PP0017

prime_field_polynomial_scale_bounded

Both the input and the constructed output of scalar multiplication have bounded coefficients.

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. ∀ k. ∀ ab. ∀ ac. ∀ bb. ∀ bc. ∀ l. FpPolyScale(p,k,ab,ac,bb,bc,l) → BetaPrefixInto(ab,ac,l,p) ∧ BetaPrefixInto(bb,bc,l,p)

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

Definition DAG

FpMul(p,a,b,c) · 2BetaPrefixInto(b,c,l,B) · 2FpPolyScale(p,k,ab,ac,bb,bc,l) · 1BetaAt(b,c,i,x) · 4

Actual proof prerequisites

none
Original expanded first-order statement 
forall p k ab ac bb bc l. (((exists pfa_gap_scale_bounded_operationscalar. pfa_gap_scale_bounded_operationscalar + S (k) = (p)) /\ ((forall pfp_index_scale_bounded_operation. (exists pfa_gap_scale_bounded_operationindex. pfa_gap_scale_bounded_operationindex + S (pfp_index_scale_bounded_operation) = (l)) -> exists pfp_source_scale_bounded_operation pfp_value_scale_bounded_operation. ((((exists ff_h_pfp_scale_bounded_operationsource. ff_h_pfp_scale_bounded_operationsource + S (pfp_source_scale_bounded_operation) = S ((S (pfp_index_scale_bounded_operation)) * ac)) /\ exists ff_q_pfp_scale_bounded_operationsource. ab = ff_q_pfp_scale_bounded_operationsource * S ((S (pfp_index_scale_bounded_operation)) * ac) + (pfp_source_scale_bounded_operation))) /\ (((((exists ff_h_pfp_scale_bounded_operationtarget. ff_h_pfp_scale_bounded_operationtarget + S (pfp_value_scale_bounded_operation) = S ((S (pfp_index_scale_bounded_operation)) * bc)) /\ exists ff_q_pfp_scale_bounded_operationtarget. bb = ff_q_pfp_scale_bounded_operationtarget * S ((S (pfp_index_scale_bounded_operation)) * bc) + (pfp_value_scale_bounded_operation))) /\ ((((exists pfa_gap_scale_bounded_operationoperationleft. pfa_gap_scale_bounded_operationoperationleft + S (k) = (p)) /\ (((exists pfa_gap_scale_bounded_operationoperationright. pfa_gap_scale_bounded_operationoperationright + S (pfp_source_scale_bounded_operation) = (p)) /\ ((((exists pfa_gap_scale_bounded_operationoperationresultbound. pfa_gap_scale_bounded_operationoperationresultbound + S (pfp_value_scale_bounded_operation) = (p)) /\ ((exists pfa_offset_left_scale_bounded_operationoperationresultcongruence pfa_offset_right_scale_bounded_operationoperationresultcongruence. ((k) * (pfp_source_scale_bounded_operation)) + (p) * pfa_offset_left_scale_bounded_operationoperationresultcongruence = (pfp_value_scale_bounded_operation) + (p) * pfa_offset_right_scale_bounded_operationoperationresultcongruence))))))))))))))))) -> ((forall fom_index_pfp_scale_bounded_source. (exists fom_gap_pfp_scale_bounded_source_index_bound. fom_gap_pfp_scale_bounded_source_index_bound + S (fom_index_pfp_scale_bounded_source) = l) -> exists fom_value_pfp_scale_bounded_source. ((((exists fom_beta_height_pfp_scale_bounded_source_entry. fom_beta_height_pfp_scale_bounded_source_entry + S (fom_value_pfp_scale_bounded_source) = S ((S (fom_index_pfp_scale_bounded_source)) * ac)) /\ exists fom_beta_quotient_pfp_scale_bounded_source_entry. ab = fom_beta_quotient_pfp_scale_bounded_source_entry * S ((S (fom_index_pfp_scale_bounded_source)) * ac) + (fom_value_pfp_scale_bounded_source))) /\ (exists fom_gap_pfp_scale_bounded_source_value_bound. fom_gap_pfp_scale_bounded_source_value_bound + S (fom_value_pfp_scale_bounded_source) = p))) /\ ((forall fom_index_pfp_scale_bounded_target. (exists fom_gap_pfp_scale_bounded_target_index_bound. fom_gap_pfp_scale_bounded_target_index_bound + S (fom_index_pfp_scale_bounded_target) = l) -> exists fom_value_pfp_scale_bounded_target. ((((exists fom_beta_height_pfp_scale_bounded_target_entry. fom_beta_height_pfp_scale_bounded_target_entry + S (fom_value_pfp_scale_bounded_target) = S ((S (fom_index_pfp_scale_bounded_target)) * bc)) /\ exists fom_beta_quotient_pfp_scale_bounded_target_entry. bb = fom_beta_quotient_pfp_scale_bounded_target_entry * S ((S (fom_index_pfp_scale_bounded_target)) * bc) + (fom_value_pfp_scale_bounded_target))) /\ (exists fom_gap_pfp_scale_bounded_target_value_bound. fom_gap_pfp_scale_bounded_target_value_bound + S (fom_value_pfp_scale_bounded_target) = p)))))

Complete tactic proof in conservative notation

All 44 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

44 script commands · 14 reading checkpoints · 2 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 assumptionsL1–8

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

  1. L1
    intro p
  2. L2
    intro k
  3. L3
    intro ab
  4. L4
    intro ac
  5. L5
    intro bb
  6. L6
    intro bc
  7. L7
    intro l
  8. L8
    intro h
02Separate the logical casesL9–10

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

  1. L9
    cases h
  2. L10
    split
03Fix variables and assumptionsL11–12

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

  1. L11
    intro i
  2. L12
    intro hi
04Establish hvL13–16

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

  1. L13
    have hv : ∃ a. ∃ r. BetaAt(ab,ac,i,a) ∧ (BetaAt(bb,bc,i,r) ∧ FpMul(p,k,a,r))Definitions: BetaAt(ab,ac,i,a)BetaAt(bb,bc,i,r)FpMul(p,k,a,r)Original native command in the exact edition
  2. L14
    specialize h_right (i)
  3. L15
    apply h_right
  4. L16
    exact hi
05Separate the logical casesL17–23

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

  1. L17
    cases hv
  2. L18
    cases hv_witness
  3. L19
    cases hv_witness_witness
  4. L20
    cases hv_witness_witness_right
  5. L21
    cases hv_witness_witness_right_right
  6. L22
    cases hv_witness_witness_right_right_right
  7. L23
    cases hv_witness_witness_right_right_right_right
06Construct an explicit witnessL24–24

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

  1. L24
    exists x
07Separate the logical casesL25–25

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

  1. L25
    split
08Use earlier factsL26–27

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

  1. L26
    exact hv_witness_witness_left
  2. L27
    exact hv_witness_witness_right_right_right_left
09Fix variables and assumptionsL28–29

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

  1. L28
    intro i
  2. L29
    intro hi
10Establish hvL30–33

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

  1. L30
    have hv : ∃ a. ∃ r. BetaAt(ab,ac,i,a) ∧ (BetaAt(bb,bc,i,r) ∧ FpMul(p,k,a,r))Definitions: BetaAt(ab,ac,i,a)BetaAt(bb,bc,i,r)FpMul(p,k,a,r)Original native command in the exact edition
  2. L31
    specialize h_right (i)
  3. L32
    apply h_right
  4. L33
    exact hi
11Separate the logical casesL34–40

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

  1. L34
    cases hv
  2. L35
    cases hv_witness
  3. L36
    cases hv_witness_witness
  4. L37
    cases hv_witness_witness_right
  5. L38
    cases hv_witness_witness_right_right
  6. L39
    cases hv_witness_witness_right_right_right
  7. L40
    cases hv_witness_witness_right_right_right_right
12Construct an explicit witnessL41–41

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

  1. L41
    exists x1
13Separate the logical casesL42–42

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

  1. L42
    split
14Use earlier factsL43–44

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

  1. L43
    exact hv_witness_witness_right_left
  2. L44
    exact hv_witness_witness_right_right_right_right_left

Library-wide reading audit

Original defined command ledger · 44 lines
  1. 0001intro p
  2. 0002intro k
  3. 0003intro ab
  4. 0004intro ac
  5. 0005intro bb
  6. 0006intro bc
  7. 0007intro l
  8. 0008intro h
  9. 0009cases h
  10. 0010split
  11. 0011intro i
  12. 0012intro hi
  13. 0013have hv : ∃ a. ∃ r. BetaAt(ab,ac,i,a) ∧ (BetaAt(bb,bc,i,r) ∧ FpMul(p,k,a,r))
  14. 0014specialize h_right (i)
  15. 0015apply h_right
  16. 0016exact hi
  17. 0017cases hv
  18. 0018cases hv_witness
  19. 0019cases hv_witness_witness
  20. 0020cases hv_witness_witness_right
  21. 0021cases hv_witness_witness_right_right
  22. 0022cases hv_witness_witness_right_right_right
  23. 0023cases hv_witness_witness_right_right_right_right
  24. 0024exists x
  25. 0025split
  26. 0026exact hv_witness_witness_left
  27. 0027exact hv_witness_witness_right_right_right_left
  28. 0028intro i
  29. 0029intro hi
  30. 0030have hv : ∃ a. ∃ r. BetaAt(ab,ac,i,a) ∧ (BetaAt(bb,bc,i,r) ∧ FpMul(p,k,a,r))
  31. 0031specialize h_right (i)
  32. 0032apply h_right
  33. 0033exact hi
  34. 0034cases hv
  35. 0035cases hv_witness
  36. 0036cases hv_witness_witness
  37. 0037cases hv_witness_witness_right
  38. 0038cases hv_witness_witness_right_right
  39. 0039cases hv_witness_witness_right_right_right
  40. 0040cases hv_witness_witness_right_right_right_right
  41. 0041exists x1
  42. 0042split
  43. 0043exact hv_witness_witness_right_left
  44. 0044exact hv_witness_witness_right_right_right_right_left