PQ0048

prime_field_polynomial_synthetic_remainder_execution

The synthetic remainder is the actual evaluation of the original input at the divisor root, not an assumed result certificate.

Alpha v34 checked-use · first admitted v32 · 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.

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. ∀ b. ∀ c. ∀ a. ∀ n. ∀ qb. ∀ qc. ∀ r. FpSyntheticDivision(p,b,c,a,n,qb,qc,r) → FpHorner(p,b,c,a,S n,r)

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

Definition DAG

FpHorner(p,b,c,x,l,r) · 1FpSyntheticDivision(p,b,c,a,n,qb,qc,r) · 1

Actual proof prerequisites

none
Original expanded first-order statement 
forall p b c a n qb qc r. (exists pfs_history_code_remainder_source pfs_history_scale_remainder_source. ((((exists pfa_gap_remainder_sourcetracebase. pfa_gap_remainder_sourcetracebase + S (a) = (p)) /\ (((((exists ff_h_pfp_remainder_sourcetraceinitial. ff_h_pfp_remainder_sourcetraceinitial + S (0) = S ((S (0)) * pfs_history_scale_remainder_source)) /\ exists ff_q_pfp_remainder_sourcetraceinitial. pfs_history_code_remainder_source = ff_q_pfp_remainder_sourcetraceinitial * S ((S (0)) * pfs_history_scale_remainder_source) + (0))) /\ (((((exists ff_h_pfp_remainder_sourcetraceterminal. ff_h_pfp_remainder_sourcetraceterminal + S (r) = S ((S (S (n))) * pfs_history_scale_remainder_source)) /\ exists ff_q_pfp_remainder_sourcetraceterminal. pfs_history_code_remainder_source = ff_q_pfp_remainder_sourcetraceterminal * S ((S (S (n))) * pfs_history_scale_remainder_source) + (r))) /\ ((forall pfh_index_remainder_sourcetracesteps. (exists pfa_gap_remainder_sourcetracestepsindex. pfa_gap_remainder_sourcetracestepsindex + S (pfh_index_remainder_sourcetracesteps) = (S (n))) -> (exists pfh_coefficient_remainder_sourcetracestepsstep pfh_before_remainder_sourcetracestepsstep pfh_after_remainder_sourcetracestepsstep pfh_product_remainder_sourcetracestepsstep. ((((exists ff_h_pfp_remainder_sourcetracestepsstepcoefficient. ff_h_pfp_remainder_sourcetracestepsstepcoefficient + S (pfh_coefficient_remainder_sourcetracestepsstep) = S ((S (pfh_index_remainder_sourcetracesteps)) * c)) /\ exists ff_q_pfp_remainder_sourcetracestepsstepcoefficient. b = ff_q_pfp_remainder_sourcetracestepsstepcoefficient * S ((S (pfh_index_remainder_sourcetracesteps)) * c) + (pfh_coefficient_remainder_sourcetracestepsstep))) /\ (((((exists ff_h_pfp_remainder_sourcetracestepsstepbefore. ff_h_pfp_remainder_sourcetracestepsstepbefore + S (pfh_before_remainder_sourcetracestepsstep) = S ((S (pfh_index_remainder_sourcetracesteps)) * pfs_history_scale_remainder_source)) /\ exists ff_q_pfp_remainder_sourcetracestepsstepbefore. pfs_history_code_remainder_source = ff_q_pfp_remainder_sourcetracestepsstepbefore * S ((S (pfh_index_remainder_sourcetracesteps)) * pfs_history_scale_remainder_source) + (pfh_before_remainder_sourcetracestepsstep))) /\ (((((exists ff_h_pfp_remainder_sourcetracestepsstepafter. ff_h_pfp_remainder_sourcetracestepsstepafter + S (pfh_after_remainder_sourcetracestepsstep) = S ((S (S (pfh_index_remainder_sourcetracesteps))) * pfs_history_scale_remainder_source)) /\ exists ff_q_pfp_remainder_sourcetracestepsstepafter. pfs_history_code_remainder_source = ff_q_pfp_remainder_sourcetracestepsstepafter * S ((S (S (pfh_index_remainder_sourcetracesteps))) * pfs_history_scale_remainder_source) + (pfh_after_remainder_sourcetracestepsstep))) /\ (((((exists pfa_gap_remainder_sourcetracestepsstepmultiplyleft. pfa_gap_remainder_sourcetracestepsstepmultiplyleft + S (pfh_before_remainder_sourcetracestepsstep) = (p)) /\ (((exists pfa_gap_remainder_sourcetracestepsstepmultiplyright. pfa_gap_remainder_sourcetracestepsstepmultiplyright + S (a) = (p)) /\ ((((exists pfa_gap_remainder_sourcetracestepsstepmultiplyresultbound. pfa_gap_remainder_sourcetracestepsstepmultiplyresultbound + S (pfh_product_remainder_sourcetracestepsstep) = (p)) /\ ((exists pfa_offset_left_remainder_sourcetracestepsstepmultiplyresultcongruence pfa_offset_right_remainder_sourcetracestepsstepmultiplyresultcongruence. ((pfh_before_remainder_sourcetracestepsstep) * (a)) + (p) * pfa_offset_left_remainder_sourcetracestepsstepmultiplyresultcongruence = (pfh_product_remainder_sourcetracestepsstep) + (p) * pfa_offset_right_remainder_sourcetracestepsstepmultiplyresultcongruence))))))))) /\ ((((exists pfa_gap_remainder_sourcetracestepsstepaddleft. pfa_gap_remainder_sourcetracestepsstepaddleft + S (pfh_product_remainder_sourcetracestepsstep) = (p)) /\ (((exists pfa_gap_remainder_sourcetracestepsstepaddright. pfa_gap_remainder_sourcetracestepsstepaddright + S (pfh_coefficient_remainder_sourcetracestepsstep) = (p)) /\ ((((exists pfa_gap_remainder_sourcetracestepsstepaddresultbound. pfa_gap_remainder_sourcetracestepsstepaddresultbound + S (pfh_after_remainder_sourcetracestepsstep) = (p)) /\ ((exists pfa_offset_left_remainder_sourcetracestepsstepaddresultcongruence pfa_offset_right_remainder_sourcetracestepsstepaddresultcongruence. ((pfh_product_remainder_sourcetracestepsstep) + (pfh_coefficient_remainder_sourcetracestepsstep)) + (p) * pfa_offset_left_remainder_sourcetracestepsstepaddresultcongruence = (pfh_after_remainder_sourcetracestepsstep) + (p) * pfa_offset_right_remainder_sourcetracestepsstepaddresultcongruence)))))))))))))))))))))))))) /\ ((forall ff_index_mcp_pfs_remainder_sourcequotient ff_source_mcp_pfs_remainder_sourcequotient ff_target_mcp_pfs_remainder_sourcequotient. (exists mcp_gap_pfs_remainder_sourcequotient_bound. mcp_gap_pfs_remainder_sourcequotient_bound + S (ff_index_mcp_pfs_remainder_sourcequotient) = (n)) -> (((exists fs_h_mcp_pfs_remainder_sourcequotient_source. fs_h_mcp_pfs_remainder_sourcequotient_source + S (ff_source_mcp_pfs_remainder_sourcequotient) = S ((S ((1) + (1) * ff_index_mcp_pfs_remainder_sourcequotient)) * pfs_history_scale_remainder_source)) /\ exists fs_q_mcp_pfs_remainder_sourcequotient_source. pfs_history_code_remainder_source = fs_q_mcp_pfs_remainder_sourcequotient_source * S ((S ((1) + (1) * ff_index_mcp_pfs_remainder_sourcequotient)) * pfs_history_scale_remainder_source) + (ff_source_mcp_pfs_remainder_sourcequotient))) -> (((exists fs_h_mcp_pfs_remainder_sourcequotient_target. fs_h_mcp_pfs_remainder_sourcequotient_target + S (ff_target_mcp_pfs_remainder_sourcequotient) = S ((S (ff_index_mcp_pfs_remainder_sourcequotient)) * qc)) /\ exists fs_q_mcp_pfs_remainder_sourcequotient_target. qb = fs_q_mcp_pfs_remainder_sourcequotient_target * S ((S (ff_index_mcp_pfs_remainder_sourcequotient)) * qc) + (ff_target_mcp_pfs_remainder_sourcequotient))) -> ff_target_mcp_pfs_remainder_sourcequotient = ff_source_mcp_pfs_remainder_sourcequotient)))) -> (exists pfh_trace_code_remainder_execution pfh_trace_scale_remainder_execution. (((exists pfa_gap_remainder_executiontracebase. pfa_gap_remainder_executiontracebase + S (a) = (p)) /\ (((((exists ff_h_pfp_remainder_executiontraceinitial. ff_h_pfp_remainder_executiontraceinitial + S (0) = S ((S (0)) * pfh_trace_scale_remainder_execution)) /\ exists ff_q_pfp_remainder_executiontraceinitial. pfh_trace_code_remainder_execution = ff_q_pfp_remainder_executiontraceinitial * S ((S (0)) * pfh_trace_scale_remainder_execution) + (0))) /\ (((((exists ff_h_pfp_remainder_executiontraceterminal. ff_h_pfp_remainder_executiontraceterminal + S (r) = S ((S (S n)) * pfh_trace_scale_remainder_execution)) /\ exists ff_q_pfp_remainder_executiontraceterminal. pfh_trace_code_remainder_execution = ff_q_pfp_remainder_executiontraceterminal * S ((S (S n)) * pfh_trace_scale_remainder_execution) + (r))) /\ ((forall pfh_index_remainder_executiontracesteps. (exists pfa_gap_remainder_executiontracestepsindex. pfa_gap_remainder_executiontracestepsindex + S (pfh_index_remainder_executiontracesteps) = (S n)) -> (exists pfh_coefficient_remainder_executiontracestepsstep pfh_before_remainder_executiontracestepsstep pfh_after_remainder_executiontracestepsstep pfh_product_remainder_executiontracestepsstep. ((((exists ff_h_pfp_remainder_executiontracestepsstepcoefficient. ff_h_pfp_remainder_executiontracestepsstepcoefficient + S (pfh_coefficient_remainder_executiontracestepsstep) = S ((S (pfh_index_remainder_executiontracesteps)) * c)) /\ exists ff_q_pfp_remainder_executiontracestepsstepcoefficient. b = ff_q_pfp_remainder_executiontracestepsstepcoefficient * S ((S (pfh_index_remainder_executiontracesteps)) * c) + (pfh_coefficient_remainder_executiontracestepsstep))) /\ (((((exists ff_h_pfp_remainder_executiontracestepsstepbefore. ff_h_pfp_remainder_executiontracestepsstepbefore + S (pfh_before_remainder_executiontracestepsstep) = S ((S (pfh_index_remainder_executiontracesteps)) * pfh_trace_scale_remainder_execution)) /\ exists ff_q_pfp_remainder_executiontracestepsstepbefore. pfh_trace_code_remainder_execution = ff_q_pfp_remainder_executiontracestepsstepbefore * S ((S (pfh_index_remainder_executiontracesteps)) * pfh_trace_scale_remainder_execution) + (pfh_before_remainder_executiontracestepsstep))) /\ (((((exists ff_h_pfp_remainder_executiontracestepsstepafter. ff_h_pfp_remainder_executiontracestepsstepafter + S (pfh_after_remainder_executiontracestepsstep) = S ((S (S (pfh_index_remainder_executiontracesteps))) * pfh_trace_scale_remainder_execution)) /\ exists ff_q_pfp_remainder_executiontracestepsstepafter. pfh_trace_code_remainder_execution = ff_q_pfp_remainder_executiontracestepsstepafter * S ((S (S (pfh_index_remainder_executiontracesteps))) * pfh_trace_scale_remainder_execution) + (pfh_after_remainder_executiontracestepsstep))) /\ (((((exists pfa_gap_remainder_executiontracestepsstepmultiplyleft. pfa_gap_remainder_executiontracestepsstepmultiplyleft + S (pfh_before_remainder_executiontracestepsstep) = (p)) /\ (((exists pfa_gap_remainder_executiontracestepsstepmultiplyright. pfa_gap_remainder_executiontracestepsstepmultiplyright + S (a) = (p)) /\ ((((exists pfa_gap_remainder_executiontracestepsstepmultiplyresultbound. pfa_gap_remainder_executiontracestepsstepmultiplyresultbound + S (pfh_product_remainder_executiontracestepsstep) = (p)) /\ ((exists pfa_offset_left_remainder_executiontracestepsstepmultiplyresultcongruence pfa_offset_right_remainder_executiontracestepsstepmultiplyresultcongruence. ((pfh_before_remainder_executiontracestepsstep) * (a)) + (p) * pfa_offset_left_remainder_executiontracestepsstepmultiplyresultcongruence = (pfh_product_remainder_executiontracestepsstep) + (p) * pfa_offset_right_remainder_executiontracestepsstepmultiplyresultcongruence))))))))) /\ ((((exists pfa_gap_remainder_executiontracestepsstepaddleft. pfa_gap_remainder_executiontracestepsstepaddleft + S (pfh_product_remainder_executiontracestepsstep) = (p)) /\ (((exists pfa_gap_remainder_executiontracestepsstepaddright. pfa_gap_remainder_executiontracestepsstepaddright + S (pfh_coefficient_remainder_executiontracestepsstep) = (p)) /\ ((((exists pfa_gap_remainder_executiontracestepsstepaddresultbound. pfa_gap_remainder_executiontracestepsstepaddresultbound + S (pfh_after_remainder_executiontracestepsstep) = (p)) /\ ((exists pfa_offset_left_remainder_executiontracestepsstepaddresultcongruence pfa_offset_right_remainder_executiontracestepsstepaddresultcongruence. ((pfh_product_remainder_executiontracestepsstep) + (pfh_coefficient_remainder_executiontracestepsstep)) + (p) * pfa_offset_left_remainder_executiontracestepsstepaddresultcongruence = (pfh_after_remainder_executiontracestepsstep) + (p) * pfa_offset_right_remainder_executiontracestepsstepaddresultcongruence)))))))))))))))))))))))))))

Complete tactic proof in conservative notation

All 15 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

15 script commands · 4 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.

01Fix variables and assumptionsL1–9

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

  1. L1
    intro p
  2. L2
    intro b
  3. L3
    intro c
  4. L4
    intro a
  5. L5
    intro n
  6. L6
    intro qb
  7. L7
    intro qc
  8. L8
    intro r
  9. L9
    intro hs
02Separate the logical casesL10–12

Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.

  1. L10
    cases hs
  2. L11
    cases hs_witness
  3. L12
    cases hs_witness_witness
03Construct an explicit witnessL13–14

Supply the displayed value, then prove that it has the required property.

  1. L13
    exists x
  2. L14
    exists x1
04Use earlier factsL15–15

Instantiate or apply named facts and discharge the corresponding proof obligations.

  1. L15
    exact hs_witness_witness_left

Library-wide reading audit

Original defined command ledger · 15 lines
  1. 0001intro p
  2. 0002intro b
  3. 0003intro c
  4. 0004intro a
  5. 0005intro n
  6. 0006intro qb
  7. 0007intro qc
  8. 0008intro r
  9. 0009intro hs
  10. 0010cases hs
  11. 0011cases hs_witness
  12. 0012cases hs_witness_witness
  13. 0013exists x
  14. 0014exists x1
  15. 0015exact hs_witness_witness_left