sub_cancel_resulto * S ((S (mdr_i_pfp_sub_cancel_result)) * bc) + (mdr_a_pfp_sub_cancel_result))) -> (((exists ff_h_mdr_pfp_sub_cancel_resultn. ff_h_mdr_pfp_sub_cancel_resultn + S (mdr_a_pfp_sub_cancel_result) = S ((S (mdr_i_pfp_sub_cancel_result)) * rc)) /\ exists ff_q_mdr_pfp_sub_cancel_resultn. rb = ff_q_mdr_pfp_sub_cancel_resultn * S ((S (mdr_i_pfp_sub_cancel_result)) * rc) + (mdr_a_pfp_sub_cancel_result))))

Complete tactic proof in conservative notation

All 34 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

34 script commands · 5 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.

Named ingredients (2)

prime field polynomial subtract functional prime field polynomial subtract from add

01Fix variables and assumptionsL1–10

Work with arbitrary variables or the premises of the current implication.