PQ0009

prime_field_polynomial_negate_involutive

Reversing a genuine coefficientwise additive inverse gives the original values, without identifying encodings.

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. ∀ rb. ∀ rc. ∀ l. FpCoefficientNegation(p,ab,ac,rb,rc,l) → FpCoefficientNegation(p,rb,rc,ab,ac,l)

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

Definition DAG

FpAdd(p,a,b,c) · 1FpCoefficientNegation(p,ab,ac,rb,rc,L) · 2BetaAt(b,c,i,x) · 2

Actual proof prerequisites

prime_field_add_commutative · checked external prerequisite
Original expanded first-order statement 
forall p ab ac rb rc l. (forall pfs_index_neg_inv_old. (exists pfa_gap_neg_inv_oldindex. pfa_gap_neg_inv_oldindex + S (pfs_index_neg_inv_old) = (l)) -> exists pfs_source_neg_inv_old pfs_result_neg_inv_old. ((((exists ff_h_pfp_neg_inv_oldsource. ff_h_pfp_neg_inv_oldsource + S (pfs_source_neg_inv_old) = S ((S (pfs_index_neg_inv_old)) * ac)) /\ exists ff_q_pfp_neg_inv_oldsource. ab = ff_q_pfp_neg_inv_oldsource * S ((S (pfs_index_neg_inv_old)) * ac) + (pfs_source_neg_inv_old))) /\ (((((exists ff_h_pfp_neg_inv_oldresult. ff_h_pfp_neg_inv_oldresult + S (pfs_result_neg_inv_old) = S ((S (pfs_index_neg_inv_old)) * rc)) /\ exists ff_q_pfp_neg_inv_oldresult. rb = ff_q_pfp_neg_inv_oldresult * S ((S (pfs_index_neg_inv_old)) * rc) + (pfs_result_neg_inv_old))) /\ ((((exists pfa_gap_neg_inv_oldoperationadditionleft. pfa_gap_neg_inv_oldoperationadditionleft + S (pfs_source_neg_inv_old) = (p)) /\ (((exists pfa_gap_neg_inv_oldoperationadditionright. pfa_gap_neg_inv_oldoperationadditionright + S (pfs_result_neg_inv_old) = (p)) /\ ((((exists pfa_gap_neg_inv_oldoperationadditionresultbound. pfa_gap_neg_inv_oldoperationadditionresultbound + S (0) = (p)) /\ ((exists pfa_offset_left_neg_inv_oldoperationadditionresultcongruence pfa_offset_right_neg_inv_oldoperationadditionresultcongruence. ((pfs_source_neg_inv_old) + (pfs_result_neg_inv_old)) + (p) * pfa_offset_left_neg_inv_oldoperationadditionresultcongruence = (0) + (p) * pfa_offset_right_neg_inv_oldoperationadditionresultcongruence)))))))))))))) -> (forall pfs_index_neg_inv_new. (exists pfa_gap_neg_inv_newindex. pfa_gap_neg_inv_newindex + S (pfs_index_neg_inv_new) = (l)) -> exists pfs_source_neg_inv_new pfs_result_neg_inv_new. ((((exists ff_h_pfp_neg_inv_newsource. ff_h_pfp_neg_inv_newsource + S (pfs_source_neg_inv_new) = S ((S (pfs_index_neg_inv_new)) * rc)) /\ exists ff_q_pfp_neg_inv_newsource. rb = ff_q_pfp_neg_inv_newsource * S ((S (pfs_index_neg_inv_new)) * rc) + (pfs_source_neg_inv_new))) /\ (((((exists ff_h_pfp_neg_inv_newresult. ff_h_pfp_neg_inv_newresult + S (pfs_result_neg_inv_new) = S ((S (pfs_index_neg_inv_new)) * ac)) /\ exists ff_q_pfp_neg_inv_newresult. ab = ff_q_pfp_neg_inv_newresult * S ((S (pfs_index_neg_inv_new)) * ac) + (pfs_result_neg_inv_new))) /\ ((((exists pfa_gap_neg_inv_newoperationadditionleft. pfa_gap_neg_inv_newoperationadditionleft + S (pfs_source_neg_inv_new) = (p)) /\ (((exists pfa_gap_neg_inv_newoperationadditionright. pfa_gap_neg_inv_newoperationadditionright + S (pfs_result_neg_inv_new) = (p)) /\ ((((exists pfa_gap_neg_inv_newoperationadditionresultbound. pfa_gap_neg_inv_newoperationadditionresultbound + S (0) = (p)) /\ ((exists pfa_offset_left_neg_inv_newoperationadditionresultcongruence pfa_offset_right_neg_inv_newoperationadditionresultcongruence. ((pfs_source_neg_inv_new) + (pfs_result_neg_inv_new)) + (p) * pfa_offset_left_neg_inv_newoperationadditionresultcongruence = (0) + (p) * pfa_offset_right_neg_inv_newoperationadditionresultcongruence))))))))))))))

Complete tactic proof in conservative notation

All 29 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

29 script commands · 8 reading checkpoints · 1 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 ab
  3. L3
    intro ac
  4. L4
    intro rb
  5. L5
    intro rc
  6. L6
    intro l
  7. L7
    intro h
  8. L8
    intro i
  9. L9
    intro hi
02Establish hvL10–13

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply h.

  1. L10
    have hv : ∃ a. ∃ r. BetaAt(ab,ac,i,a) ∧ (BetaAt(rb,rc,i,r) ∧ FpAdd(p,a,r,0))Definitions: BetaAt(ab,ac,i,a)BetaAt(rb,rc,i,r)FpAdd(p,a,r,0)Original native command in the exact edition
  2. L11
    specialize h (i)
  3. L12
    apply h
  4. L13
    exact hi
03Separate the logical casesL14–17

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

  1. L14
    cases hv
  2. L15
    cases hv_witness
  3. L16
    cases hv_witness_witness
  4. L17
    cases hv_witness_witness_right
04Construct an explicit witnessL18–19

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

  1. L18
    exists x1
  2. L19
    exists x
05Separate the logical casesL20–20

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

  1. L20
    split
06Use earlier factsL21–21

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

  1. L21
    exact hv_witness_witness_right_left
07Separate the logical casesL22–22

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

  1. L22
    split
08Use earlier factsL23–29

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

  1. L23
    exact hv_witness_witness_left
  2. L24
    specialize prime_field_add_commutative (p)
  3. L25
    specialize prime_field_add_commutative (x)
  4. L26
    specialize prime_field_add_commutative (x1)
  5. L27
    specialize prime_field_add_commutative (0)
  6. L28
    apply prime_field_add_commutative
  7. L29
    exact hv_witness_witness_right_right

Library-wide reading audit

Original defined command ledger · 29 lines
  1. 0001intro p
  2. 0002intro ab
  3. 0003intro ac
  4. 0004intro rb
  5. 0005intro rc
  6. 0006intro l
  7. 0007intro h
  8. 0008intro i
  9. 0009intro hi
  10. 0010have hv : ∃ a. ∃ r. BetaAt(ab,ac,i,a) ∧ (BetaAt(rb,rc,i,r) ∧ FpAdd(p,a,r,0))
  11. 0011specialize h (i)
  12. 0012apply h
  13. 0013exact hi
  14. 0014cases hv
  15. 0015cases hv_witness
  16. 0016cases hv_witness_witness
  17. 0017cases hv_witness_witness_right
  18. 0018exists x1
  19. 0019exists x
  20. 0020split
  21. 0021exact hv_witness_witness_right_left
  22. 0022split
  23. 0023exact hv_witness_witness_left
  24. 0024specialize prime_field_add_commutative (p)
  25. 0025specialize prime_field_add_commutative (x)
  26. 0026specialize prime_field_add_commutative (x1)
  27. 0027specialize prime_field_add_commutative (0)
  28. 0028apply prime_field_add_commutative
  29. 0029exact hv_witness_witness_right_right