PQ0015

prime_field_polynomial_subtract_zero_right

Subtracting an actual zero prefix leaves the represented canonical coefficients unchanged.

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,zb,zc,ab,ac,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_commutative · checked external prerequisiteprime_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_zero_prime pfa_factor_right_sub_zero_prime. (p) = pfa_factor_left_sub_zero_prime * pfa_factor_right_sub_zero_prime -> pfa_factor_left_sub_zero_prime = 1 \/ pfa_factor_right_sub_zero_prime = 1) -> (forall fom_index_pfp_sub_zero_coeff. (exists fom_gap_pfp_sub_zero_coeff_index_bound. fom_gap_pfp_sub_zero_coeff_index_bound + S (fom_index_pfp_sub_zero_coeff) = l) -> exists fom_value_pfp_sub_zero_coeff. ((((exists fom_beta_height_pfp_sub_zero_coeff_entry. fom_beta_height_pfp_sub_zero_coeff_entry + S (fom_value_pfp_sub_zero_coeff) = S ((S (fom_index_pfp_sub_zero_coeff)) * ac)) /\ exists fom_beta_quotient_pfp_sub_zero_coeff_entry. ab = fom_beta_quotient_pfp_sub_zero_coeff_entry * S ((S (fom_index_pfp_sub_zero_coeff)) * ac) + (fom_value_pfp_sub_zero_coeff))) /\ (exists fom_gap_pfp_sub_zero_coeff_value_bound. fom_gap_pfp_sub_zero_coeff_value_bound + S (fom_value_pfp_sub_zero_coeff) = p))) -> (forall pfp_repeat_index_sub_zero_prefix. (exists pfa_gap_sub_zero_prefixindex. pfa_gap_sub_zero_prefixindex + S (pfp_repeat_index_sub_zero_prefix) = (l)) -> (((exists ff_h_pfp_sub_zero_prefixentry. ff_h_pfp_sub_zero_prefixentry + S (0) = S ((S (pfp_repeat_index_sub_zero_prefix)) * zc)) /\ exists ff_q_pfp_sub_zero_prefixentry. zb = ff_q_pfp_sub_zero_prefixentry * S ((S (pfp_repeat_index_sub_zero_prefix)) * zc) + (0)))) -> (forall pfs_index_sub_zero_result. (exists pfa_gap_sub_zero_resultindex. pfa_gap_sub_zero_resultindex + S (pfs_index_sub_zero_result) = (l)) -> exists pfs_left_sub_zero_result pfs_right_sub_zero_result pfs_result_sub_zero_result. ((((exists ff_h_pfp_sub_zero_resultleft. ff_h_pfp_sub_zero_resultleft + S (pfs_left_sub_zero_result) = S ((S (pfs_index_sub_zero_result)) * ac)) /\ exists ff_q_pfp_sub_zero_resultleft. ab = ff_q_pfp_sub_zero_resultleft * S ((S (pfs_index_sub_zero_result)) * ac) + (pfs_left_sub_zero_result))) /\ (((((exists ff_h_pfp_sub_zero_resultright. ff_h_pfp_sub_zero_resultright + S (pfs_right_sub_zero_result) = S ((S (pfs_index_sub_zero_result)) * zc)) /\ exists ff_q_pfp_sub_zero_resultright. zb = ff_q_pfp_sub_zero_resultright * S ((S (pfs_index_sub_zero_result)) * zc) + (pfs_right_sub_zero_result))) /\ (((((exists ff_h_pfp_sub_zero_resultresult. ff_h_pfp_sub_zero_resultresult + S (pfs_result_sub_zero_result) = S ((S (pfs_index_sub_zero_result)) * ac)) /\ exists ff_q_pfp_sub_zero_resultresult. ab = ff_q_pfp_sub_zero_resultresult * S ((S (pfs_index_sub_zero_result)) * ac) + (pfs_result_sub_zero_result))) /\ ((((exists pfa_gap_sub_zero_resultoperationleft. pfa_gap_sub_zero_resultoperationleft + S (pfs_right_sub_zero_result) = (p)) /\ (((exists pfa_gap_sub_zero_resultoperationright. pfa_gap_sub_zero_resultoperationright + S (pfs_result_sub_zero_result) = (p)) /\ ((((exists pfa_gap_sub_zero_resultoperationresultbound. pfa_gap_sub_zero_resultoperationresultbound + S (pfs_left_sub_zero_result) = (p)) /\ ((exists pfa_offset_left_sub_zero_resultoperationresultcongruence pfa_offset_right_sub_zero_resultoperationresultcongruence. ((pfs_right_sub_zero_result) + (pfs_result_sub_zero_result)) + (p) * pfa_offset_left_sub_zero_resultoperationresultcongruence = (pfs_left_sub_zero_result) + (p) * pfa_offset_right_sub_zero_resultoperationresultcongruence))))))))))))))))

Complete tactic proof in conservative notation

All 37 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

37 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.

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 (zb)
  5. L14
    specialize prime_field_polynomial_subtract_from_add (zc)
  6. L15
    specialize prime_field_polynomial_subtract_from_add (ab)
  7. L16
    specialize prime_field_polynomial_subtract_from_add (ac)
  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_commutative (p)
03Use earlier factsL20–29

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

  1. L20
    specialize prime_field_polynomial_add_commutative (ab)
  2. L21
    specialize prime_field_polynomial_add_commutative (ac)
  3. L22
    specialize prime_field_polynomial_add_commutative (zb)
  4. L23
    specialize prime_field_polynomial_add_commutative (zc)
  5. L24
    specialize prime_field_polynomial_add_commutative (ab)
  6. L25
    specialize prime_field_polynomial_add_commutative (ac)
  7. L26
    specialize prime_field_polynomial_add_commutative (l)
  8. L27
    apply prime_field_polynomial_add_commutative
  9. L28
    specialize prime_field_polynomial_add_zero_right (p)
  10. L29
    specialize prime_field_polynomial_add_zero_right (ab)
04Use earlier factsL30–37

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

  1. L30
    specialize prime_field_polynomial_add_zero_right (ac)
  2. L31
    specialize prime_field_polynomial_add_zero_right (zb)
  3. L32
    specialize prime_field_polynomial_add_zero_right (zc)
  4. L33
    specialize prime_field_polynomial_add_zero_right (l)
  5. L34
    apply prime_field_polynomial_add_zero_right
  6. L35
    exact hp
  7. L36
    exact ha
  8. L37
    exact hz

Library-wide reading audit

Original defined command ledger · 37 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 (zb)
  14. 0014specialize prime_field_polynomial_subtract_from_add (zc)
  15. 0015specialize prime_field_polynomial_subtract_from_add (ab)
  16. 0016specialize prime_field_polynomial_subtract_from_add (ac)
  17. 0017specialize prime_field_polynomial_subtract_from_add (l)
  18. 0018apply prime_field_polynomial_subtract_from_add
  19. 0019specialize prime_field_polynomial_add_commutative (p)
  20. 0020specialize prime_field_polynomial_add_commutative (ab)
  21. 0021specialize prime_field_polynomial_add_commutative (ac)
  22. 0022specialize prime_field_polynomial_add_commutative (zb)
  23. 0023specialize prime_field_polynomial_add_commutative (zc)
  24. 0024specialize prime_field_polynomial_add_commutative (ab)
  25. 0025specialize prime_field_polynomial_add_commutative (ac)
  26. 0026specialize prime_field_polynomial_add_commutative (l)
  27. 0027apply prime_field_polynomial_add_commutative
  28. 0028specialize prime_field_polynomial_add_zero_right (p)
  29. 0029specialize prime_field_polynomial_add_zero_right (ab)
  30. 0030specialize prime_field_polynomial_add_zero_right (ac)
  31. 0031specialize prime_field_polynomial_add_zero_right (zb)
  32. 0032specialize prime_field_polynomial_add_zero_right (zc)
  33. 0033specialize prime_field_polynomial_add_zero_right (l)
  34. 0034apply prime_field_polynomial_add_zero_right
  35. 0035exact hp
  36. 0036exact ha
  37. 0037exact hz