PQ0011

prime_field_polynomial_subtract_transport

Alpha v34 independently verified · alpha_closed; checked-use authorized; not Stable

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

Current library: Alpha v34, 4,223 checked-use theorems; Stable remains 432. Historical first admissions, original proof editions, and non-admitted aliases are preserved.

Exact expanded first-order arithmetic statement

forall p ab ac bb bc rb rc AB AC BB BC RB RC l. (forall mdr_i_pfp_subtract_transport_ab mdr_a_pfp_subtract_transport_ab. (exists mdr_gap_pfp_subtract_transport_abb. mdr_gap_pfp_subtract_transport_abb + S (mdr_i_pfp_subtract_transport_ab) = (l)) -> (((exists ff_h_mdr_pfp_subtract_transport_abo. ff_h_mdr_pfp_subtract_transport_abo + S (mdr_a_pfp_subtract_transport_ab) = S ((S (mdr_i_pfp_subtract_transport_ab)) * ac)) /\ exists ff_q_mdr_pfp_subtract_transport_abo. ab = ff_q_mdr_pfp_subtract_transport_abo * S ((S (mdr_i_pfp_subtract_transport_ab)) * ac) + (mdr_a_pfp_subtract_transport_ab))) -> (((exists ff_h_mdr_pfp_subtract_transport_abn. ff_h_mdr_pfp_subtract_transport_abn + S (mdr_a_pfp_subtract_transport_ab) = S ((S (mdr_i_pfp_subtract_transport_ab)) * AC)) /\ exists ff_q_mdr_pfp_subtract_transport_abn. AB = ff_q_mdr_pfp_subtract_transport_abn * S ((S (mdr_i_pfp_subtract_transport_ab)) * AC) + (mdr_a_pfp_subtract_transport_ab)))) -> (forall mdr_i_pfp_subtract_transport_bb mdr_a_pfp_subtract_transport_bb. (exists mdr_gap_pfp_subtract_transport_bbb. mdr_gap_pfp_subtract_transport_bbb + S (mdr_i_pfp_subtract_transport_bb) = (l)) -> (((exists ff_h_mdr_pfp_subtract_transport_bbo. ff_h_mdr_pfp_subtract_transport_bbo + S (mdr_a_pfp_subtract_transport_bb) = S ((S (mdr_i_pfp_subtract_transport_bb)) * bc)) /\ exists ff_q_mdr_pfp_subtract_transport_bbo. bb = ff_q_mdr_pfp_subtract_transport_bbo * S ((S (mdr_i_pfp_subtract_transport_bb)) * bc) + (mdr_a_pfp_subtract_transport_bb))) -> (((exists ff_h_mdr_pfp_subtract_transport_bbn. ff_h_mdr_pfp_subtract_transport_bbn + S (mdr_a_pfp_subtract_transport_bb) = S ((S (mdr_i_pfp_subtract_transport_bb)) * BC)) /\ exists ff_q_mdr_pfp_subtract_transport_bbn. BB = ff_q_mdr_pfp_subtract_transport_bbn * S ((S (mdr_i_pfp_subtract_transport_bb)) * BC) + (mdr_a_pfp_subtract_transport_bb)))) -> (forall mdr_i_pfp_subtract_transport_rb mdr_a_pfp_subtract_transport_rb. (exists mdr_gap_pfp_subtract_transport_rbb. mdr_gap_pfp_subtract_transport_rbb + S (mdr_i_pfp_subtract_transport_rb) = (l)) -> (((exists ff_h_mdr_pfp_subtract_transport_rbo. ff_h_mdr_pfp_subtract_transport_rbo + S (mdr_a_pfp_subtract_transport_rb) = S ((S (mdr_i_pfp_subtract_transport_rb)) * rc)) /\ exists ff_q_mdr_pfp_subtract_transport_rbo. rb = ff_q_mdr_pfp_subtract_transport_rbo * S ((S (mdr_i_pfp_subtract_transport_rb)) * rc) + (mdr_a_pfp_subtract_transport_rb))) -> (((exists ff_h_mdr_pfp_subtract_transport_rbn. ff_h_mdr_pfp_subtract_transport_rbn + S (mdr_a_pfp_subtract_transport_rb) = S ((S (mdr_i_pfp_subtract_transport_rb)) * RC)) /\ exists ff_q_mdr_pfp_subtract_transport_rbn. RB = ff_q_mdr_pfp_subtract_transport_rbn * S ((S (mdr_i_pfp_subtract_transport_rb)) * RC) + (mdr_a_pfp_subtract_transport_rb)))) -> (forall pfs_index_subtract_transport_old. (exists pfa_gap_subtract_transport_oldindex. pfa_gap_subtract_transport_oldindex + S (pfs_index_subtract_transport_old) = (l)) -> exists pfs_left_subtract_transport_old pfs_right_subtract_transport_old pfs_result_subtract_transport_old. ((((exists ff_h_pfp_subtract_transport_oldleft. ff_h_pfp_subtract_transport_oldleft + S (pfs_left_subtract_transport_old) = S ((S (pfs_index_subtract_transport_old)) * ac)) /\ exists ff_q_pfp_subtract_transport_oldleft. ab = ff_q_pfp_subtract_transport_oldleft * S ((S (pfs_index_subtract_transport_old)) * ac) + (pfs_left_subtract_transport_old))) /\ (((((exists ff_h_pfp_subtract_transport_oldright. ff_h_pfp_subtract_transport_oldright + S (pfs_right_subtract_transport_old) = S ((S (pfs_index_subtract_transport_old)) * bc)) /\ exists ff_q_pfp_subtract_transport_oldright. bb = ff_q_pfp_subtract_transport_oldright * S ((S (pfs_index_subtract_transport_old)) * bc) + (pfs_right_subtract_transport_old))) /\ (((((exists ff_h_pfp_subtract_transport_oldresult. ff_h_pfp_subtract_transport_oldresult + S (pfs_result_subtract_transport_old) = S ((S (pfs_index_subtract_transport_old)) * rc)) /\ exists ff_q_pfp_subtract_transport_oldresult. rb = ff_q_pfp_subtract_transport_oldresult * S ((S (pfs_index_subtract_transport_old)) * rc) + (pfs_result_subtract_transport_old))) /\ ((((exists pfa_gap_subtract_transport_oldoperationleft. pfa_gap_subtract_transport_oldoperationleft + S (pfs_right_subtract_transport_old) = (p)) /\ (((exists pfa_gap_subtract_transport_oldoperationright. pfa_gap_subtract_transport_oldoperationright + S (pfs_result_subtract_transport_old) = (p)) /\ ((((exists pfa_gap_subtract_transport_oldoperationresultbound. pfa_gap_subtract_transport_oldoperationresultbound + S (pfs_left_subtract_transport_old) = (p)) /\ ((exists pfa_offset_left_subtract_transport_oldoperationresultcongruence pfa_offset_right_subtract_transport_oldoperationresultcongruence. ((pfs_right_subtract_transport_old) + (pfs_result_subtract_transport_old)) + (p) * pfa_offset_left_subtract_transport_oldoperationresultcongruence = (pfs_left_subtract_transport_old) + (p) * pfa_offset_right_subtract_transport_oldoperationresultcongruence)))))))))))))))) -> (forall pfs_index_subtract_transport_new. (exists pfa_gap_subtract_transport_newindex. pfa_gap_subtract_transport_newindex + S (pfs_index_subtract_transport_new) = (l)) -> exists pfs_left_subtract_transport_new pfs_right_subtract_transport_new pfs_result_subtract_transport_new. ((((exists ff_h_pfp_subtract_transport_newleft. ff_h_pfp_subtract_transport_newleft + S (pfs_left_subtract_transport_new) = S ((S (pfs_index_subtract_transport_new)) * AC)) /\ exists ff_q_pfp_subtract_transport_newleft. AB = ff_q_pfp_subtract_transport_newleft * S ((S (pfs_index_subtract_transport_new)) * AC) + (pfs_left_subtract_transport_new))) /\ (((((exists ff_h_pfp_subtract_transport_newright. ff_h_pfp_subtract_transport_newright + S (pfs_right_subtract_transport_new) = S ((S (pfs_index_subtract_transport_new)) * BC)) /\ exists ff_q_pfp_subtract_transport_newright. BB = ff_q_pfp_subtract_transport_newright * S ((S (pfs_index_subtract_transport_new)) * BC) + (pfs_right_subtract_transport_new))) /\ (((((exists ff_h_pfp_subtract_transport_newresult. ff_h_pfp_subtract_transport_newresult + S (pfs_result_subtract_transport_new) = S ((S (pfs_index_subtract_transport_new)) * RC)) /\ exists ff_q_pfp_subtract_transport_newresult. RB = ff_q_pfp_subtract_transport_newresult * S ((S (pfs_index_subtract_transport_new)) * RC) + (pfs_result_subtract_transport_new))) /\ ((((exists pfa_gap_subtract_transport_newoperationleft. pfa_gap_subtract_transport_newoperationleft + S (pfs_right_subtract_transport_new) = (p)) /\ (((exists pfa_gap_subtract_transport_newoperationright. pfa_gap_subtract_transport_newoperationright + S (pfs_result_subtract_transport_new) = (p)) /\ ((((exists pfa_gap_subtract_transport_newoperationresultbound. pfa_gap_subtract_transport_newoperationresultbound + S (pfs_left_subtract_transport_new) = (p)) /\ ((exists pfa_offset_left_subtract_transport_newoperationresultcongruence pfa_offset_right_subtract_transport_newoperationresultcongruence. ((pfs_right_subtract_transport_new) + (pfs_result_subtract_transport_new)) + (p) * pfa_offset_left_subtract_transport_newoperationresultcongruence = (pfs_left_subtract_transport_new) + (p) * pfa_offset_right_subtract_transport_newoperationresultcongruence))))))))))))))))

Constructive proof overview

Generated structural guide

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

The unchanged tactic script uses 0 declared prerequisites and contains 52 exact native proof lines.

Alpha v34 checked-use · first admitted v32 · independently kernel and Lean verified; not Stable

Proof neighborhood

Direct dependencies

none

Direct dependents

none

Formal native tactic body

Dependencies are introduced as named hypotheses before line 1. Local theorem links identify exact declared prerequisites. This exact body belongs to a complete independently kernel-checked constructive proof bundle and has Alpha checked-use authority; it does not imply Stable membership.

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.

Long local formulas use this family’s existing definitions. Each new abbreviation was expanded back to the identical native formula, including its free-variable context. The original edition is preserved below.

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 rb
  7. L7
    intro rc
  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 RB
  3. L13
    intro RC
  4. L14
    intro l
  5. L15
    intro h0
  6. L16
    intro h1
  7. L17
    intro h2
  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(rb,rc,i,r) ∧ FpAdd(p,b,r,a)))Definitions: FpAddBetaAt
  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 h0 (i)
  2. L36
    specialize h0 (x)
  3. L37
    apply h0
  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 h1 (i)
  2. L42
    specialize h1 (x1)
  3. L43
    apply h1
  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 h2 (i)
  2. L48
    specialize h2 (x2)
  3. L49
    apply h2
  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 exact command ledger · 52 lines
  1. 0001intro p
  2. 0002intro ab
  3. 0003intro ac
  4. 0004intro bb
  5. 0005intro bc
  6. 0006intro rb
  7. 0007intro rc
  8. 0008intro AB
  9. 0009intro AC
  10. 0010intro BB
  11. 0011intro BC
  12. 0012intro RB
  13. 0013intro RC
  14. 0014intro l
  15. 0015intro h0
  16. 0016intro h1
  17. 0017intro h2
  18. 0018intro h
  19. 0019intro i
  20. 0020intro hi
  21. 0021have hv : exists a b r. (((((exists ff_h_pfp_subtract_transport_chosen0. ff_h_pfp_subtract_transport_chosen0 + S (a) = S ((S (i)) * ac)) /\ exists ff_q_pfp_subtract_transport_chosen0. ab = ff_q_pfp_subtract_transport_chosen0 * S ((S (i)) * ac) + (a))) /\ (((((exists ff_h_pfp_subtract_transport_chosen1. ff_h_pfp_subtract_transport_chosen1 + S (b) = S ((S (i)) * bc)) /\ exists ff_q_pfp_subtract_transport_chosen1. bb = ff_q_pfp_subtract_transport_chosen1 * S ((S (i)) * bc) + (b))) /\ (((((exists ff_h_pfp_subtract_transport_chosen2. ff_h_pfp_subtract_transport_chosen2 + S (r) = S ((S (i)) * rc)) /\ exists ff_q_pfp_subtract_transport_chosen2. rb = ff_q_pfp_subtract_transport_chosen2 * S ((S (i)) * rc) + (r))) /\ ((((exists pfa_gap_subtract_transport_chosenoperationleft. pfa_gap_subtract_transport_chosenoperationleft + S (b) = (p)) /\ (((exists pfa_gap_subtract_transport_chosenoperationright. pfa_gap_subtract_transport_chosenoperationright + S (r) = (p)) /\ ((((exists pfa_gap_subtract_transport_chosenoperationresultbound. pfa_gap_subtract_transport_chosenoperationresultbound + S (a) = (p)) /\ ((exists pfa_offset_left_subtract_transport_chosenoperationresultcongruence pfa_offset_right_subtract_transport_chosenoperationresultcongruence. ((b) + (r)) + (p) * pfa_offset_left_subtract_transport_chosenoperationresultcongruence = (a) + (p) * pfa_offset_right_subtract_transport_chosenoperationresultcongruence))))))))))))))))
  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 h0 (i)
  36. 0036specialize h0 (x)
  37. 0037apply h0
  38. 0038exact hi
  39. 0039exact hv_witness_witness_witness_left
  40. 0040split
  41. 0041specialize h1 (i)
  42. 0042specialize h1 (x1)
  43. 0043apply h1
  44. 0044exact hi
  45. 0045exact hv_witness_witness_witness_right_left
  46. 0046split
  47. 0047specialize h2 (i)
  48. 0048specialize h2 (x2)
  49. 0049apply h2
  50. 0050exact hi
  51. 0051exact hv_witness_witness_witness_right_right_left
  52. 0052exact hv_witness_witness_witness_right_right_right