PG000F

prime_field_polynomial_shift_power_successor

Each actual coefficient at power k becomes the same coefficient at power S k; together with constant zero this is genuine multiplication by X, not evaluation equality.

Alpha v34 checked-use · first admitted v34 · 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 products and sums use actual beta-coded coefficients; equality is formal coefficient equivalence, not equality of evaluations or raw codes. The zero gcd is included. Uniqueness is up to formal equivalence, not unique Bézout coefficients. This proves the polynomial gcd component, not G091 irreducible-polynomial existence or arbitrary prime-power-field construction.

Exact theorem in conservative defined notation

∀ b. ∀ c. ∀ L. ∀ d. ∀ e. ∀ k. ∀ a. PolynomialShift(b,c,L,d,e) → PolynomialPowerCoefficient(b,c,L,k,a) → PolynomialPowerCoefficient(d,e,S L,S k,a)

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

Definition DAG

PolynomialPowerCoefficient(b,c,L,k,a) · 2PolynomialShift(b,c,L,d,e) · 1

Actual proof prerequisites

prime_field_polynomial_power_index_bound · checked external prerequisitesucc_le_succ · checked external prerequisite
Original expanded first-order statement 
forall b c L d e k a. (((forall mdr_i_pfp_shift_power_successor_relationprefix mdr_a_pfp_shift_power_successor_relationprefix. (exists mdr_gap_pfp_shift_power_successor_relationprefixb. mdr_gap_pfp_shift_power_successor_relationprefixb + S (mdr_i_pfp_shift_power_successor_relationprefix) = (L)) -> (((exists ff_h_mdr_pfp_shift_power_successor_relationprefixo. ff_h_mdr_pfp_shift_power_successor_relationprefixo + S (mdr_a_pfp_shift_power_successor_relationprefix) = S ((S (mdr_i_pfp_shift_power_successor_relationprefix)) * c)) /\ exists ff_q_mdr_pfp_shift_power_successor_relationprefixo. b = ff_q_mdr_pfp_shift_power_successor_relationprefixo * S ((S (mdr_i_pfp_shift_power_successor_relationprefix)) * c) + (mdr_a_pfp_shift_power_successor_relationprefix))) -> (((exists ff_h_mdr_pfp_shift_power_successor_relationprefixn. ff_h_mdr_pfp_shift_power_successor_relationprefixn + S (mdr_a_pfp_shift_power_successor_relationprefix) = S ((S (mdr_i_pfp_shift_power_successor_relationprefix)) * e)) /\ exists ff_q_mdr_pfp_shift_power_successor_relationprefixn. d = ff_q_mdr_pfp_shift_power_successor_relationprefixn * S ((S (mdr_i_pfp_shift_power_successor_relationprefix)) * e) + (mdr_a_pfp_shift_power_successor_relationprefix)))) /\ ((((exists ff_h_pfp_shift_power_successor_relationlast. ff_h_pfp_shift_power_successor_relationlast + S (0) = S ((S (L)) * e)) /\ exists ff_q_pfp_shift_power_successor_relationlast. d = ff_q_pfp_shift_power_successor_relationlast * S ((S (L)) * e) + (0)))))) -> ((exists pfrep_position_shift_power_successor_source. ((pfrep_position_shift_power_successor_source+S (k)=(L)) /\ ((((exists ff_h_pfp_shift_power_successor_sourceentry. ff_h_pfp_shift_power_successor_sourceentry + S (a) = S ((S (pfrep_position_shift_power_successor_source)) * c)) /\ exists ff_q_pfp_shift_power_successor_sourceentry. b = ff_q_pfp_shift_power_successor_sourceentry * S ((S (pfrep_position_shift_power_successor_source)) * c) + (a)))))) \/ (((exists pfrep_gap_shift_power_successor_sourceoutside. pfrep_gap_shift_power_successor_sourceoutside+(L)=(k)) /\ (((a)=0))))) -> ((exists pfrep_position_shift_power_successor_result. ((pfrep_position_shift_power_successor_result+S (S k)=(S L)) /\ ((((exists ff_h_pfp_shift_power_successor_resultentry. ff_h_pfp_shift_power_successor_resultentry + S (a) = S ((S (pfrep_position_shift_power_successor_result)) * e)) /\ exists ff_q_pfp_shift_power_successor_resultentry. d = ff_q_pfp_shift_power_successor_resultentry * S ((S (pfrep_position_shift_power_successor_result)) * e) + (a)))))) \/ (((exists pfrep_gap_shift_power_successor_resultoutside. pfrep_gap_shift_power_successor_resultoutside+(S L)=(S k)) /\ (((a)=0)))))

Complete tactic proof in conservative notation

All 37 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

37 script commands · 10 reading checkpoints · 0 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 b
  2. L2
    intro c
  3. L3
    intro L
  4. L4
    intro d
  5. L5
    intro e
  6. L6
    intro k
  7. L7
    intro a
  8. L8
    intro hs
  9. L9
    intro ha
02Separate the logical casesL10–14

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

  1. L10
    cases hs
  2. L11
    cases ha
  3. L12
    cases ha_left
  4. L13
    cases ha_left_witness
  5. L14
    left
03Construct an explicit witnessL15–15

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

  1. L15
    exists x
04Separate the logical casesL16–16

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

  1. L16
    split
05Calculate and transport equalitiesL17–17

Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.

  1. L17
    trans S (x+S k)
06Use earlier factsL18–18

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

  1. L18
    apply PA4
07Calculate and transport equalitiesL19–19

Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.

  1. L19
    congr
08Use earlier factsL20–29

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

  1. L20
    exact ha_left_witness_left
  2. L21
    specialize hs_left (x)
  3. L22
    specialize hs_left (a)
  4. L23
    apply hs_left
  5. L24
    specialize prime_field_polynomial_power_index_bound (x)
  6. L25
    specialize prime_field_polynomial_power_index_bound (k)
  7. L26
    specialize prime_field_polynomial_power_index_bound (L)
  8. L27
    apply prime_field_polynomial_power_index_bound
  9. L28
    exact ha_left_witness_left
  10. L29
    exact ha_left_witness_right
09Separate the logical casesL30–32

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

  1. L30
    cases ha_right
  2. L31
    right
  3. L32
    split
10Use earlier factsL33–37

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

  1. L33
    specialize succ_le_succ (L)
  2. L34
    specialize succ_le_succ (k)
  3. L35
    apply succ_le_succ
  4. L36
    exact ha_right_left
  5. L37
    exact ha_right_right

Library-wide reading audit

Original defined command ledger · 37 lines
  1. 0001intro b
  2. 0002intro c
  3. 0003intro L
  4. 0004intro d
  5. 0005intro e
  6. 0006intro k
  7. 0007intro a
  8. 0008intro hs
  9. 0009intro ha
  10. 0010cases hs
  11. 0011cases ha
  12. 0012cases ha_left
  13. 0013cases ha_left_witness
  14. 0014left
  15. 0015exists x
  16. 0016split
  17. 0017trans S (x+S k)
  18. 0018apply PA4
  19. 0019congr
  20. 0020exact ha_left_witness_left
  21. 0021specialize hs_left (x)
  22. 0022specialize hs_left (a)
  23. 0023apply hs_left
  24. 0024specialize prime_field_polynomial_power_index_bound (x)
  25. 0025specialize prime_field_polynomial_power_index_bound (k)
  26. 0026specialize prime_field_polynomial_power_index_bound (L)
  27. 0027apply prime_field_polynomial_power_index_bound
  28. 0028exact ha_left_witness_left
  29. 0029exact ha_left_witness_right
  30. 0030cases ha_right
  31. 0031right
  32. 0032split
  33. 0033specialize succ_le_succ (L)
  34. 0034specialize succ_le_succ (k)
  35. 0035apply succ_le_succ
  36. 0036exact ha_right_left
  37. 0037exact ha_right_right