PQ0008

prime_field_polynomial_negate_transport

Independent beta recodings of every input and output preserve the actual aligned coefficient operation.

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. ∀ AB. ∀ AC. ∀ RB. ∀ RC. ∀ l. (∀ x. ∀ y. Lt(x,l) → BetaAt(ab,ac,x,y) → BetaAt(AB,AC,x,y)) → (∀ x. ∀ y. Lt(x,l) → BetaAt(rb,rc,x,y) → BetaAt(RB,RC,x,y)) → FpCoefficientNegation(p,ab,ac,rb,rc,l) → FpCoefficientNegation(p,AB,AC,RB,RC,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) · 2Lt(a,b) · 2BetaAt(b,c,i,x) · 6

Actual proof prerequisites

none
Original expanded first-order statement 
forall p ab ac rb rc AB AC RB RC l. (forall mdr_i_pfp_negate_transport_ab mdr_a_pfp_negate_transport_ab. (exists mdr_gap_pfp_negate_transport_abb. mdr_gap_pfp_negate_transport_abb + S (mdr_i_pfp_negate_transport_ab) = (l)) -> (((exists ff_h_mdr_pfp_negate_transport_abo. ff_h_mdr_pfp_negate_transport_abo + S (mdr_a_pfp_negate_transport_ab) = S ((S (mdr_i_pfp_negate_transport_ab)) * ac)) /\ exists ff_q_mdr_pfp_negate_transport_abo. ab = ff_q_mdr_pfp_negate_transport_abo * S ((S (mdr_i_pfp_negate_transport_ab)) * ac) + (mdr_a_pfp_negate_transport_ab))) -> (((exists ff_h_mdr_pfp_negate_transport_abn. ff_h_mdr_pfp_negate_transport_abn + S (mdr_a_pfp_negate_transport_ab) = S ((S (mdr_i_pfp_negate_transport_ab)) * AC)) /\ exists ff_q_mdr_pfp_negate_transport_abn. AB = ff_q_mdr_pfp_negate_transport_abn * S ((S (mdr_i_pfp_negate_transport_ab)) * AC) + (mdr_a_pfp_negate_transport_ab)))) -> (forall mdr_i_pfp_negate_transport_rb mdr_a_pfp_negate_transport_rb. (exists mdr_gap_pfp_negate_transport_rbb. mdr_gap_pfp_negate_transport_rbb + S (mdr_i_pfp_negate_transport_rb) = (l)) -> (((exists ff_h_mdr_pfp_negate_transport_rbo. ff_h_mdr_pfp_negate_transport_rbo + S (mdr_a_pfp_negate_transport_rb) = S ((S (mdr_i_pfp_negate_transport_rb)) * rc)) /\ exists ff_q_mdr_pfp_negate_transport_rbo. rb = ff_q_mdr_pfp_negate_transport_rbo * S ((S (mdr_i_pfp_negate_transport_rb)) * rc) + (mdr_a_pfp_negate_transport_rb))) -> (((exists ff_h_mdr_pfp_negate_transport_rbn. ff_h_mdr_pfp_negate_transport_rbn + S (mdr_a_pfp_negate_transport_rb) = S ((S (mdr_i_pfp_negate_transport_rb)) * RC)) /\ exists ff_q_mdr_pfp_negate_transport_rbn. RB = ff_q_mdr_pfp_negate_transport_rbn * S ((S (mdr_i_pfp_negate_transport_rb)) * RC) + (mdr_a_pfp_negate_transport_rb)))) -> (forall pfs_index_negate_transport_old. (exists pfa_gap_negate_transport_oldindex. pfa_gap_negate_transport_oldindex + S (pfs_index_negate_transport_old) = (l)) -> exists pfs_source_negate_transport_old pfs_result_negate_transport_old. ((((exists ff_h_pfp_negate_transport_oldsource. ff_h_pfp_negate_transport_oldsource + S (pfs_source_negate_transport_old) = S ((S (pfs_index_negate_transport_old)) * ac)) /\ exists ff_q_pfp_negate_transport_oldsource. ab = ff_q_pfp_negate_transport_oldsource * S ((S (pfs_index_negate_transport_old)) * ac) + (pfs_source_negate_transport_old))) /\ (((((exists ff_h_pfp_negate_transport_oldresult. ff_h_pfp_negate_transport_oldresult + S (pfs_result_negate_transport_old) = S ((S (pfs_index_negate_transport_old)) * rc)) /\ exists ff_q_pfp_negate_transport_oldresult. rb = ff_q_pfp_negate_transport_oldresult * S ((S (pfs_index_negate_transport_old)) * rc) + (pfs_result_negate_transport_old))) /\ ((((exists pfa_gap_negate_transport_oldoperationadditionleft. pfa_gap_negate_transport_oldoperationadditionleft + S (pfs_source_negate_transport_old) = (p)) /\ (((exists pfa_gap_negate_transport_oldoperationadditionright. pfa_gap_negate_transport_oldoperationadditionright + S (pfs_result_negate_transport_old) = (p)) /\ ((((exists pfa_gap_negate_transport_oldoperationadditionresultbound. pfa_gap_negate_transport_oldoperationadditionresultbound + S (0) = (p)) /\ ((exists pfa_offset_left_negate_transport_oldoperationadditionresultcongruence pfa_offset_right_negate_transport_oldoperationadditionresultcongruence. ((pfs_source_negate_transport_old) + (pfs_result_negate_transport_old)) + (p) * pfa_offset_left_negate_transport_oldoperationadditionresultcongruence = (0) + (p) * pfa_offset_right_negate_transport_oldoperationadditionresultcongruence)))))))))))))) -> (forall pfs_index_negate_transport_new. (exists pfa_gap_negate_transport_newindex. pfa_gap_negate_transport_newindex + S (pfs_index_negate_transport_new) = (l)) -> exists pfs_source_negate_transport_new pfs_result_negate_transport_new. ((((exists ff_h_pfp_negate_transport_newsource. ff_h_pfp_negate_transport_newsource + S (pfs_source_negate_transport_new) = S ((S (pfs_index_negate_transport_new)) * AC)) /\ exists ff_q_pfp_negate_transport_newsource. AB = ff_q_pfp_negate_transport_newsource * S ((S (pfs_index_negate_transport_new)) * AC) + (pfs_source_negate_transport_new))) /\ (((((exists ff_h_pfp_negate_transport_newresult. ff_h_pfp_negate_transport_newresult + S (pfs_result_negate_transport_new) = S ((S (pfs_index_negate_transport_new)) * RC)) /\ exists ff_q_pfp_negate_transport_newresult. RB = ff_q_pfp_negate_transport_newresult * S ((S (pfs_index_negate_transport_new)) * RC) + (pfs_result_negate_transport_new))) /\ ((((exists pfa_gap_negate_transport_newoperationadditionleft. pfa_gap_negate_transport_newoperationadditionleft + S (pfs_source_negate_transport_new) = (p)) /\ (((exists pfa_gap_negate_transport_newoperationadditionright. pfa_gap_negate_transport_newoperationadditionright + S (pfs_result_negate_transport_new) = (p)) /\ ((((exists pfa_gap_negate_transport_newoperationadditionresultbound. pfa_gap_negate_transport_newoperationadditionresultbound + S (0) = (p)) /\ ((exists pfa_offset_left_negate_transport_newoperationadditionresultcongruence pfa_offset_right_negate_transport_newoperationadditionresultcongruence. ((pfs_source_negate_transport_new) + (pfs_result_negate_transport_new)) + (p) * pfa_offset_left_negate_transport_newoperationadditionresultcongruence = (0) + (p) * pfa_offset_right_negate_transport_newoperationadditionresultcongruence))))))))))))))

Complete tactic proof in conservative notation

All 38 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

38 script commands · 9 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 assumptionsL1–10

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 AB
  7. L7
    intro AC
  8. L8
    intro RB
  9. L9
    intro RC
  10. L10
    intro l
02Fix variables and assumptionsL11–15

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

  1. L11
    intro h0
  2. L12
    intro h1
  3. L13
    intro h
  4. L14
    intro i
  5. L15
    intro hi
03Establish hvL16–19

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

  1. L16
    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. L17
    specialize h (i)
  3. L18
    apply h
  4. L19
    exact hi
04Separate the logical casesL20–23

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

  1. L20
    cases hv
  2. L21
    cases hv_witness
  3. L22
    cases hv_witness_witness
  4. L23
    cases hv_witness_witness_right
05Construct an explicit witnessL24–25

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

  1. L24
    exists x
  2. L25
    exists x1
06Separate the logical casesL26–26

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

  1. L26
    split
07Use earlier factsL27–31

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

  1. L27
    specialize h0 (i)
  2. L28
    specialize h0 (x)
  3. L29
    apply h0
  4. L30
    exact hi
  5. L31
    exact hv_witness_witness_left
08Separate the logical casesL32–32

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

  1. L32
    split
09Use earlier factsL33–38

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

  1. L33
    specialize h1 (i)
  2. L34
    specialize h1 (x1)
  3. L35
    apply h1
  4. L36
    exact hi
  5. L37
    exact hv_witness_witness_right_left
  6. L38
    exact hv_witness_witness_right_right

Library-wide reading audit

Original defined command ledger · 38 lines
  1. 0001intro p
  2. 0002intro ab
  3. 0003intro ac
  4. 0004intro rb
  5. 0005intro rc
  6. 0006intro AB
  7. 0007intro AC
  8. 0008intro RB
  9. 0009intro RC
  10. 0010intro l
  11. 0011intro h0
  12. 0012intro h1
  13. 0013intro h
  14. 0014intro i
  15. 0015intro hi
  16. 0016have hv : ∃ a. ∃ r. BetaAt(ab,ac,i,a) ∧ (BetaAt(rb,rc,i,r) ∧ FpAdd(p,a,r,0))
  17. 0017specialize h (i)
  18. 0018apply h
  19. 0019exact hi
  20. 0020cases hv
  21. 0021cases hv_witness
  22. 0022cases hv_witness_witness
  23. 0023cases hv_witness_witness_right
  24. 0024exists x
  25. 0025exists x1
  26. 0026split
  27. 0027specialize h0 (i)
  28. 0028specialize h0 (x)
  29. 0029apply h0
  30. 0030exact hi
  31. 0031exact hv_witness_witness_left
  32. 0032split
  33. 0033specialize h1 (i)
  34. 0034specialize h1 (x1)
  35. 0035apply h1
  36. 0036exact hi
  37. 0037exact hv_witness_witness_right_left
  38. 0038exact hv_witness_witness_right_right