PX0021

prime_field_polynomial_zero_power_coefficient

Every formal power coefficient of an actual zero prefix is zero, including exterior powers.

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. ∀ L. ∀ k. ∀ a. Repeat(b,c,0,L) → PolynomialPowerCoefficient(b,c,L,k,a) → a = 0

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

Definition DAG

PolynomialPowerCoefficient(b,c,L,k,a) · 1Repeat(b,c,a,l) · 1

Actual proof prerequisites

beta_repeat_entry_eq · checked external prerequisiteprime_field_polynomial_power_index_bound
Original expanded first-order statement 
forall b c L k a. (forall pfp_repeat_index_zero_power_source. (exists pfa_gap_zero_power_sourceindex. pfa_gap_zero_power_sourceindex + S (pfp_repeat_index_zero_power_source) = (L)) -> (((exists ff_h_pfp_zero_power_sourceentry. ff_h_pfp_zero_power_sourceentry + S (0) = S ((S (pfp_repeat_index_zero_power_source)) * c)) /\ exists ff_q_pfp_zero_power_sourceentry. b = ff_q_pfp_zero_power_sourceentry * S ((S (pfp_repeat_index_zero_power_source)) * c) + (0)))) -> ((exists pfrep_position_zero_power_entry. ((pfrep_position_zero_power_entry+S (k)=(L)) /\ ((((exists ff_h_pfp_zero_power_entryentry. ff_h_pfp_zero_power_entryentry + S (a) = S ((S (pfrep_position_zero_power_entry)) * c)) /\ exists ff_q_pfp_zero_power_entryentry. b = ff_q_pfp_zero_power_entryentry * S ((S (pfrep_position_zero_power_entry)) * c) + (a)))))) \/ (((exists pfrep_gap_zero_power_entryoutside. pfrep_gap_zero_power_entryoutside+(L)=(k)) /\ (((a)=0))))) -> a=0

Complete tactic proof in conservative notation

All 26 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

26 script commands · 6 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 assumptionsL1–7

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 k
  5. L5
    intro a
  6. L6
    intro hz
  7. L7
    intro ha
02Separate the logical casesL8–10

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

  1. L8
    cases ha
  2. L9
    cases ha_left
  3. L10
    cases ha_left_witness
03Use earlier factsL11–20

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

  1. L11
    specialize beta_repeat_entry_eq (b)
  2. L12
    specialize beta_repeat_entry_eq (c)
  3. L13
    specialize beta_repeat_entry_eq (0)
  4. L14
    specialize beta_repeat_entry_eq (L)
  5. L15
    specialize beta_repeat_entry_eq (x)
  6. L16
    specialize beta_repeat_entry_eq (a)
  7. L17
    apply beta_repeat_entry_eq
  8. L18
    exact hz
  9. L19
    specialize prime_field_polynomial_power_index_bound (x)
  10. L20
    specialize prime_field_polynomial_power_index_bound (k)
04Use earlier factsL21–24

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

  1. L21
    specialize prime_field_polynomial_power_index_bound (L)
  2. L22
    apply prime_field_polynomial_power_index_bound
  3. L23
    exact ha_left_witness_left
  4. L24
    exact ha_left_witness_right
05Separate the logical casesL25–25

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

  1. L25
    cases ha_right
06Use earlier factsL26–26

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

  1. L26
    exact ha_right_right

Library-wide reading audit

Original defined command ledger · 26 lines
  1. 0001intro b
  2. 0002intro c
  3. 0003intro L
  4. 0004intro k
  5. 0005intro a
  6. 0006intro hz
  7. 0007intro ha
  8. 0008cases ha
  9. 0009cases ha_left
  10. 0010cases ha_left_witness
  11. 0011specialize beta_repeat_entry_eq (b)
  12. 0012specialize beta_repeat_entry_eq (c)
  13. 0013specialize beta_repeat_entry_eq (0)
  14. 0014specialize beta_repeat_entry_eq (L)
  15. 0015specialize beta_repeat_entry_eq (x)
  16. 0016specialize beta_repeat_entry_eq (a)
  17. 0017apply beta_repeat_entry_eq
  18. 0018exact hz
  19. 0019specialize prime_field_polynomial_power_index_bound (x)
  20. 0020specialize prime_field_polynomial_power_index_bound (k)
  21. 0021specialize prime_field_polynomial_power_index_bound (L)
  22. 0022apply prime_field_polynomial_power_index_bound
  23. 0023exact ha_left_witness_left
  24. 0024exact ha_left_witness_right
  25. 0025cases ha_right
  26. 0026exact ha_right_right