PX000E

prime_field_polynomial_power_coefficient_transport

An exact decoded-prefix recoding preserves every formal coefficient at the same annotated length.

Alpha v34 checked-use · first admitted v33 · 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.

Coefficients are highest-degree-first. The divisor has a nonzero decoded head; primality supplies its actual inverse. Empty quotients and remainders are included. Functionality compares the constructed execution lengths and decoded coefficients, never arbitrary beta codes. Formal polynomial equivalence compares every coefficient, not evaluations on a finite field. The formal identity and remainder-degree bound are proved separately, not assumed by the execution graph. Arbitrary quotient/remainder-pair uniqueness from a formal identity, multiplication associativity, gcd/Bezout, irreducible-polynomial existence, and the full G091 prime-power-field goal remain open. The seven displayed new names are conservative first-order notation, not new kernel primitives.

Exact theorem in conservative defined notation

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

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

Definition DAG

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

Actual proof prerequisites

prime_field_polynomial_power_index_bound
Original expanded first-order statement 
forall b c d e L k a. (forall mdr_i_pfp_power_transport_equal mdr_a_pfp_power_transport_equal. (exists mdr_gap_pfp_power_transport_equalb. mdr_gap_pfp_power_transport_equalb + S (mdr_i_pfp_power_transport_equal) = (L)) -> (((exists ff_h_mdr_pfp_power_transport_equalo. ff_h_mdr_pfp_power_transport_equalo + S (mdr_a_pfp_power_transport_equal) = S ((S (mdr_i_pfp_power_transport_equal)) * c)) /\ exists ff_q_mdr_pfp_power_transport_equalo. b = ff_q_mdr_pfp_power_transport_equalo * S ((S (mdr_i_pfp_power_transport_equal)) * c) + (mdr_a_pfp_power_transport_equal))) -> (((exists ff_h_mdr_pfp_power_transport_equaln. ff_h_mdr_pfp_power_transport_equaln + S (mdr_a_pfp_power_transport_equal) = S ((S (mdr_i_pfp_power_transport_equal)) * e)) /\ exists ff_q_mdr_pfp_power_transport_equaln. d = ff_q_mdr_pfp_power_transport_equaln * S ((S (mdr_i_pfp_power_transport_equal)) * e) + (mdr_a_pfp_power_transport_equal)))) -> ((exists pfrep_position_power_transport_old. ((pfrep_position_power_transport_old+S (k)=(L)) /\ ((((exists ff_h_pfp_power_transport_oldentry. ff_h_pfp_power_transport_oldentry + S (a) = S ((S (pfrep_position_power_transport_old)) * c)) /\ exists ff_q_pfp_power_transport_oldentry. b = ff_q_pfp_power_transport_oldentry * S ((S (pfrep_position_power_transport_old)) * c) + (a)))))) \/ (((exists pfrep_gap_power_transport_oldoutside. pfrep_gap_power_transport_oldoutside+(L)=(k)) /\ (((a)=0))))) -> ((exists pfrep_position_power_transport_new. ((pfrep_position_power_transport_new+S (k)=(L)) /\ ((((exists ff_h_pfp_power_transport_newentry. ff_h_pfp_power_transport_newentry + S (a) = S ((S (pfrep_position_power_transport_new)) * e)) /\ exists ff_q_pfp_power_transport_newentry. d = ff_q_pfp_power_transport_newentry * S ((S (pfrep_position_power_transport_new)) * e) + (a)))))) \/ (((exists pfrep_gap_power_transport_newoutside. pfrep_gap_power_transport_newoutside+(L)=(k)) /\ (((a)=0)))))

Complete tactic proof in conservative notation

All 27 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

27 script commands · 7 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.

Named ingredients (1)
prime field polynomial power index bound
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 d
  4. L4
    intro e
  5. L5
    intro L
  6. L6
    intro k
  7. L7
    intro a
  8. L8
    intro he
  9. L9
    intro ha
02Separate the logical casesL10–13

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

  1. L10
    cases ha
  2. L11
    cases ha_left
  3. L12
    cases ha_left_witness
  4. L13
    left
03Construct an explicit witnessL14–14

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

  1. L14
    exists x
04Separate the logical casesL15–15

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

  1. L15
    split
05Use earlier factsL16–25

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

  1. L16
    exact ha_left_witness_left
  2. L17
    specialize he (x)
  3. L18
    specialize he (a)
  4. L19
    apply he
  5. L20
    specialize prime_field_polynomial_power_index_bound (x)
  6. L21
    specialize prime_field_polynomial_power_index_bound (k)
  7. L22
    specialize prime_field_polynomial_power_index_bound (L)
  8. L23
    apply prime_field_polynomial_power_index_bound
  9. L24
    exact ha_left_witness_left
  10. L25
    exact ha_left_witness_right
06Separate the logical casesL26–26

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

  1. L26
    right
07Use earlier factsL27–27

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

  1. L27
    exact ha_right

Library-wide reading audit

Original defined command ledger · 27 lines
  1. 0001intro b
  2. 0002intro c
  3. 0003intro d
  4. 0004intro e
  5. 0005intro L
  6. 0006intro k
  7. 0007intro a
  8. 0008intro he
  9. 0009intro ha
  10. 0010cases ha
  11. 0011cases ha_left
  12. 0012cases ha_left_witness
  13. 0013left
  14. 0014exists x
  15. 0015split
  16. 0016exact ha_left_witness_left
  17. 0017specialize he (x)
  18. 0018specialize he (a)
  19. 0019apply he
  20. 0020specialize prime_field_polynomial_power_index_bound (x)
  21. 0021specialize prime_field_polynomial_power_index_bound (k)
  22. 0022specialize prime_field_polynomial_power_index_bound (L)
  23. 0023apply prime_field_polynomial_power_index_bound
  24. 0024exact ha_left_witness_left
  25. 0025exact ha_left_witness_right
  26. 0026right
  27. 0027exact ha_right