PP0011

prime_field_polynomial_add_transport

Independent beta recoding of both inputs and the output preserves actual coefficient addition.

Alpha v34 checked-use · first admitted v31 · 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.

Length is representation length, not polynomial degree. Leading zeros and the empty zero polynomial are allowed; the canonical argument guard x<p also applies to the empty case. Evaluation is defined by actual field-operation steps, not an assumed residue invariant. Polynomial division, gcd, irreducibles and general prime-power extension fields remain open; this does not close G091.

Exact theorem in conservative defined notation

∀ p. ∀ ab. ∀ ac. ∀ bb. ∀ bc. ∀ cb. ∀ cc. ∀ AB. ∀ AC. ∀ BB. ∀ BC. ∀ CB. ∀ CC. ∀ l. BetaPrefixEqual(ab,ac,AB,AC,l) → BetaPrefixEqual(bb,bc,BB,BC,l) → BetaPrefixEqual(cb,cc,CB,CC,l) → FpPolyAdd(p,ab,ac,bb,bc,cb,cc,l) → FpPolyAdd(p,AB,AC,BB,BC,CB,CC,l)

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

Definition DAG

FpAdd(p,a,b,c) · 1BetaPrefixEqual(b,c,d,e,l) · 3FpPolyAdd(p,ab,ac,bb,bc,cb,cc,l) · 2BetaAt(b,c,i,x) · 3

Actual proof prerequisites

none
Original expanded first-order statement 
forall p ab ac bb bc cb cc AB AC BB BC CB CC l. (forall mdr_i_pfp_add_transport_a mdr_a_pfp_add_transport_a. (exists mdr_gap_pfp_add_transport_ab. mdr_gap_pfp_add_transport_ab + S (mdr_i_pfp_add_transport_a) = (l)) -> (((exists ff_h_mdr_pfp_add_transport_ao. ff_h_mdr_pfp_add_transport_ao + S (mdr_a_pfp_add_transport_a) = S ((S (mdr_i_pfp_add_transport_a)) * ac)) /\ exists ff_q_mdr_pfp_add_transport_ao. ab = ff_q_mdr_pfp_add_transport_ao * S ((S (mdr_i_pfp_add_transport_a)) * ac) + (mdr_a_pfp_add_transport_a))) -> (((exists ff_h_mdr_pfp_add_transport_an. ff_h_mdr_pfp_add_transport_an + S (mdr_a_pfp_add_transport_a) = S ((S (mdr_i_pfp_add_transport_a)) * AC)) /\ exists ff_q_mdr_pfp_add_transport_an. AB = ff_q_mdr_pfp_add_transport_an * S ((S (mdr_i_pfp_add_transport_a)) * AC) + (mdr_a_pfp_add_transport_a)))) -> (forall mdr_i_pfp_add_transport_b mdr_a_pfp_add_transport_b. (exists mdr_gap_pfp_add_transport_bb. mdr_gap_pfp_add_transport_bb + S (mdr_i_pfp_add_transport_b) = (l)) -> (((exists ff_h_mdr_pfp_add_transport_bo. ff_h_mdr_pfp_add_transport_bo + S (mdr_a_pfp_add_transport_b) = S ((S (mdr_i_pfp_add_transport_b)) * bc)) /\ exists ff_q_mdr_pfp_add_transport_bo. bb = ff_q_mdr_pfp_add_transport_bo * S ((S (mdr_i_pfp_add_transport_b)) * bc) + (mdr_a_pfp_add_transport_b))) -> (((exists ff_h_mdr_pfp_add_transport_bn. ff_h_mdr_pfp_add_transport_bn + S (mdr_a_pfp_add_transport_b) = S ((S (mdr_i_pfp_add_transport_b)) * BC)) /\ exists ff_q_mdr_pfp_add_transport_bn. BB = ff_q_mdr_pfp_add_transport_bn * S ((S (mdr_i_pfp_add_transport_b)) * BC) + (mdr_a_pfp_add_transport_b)))) -> (forall mdr_i_pfp_add_transport_c mdr_a_pfp_add_transport_c. (exists mdr_gap_pfp_add_transport_cb. mdr_gap_pfp_add_transport_cb + S (mdr_i_pfp_add_transport_c) = (l)) -> (((exists ff_h_mdr_pfp_add_transport_co. ff_h_mdr_pfp_add_transport_co + S (mdr_a_pfp_add_transport_c) = S ((S (mdr_i_pfp_add_transport_c)) * cc)) /\ exists ff_q_mdr_pfp_add_transport_co. cb = ff_q_mdr_pfp_add_transport_co * S ((S (mdr_i_pfp_add_transport_c)) * cc) + (mdr_a_pfp_add_transport_c))) -> (((exists ff_h_mdr_pfp_add_transport_cn. ff_h_mdr_pfp_add_transport_cn + S (mdr_a_pfp_add_transport_c) = S ((S (mdr_i_pfp_add_transport_c)) * CC)) /\ exists ff_q_mdr_pfp_add_transport_cn. CB = ff_q_mdr_pfp_add_transport_cn * S ((S (mdr_i_pfp_add_transport_c)) * CC) + (mdr_a_pfp_add_transport_c)))) -> (forall pfp_index_add_transport_old. (exists pfa_gap_add_transport_oldindex. pfa_gap_add_transport_oldindex + S (pfp_index_add_transport_old) = (l)) -> exists pfp_left_add_transport_old pfp_right_add_transport_old pfp_value_add_transport_old. ((((exists ff_h_pfp_add_transport_oldleft. ff_h_pfp_add_transport_oldleft + S (pfp_left_add_transport_old) = S ((S (pfp_index_add_transport_old)) * ac)) /\ exists ff_q_pfp_add_transport_oldleft. ab = ff_q_pfp_add_transport_oldleft * S ((S (pfp_index_add_transport_old)) * ac) + (pfp_left_add_transport_old))) /\ (((((exists ff_h_pfp_add_transport_oldright. ff_h_pfp_add_transport_oldright + S (pfp_right_add_transport_old) = S ((S (pfp_index_add_transport_old)) * bc)) /\ exists ff_q_pfp_add_transport_oldright. bb = ff_q_pfp_add_transport_oldright * S ((S (pfp_index_add_transport_old)) * bc) + (pfp_right_add_transport_old))) /\ (((((exists ff_h_pfp_add_transport_oldtarget. ff_h_pfp_add_transport_oldtarget + S (pfp_value_add_transport_old) = S ((S (pfp_index_add_transport_old)) * cc)) /\ exists ff_q_pfp_add_transport_oldtarget. cb = ff_q_pfp_add_transport_oldtarget * S ((S (pfp_index_add_transport_old)) * cc) + (pfp_value_add_transport_old))) /\ ((((exists pfa_gap_add_transport_oldoperationleft. pfa_gap_add_transport_oldoperationleft + S (pfp_left_add_transport_old) = (p)) /\ (((exists pfa_gap_add_transport_oldoperationright. pfa_gap_add_transport_oldoperationright + S (pfp_right_add_transport_old) = (p)) /\ ((((exists pfa_gap_add_transport_oldoperationresultbound. pfa_gap_add_transport_oldoperationresultbound + S (pfp_value_add_transport_old) = (p)) /\ ((exists pfa_offset_left_add_transport_oldoperationresultcongruence pfa_offset_right_add_transport_oldoperationresultcongruence. ((pfp_left_add_transport_old) + (pfp_right_add_transport_old)) + (p) * pfa_offset_left_add_transport_oldoperationresultcongruence = (pfp_value_add_transport_old) + (p) * pfa_offset_right_add_transport_oldoperationresultcongruence)))))))))))))))) -> (forall pfp_index_add_transport_new. (exists pfa_gap_add_transport_newindex. pfa_gap_add_transport_newindex + S (pfp_index_add_transport_new) = (l)) -> exists pfp_left_add_transport_new pfp_right_add_transport_new pfp_value_add_transport_new. ((((exists ff_h_pfp_add_transport_newleft. ff_h_pfp_add_transport_newleft + S (pfp_left_add_transport_new) = S ((S (pfp_index_add_transport_new)) * AC)) /\ exists ff_q_pfp_add_transport_newleft. AB = ff_q_pfp_add_transport_newleft * S ((S (pfp_index_add_transport_new)) * AC) + (pfp_left_add_transport_new))) /\ (((((exists ff_h_pfp_add_transport_newright. ff_h_pfp_add_transport_newright + S (pfp_right_add_transport_new) = S ((S (pfp_index_add_transport_new)) * BC)) /\ exists ff_q_pfp_add_transport_newright. BB = ff_q_pfp_add_transport_newright * S ((S (pfp_index_add_transport_new)) * BC) + (pfp_right_add_transport_new))) /\ (((((exists ff_h_pfp_add_transport_newtarget. ff_h_pfp_add_transport_newtarget + S (pfp_value_add_transport_new) = S ((S (pfp_index_add_transport_new)) * CC)) /\ exists ff_q_pfp_add_transport_newtarget. CB = ff_q_pfp_add_transport_newtarget * S ((S (pfp_index_add_transport_new)) * CC) + (pfp_value_add_transport_new))) /\ ((((exists pfa_gap_add_transport_newoperationleft. pfa_gap_add_transport_newoperationleft + S (pfp_left_add_transport_new) = (p)) /\ (((exists pfa_gap_add_transport_newoperationright. pfa_gap_add_transport_newoperationright + S (pfp_right_add_transport_new) = (p)) /\ ((((exists pfa_gap_add_transport_newoperationresultbound. pfa_gap_add_transport_newoperationresultbound + S (pfp_value_add_transport_new) = (p)) /\ ((exists pfa_offset_left_add_transport_newoperationresultcongruence pfa_offset_right_add_transport_newoperationresultcongruence. ((pfp_left_add_transport_new) + (pfp_right_add_transport_new)) + (p) * pfa_offset_left_add_transport_newoperationresultcongruence = (pfp_value_add_transport_new) + (p) * pfa_offset_right_add_transport_newoperationresultcongruence))))))))))))))))

Complete tactic proof in conservative notation

All 52 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

52 script commands · 11 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 bb
  5. L5
    intro bc
  6. L6
    intro cb
  7. L7
    intro cc
  8. L8
    intro AB
  9. L9
    intro AC
  10. L10
    intro BB
02Fix variables and assumptionsL11–20

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

  1. L11
    intro BC
  2. L12
    intro CB
  3. L13
    intro CC
  4. L14
    intro l
  5. L15
    intro ha
  6. L16
    intro hb
  7. L17
    intro hc
  8. L18
    intro h
  9. L19
    intro i
  10. L20
    intro hi
03Establish hvL21–24

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

  1. L21
    have hv : ∃ a. ∃ b. ∃ r. BetaAt(ab,ac,i,a) ∧ (BetaAt(bb,bc,i,b) ∧ (BetaAt(cb,cc,i,r) ∧ FpAdd(p,a,b,r)))Definitions: BetaAt(ab,ac,i,a)BetaAt(bb,bc,i,b)BetaAt(cb,cc,i,r)FpAdd(p,a,b,r)Original native command in the exact edition
  2. L22
    specialize h (i)
  3. L23
    apply h
  4. L24
    exact hi
04Separate the logical casesL25–30

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

  1. L25
    cases hv
  2. L26
    cases hv_witness
  3. L27
    cases hv_witness_witness
  4. L28
    cases hv_witness_witness_witness
  5. L29
    cases hv_witness_witness_witness_right
  6. L30
    cases hv_witness_witness_witness_right_right
05Construct an explicit witnessL31–33

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

  1. L31
    exists x
  2. L32
    exists x1
  3. L33
    exists x2
06Separate the logical casesL34–34

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

  1. L34
    split
07Use earlier factsL35–39

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

  1. L35
    specialize ha (i)
  2. L36
    specialize ha (x)
  3. L37
    apply ha
  4. L38
    exact hi
  5. L39
    exact hv_witness_witness_witness_left
08Separate the logical casesL40–40

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

  1. L40
    split
09Use earlier factsL41–45

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

  1. L41
    specialize hb (i)
  2. L42
    specialize hb (x1)
  3. L43
    apply hb
  4. L44
    exact hi
  5. L45
    exact hv_witness_witness_witness_right_left
10Separate the logical casesL46–46

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

  1. L46
    split
11Use earlier factsL47–52

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

  1. L47
    specialize hc (i)
  2. L48
    specialize hc (x2)
  3. L49
    apply hc
  4. L50
    exact hi
  5. L51
    exact hv_witness_witness_witness_right_right_left
  6. L52
    exact hv_witness_witness_witness_right_right_right

Library-wide reading audit

Original defined command ledger · 52 lines
  1. 0001intro p
  2. 0002intro ab
  3. 0003intro ac
  4. 0004intro bb
  5. 0005intro bc
  6. 0006intro cb
  7. 0007intro cc
  8. 0008intro AB
  9. 0009intro AC
  10. 0010intro BB
  11. 0011intro BC
  12. 0012intro CB
  13. 0013intro CC
  14. 0014intro l
  15. 0015intro ha
  16. 0016intro hb
  17. 0017intro hc
  18. 0018intro h
  19. 0019intro i
  20. 0020intro hi
  21. 0021have hv : ∃ a. ∃ b. ∃ r. BetaAt(ab,ac,i,a) ∧ (BetaAt(bb,bc,i,b) ∧ (BetaAt(cb,cc,i,r) ∧ FpAdd(p,a,b,r)))
  22. 0022specialize h (i)
  23. 0023apply h
  24. 0024exact hi
  25. 0025cases hv
  26. 0026cases hv_witness
  27. 0027cases hv_witness_witness
  28. 0028cases hv_witness_witness_witness
  29. 0029cases hv_witness_witness_witness_right
  30. 0030cases hv_witness_witness_witness_right_right
  31. 0031exists x
  32. 0032exists x1
  33. 0033exists x2
  34. 0034split
  35. 0035specialize ha (i)
  36. 0036specialize ha (x)
  37. 0037apply ha
  38. 0038exact hi
  39. 0039exact hv_witness_witness_witness_left
  40. 0040split
  41. 0041specialize hb (i)
  42. 0042specialize hb (x1)
  43. 0043apply hb
  44. 0044exact hi
  45. 0045exact hv_witness_witness_witness_right_left
  46. 0046split
  47. 0047specialize hc (i)
  48. 0048specialize hc (x2)
  49. 0049apply hc
  50. 0050exact hi
  51. 0051exact hv_witness_witness_witness_right_right_left
  52. 0052exact hv_witness_witness_witness_right_right_right