PQ0014

prime_field_polynomial_subtract_self_zero

Subtracting a canonical prefix from itself constructs its genuine all-zero coefficient result.

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. ∀ ab. ∀ ac. ∀ zb. ∀ zc. ∀ l. Prime(p) → BetaPrefixInto(ab,ac,l,p) → Repeat(zb,zc,0,l) → FpCoefficientSubtraction(p,ab,ac,ab,ac,zb,zc,l)

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

Definition DAG

BetaPrefixInto(b,c,l,B) · 1FpCoefficientSubtraction(p,ab,ac,bb,bc,rb,rc,L) · 1Prime(p) · 1Repeat(b,c,a,l) · 1

Actual proof prerequisites

prime_field_polynomial_subtract_from_addprime_field_polynomial_add_zero_right · checked external prerequisite
Original expanded first-order statement 
forall p ab ac zb zc l. (~((p) = 1) /\ forall pfa_factor_left_sub_self_prime pfa_factor_right_sub_self_prime. (p) = pfa_factor_left_sub_self_prime * pfa_factor_right_sub_self_prime -> pfa_factor_left_sub_self_prime = 1 \/ pfa_factor_right_sub_self_prime = 1) -> (forall fom_index_pfp_sub_self_coeff. (exists fom_gap_pfp_sub_self_coeff_index_bound. fom_gap_pfp_sub_self_coeff_index_bound + S (fom_index_pfp_sub_self_coeff) = l) -> exists fom_value_pfp_sub_self_coeff. ((((exists fom_beta_height_pfp_sub_self_coeff_entry. fom_beta_height_pfp_sub_self_coeff_entry + S (fom_value_pfp_sub_self_coeff) = S ((S (fom_index_pfp_sub_self_coeff)) * ac)) /\ exists fom_beta_quotient_pfp_sub_self_coeff_entry. ab = fom_beta_quotient_pfp_sub_self_coeff_entry * S ((S (fom_index_pfp_sub_self_coeff)) * ac) + (fom_value_pfp_sub_self_coeff))) /\ (exists fom_gap_pfp_sub_self_coeff_value_bound. fom_gap_pfp_sub_self_coeff_value_bound + S (fom_value_pfp_sub_self_coeff) = p))) -> (forall pfp_repeat_index_sub_self_zero. (exists pfa_gap_sub_self_zeroindex. pfa_gap_sub_self_zeroindex + S (pfp_repeat_index_sub_self_zero) = (l)) -> (((exists ff_h_pfp_sub_self_zeroentry. ff_h_pfp_sub_self_zeroentry + S (0) = S ((S (pfp_repeat_index_sub_self_zero)) * zc)) /\ exists ff_q_pfp_sub_self_zeroentry. zb = ff_q_pfp_sub_self_zeroentry * S ((S (pfp_repeat_index_sub_self_zero)) * zc) + (0)))) -> (forall pfs_index_sub_self_result. (exists pfa_gap_sub_self_resultindex. pfa_gap_sub_self_resultindex + S (pfs_index_sub_self_result) = (l)) -> exists pfs_left_sub_self_result pfs_right_sub_self_result pfs_result_sub_self_result. ((((exists ff_h_pfp_sub_self_resultleft. ff_h_pfp_sub_self_resultleft + S (pfs_left_sub_self_result) = S ((S (pfs_index_sub_self_result)) * ac)) /\ exists ff_q_pfp_sub_self_resultleft. ab = ff_q_pfp_sub_self_resultleft * S ((S (pfs_index_sub_self_result)) * ac) + (pfs_left_sub_self_result))) /\ (((((exists ff_h_pfp_sub_self_resultright. ff_h_pfp_sub_self_resultright + S (pfs_right_sub_self_result) = S ((S (pfs_index_sub_self_result)) * ac)) /\ exists ff_q_pfp_sub_self_resultright. ab = ff_q_pfp_sub_self_resultright * S ((S (pfs_index_sub_self_result)) * ac) + (pfs_right_sub_self_result))) /\ (((((exists ff_h_pfp_sub_self_resultresult. ff_h_pfp_sub_self_resultresult + S (pfs_result_sub_self_result) = S ((S (pfs_index_sub_self_result)) * zc)) /\ exists ff_q_pfp_sub_self_resultresult. zb = ff_q_pfp_sub_self_resultresult * S ((S (pfs_index_sub_self_result)) * zc) + (pfs_result_sub_self_result))) /\ ((((exists pfa_gap_sub_self_resultoperationleft. pfa_gap_sub_self_resultoperationleft + S (pfs_right_sub_self_result) = (p)) /\ (((exists pfa_gap_sub_self_resultoperationright. pfa_gap_sub_self_resultoperationright + S (pfs_result_sub_self_result) = (p)) /\ ((((exists pfa_gap_sub_self_resultoperationresultbound. pfa_gap_sub_self_resultoperationresultbound + S (pfs_left_sub_self_result) = (p)) /\ ((exists pfa_offset_left_sub_self_resultoperationresultcongruence pfa_offset_right_sub_self_resultoperationresultcongruence. ((pfs_right_sub_self_result) + (pfs_result_sub_self_result)) + (p) * pfa_offset_left_sub_self_resultoperationresultcongruence = (pfs_left_sub_self_result) + (p) * pfa_offset_right_sub_self_resultoperationresultcongruence))))))))))))))))

Complete tactic proof in conservative notation

All 28 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

28 script commands · 3 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 (1)
prime field polynomial subtract from add
01Fix variables and assumptionsL1–9

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

  1. L1
    intro p
  2. L2
    intro ab
  3. L3
    intro ac
  4. L4
    intro zb
  5. L5
    intro zc
  6. L6
    intro l
  7. L7
    intro hp
  8. L8
    intro ha
  9. L9
    intro hz
02Use earlier factsL10–19

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

  1. L10
    specialize prime_field_polynomial_subtract_from_add (p)
  2. L11
    specialize prime_field_polynomial_subtract_from_add (ab)
  3. L12
    specialize prime_field_polynomial_subtract_from_add (ac)
  4. L13
    specialize prime_field_polynomial_subtract_from_add (ab)
  5. L14
    specialize prime_field_polynomial_subtract_from_add (ac)
  6. L15
    specialize prime_field_polynomial_subtract_from_add (zb)
  7. L16
    specialize prime_field_polynomial_subtract_from_add (zc)
  8. L17
    specialize prime_field_polynomial_subtract_from_add (l)
  9. L18
    apply prime_field_polynomial_subtract_from_add
  10. L19
    specialize prime_field_polynomial_add_zero_right (p)
03Use earlier factsL20–28

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

  1. L20
    specialize prime_field_polynomial_add_zero_right (ab)
  2. L21
    specialize prime_field_polynomial_add_zero_right (ac)
  3. L22
    specialize prime_field_polynomial_add_zero_right (zb)
  4. L23
    specialize prime_field_polynomial_add_zero_right (zc)
  5. L24
    specialize prime_field_polynomial_add_zero_right (l)
  6. L25
    apply prime_field_polynomial_add_zero_right
  7. L26
    exact hp
  8. L27
    exact ha
  9. L28
    exact hz

Library-wide reading audit

Original defined command ledger · 28 lines
  1. 0001intro p
  2. 0002intro ab
  3. 0003intro ac
  4. 0004intro zb
  5. 0005intro zc
  6. 0006intro l
  7. 0007intro hp
  8. 0008intro ha
  9. 0009intro hz
  10. 0010specialize prime_field_polynomial_subtract_from_add (p)
  11. 0011specialize prime_field_polynomial_subtract_from_add (ab)
  12. 0012specialize prime_field_polynomial_subtract_from_add (ac)
  13. 0013specialize prime_field_polynomial_subtract_from_add (ab)
  14. 0014specialize prime_field_polynomial_subtract_from_add (ac)
  15. 0015specialize prime_field_polynomial_subtract_from_add (zb)
  16. 0016specialize prime_field_polynomial_subtract_from_add (zc)
  17. 0017specialize prime_field_polynomial_subtract_from_add (l)
  18. 0018apply prime_field_polynomial_subtract_from_add
  19. 0019specialize prime_field_polynomial_add_zero_right (p)
  20. 0020specialize prime_field_polynomial_add_zero_right (ab)
  21. 0021specialize prime_field_polynomial_add_zero_right (ac)
  22. 0022specialize prime_field_polynomial_add_zero_right (zb)
  23. 0023specialize prime_field_polynomial_add_zero_right (zc)
  24. 0024specialize prime_field_polynomial_add_zero_right (l)
  25. 0025apply prime_field_polynomial_add_zero_right
  26. 0026exact hp
  27. 0027exact ha
  28. 0028exact hz