PX000C

prime_field_polynomial_power_coefficient_exists

Every natural power has an actual decoded coefficient or the proved exterior zero, for arbitrary beta encodings.

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. PolynomialPowerCoefficient(b,c,L,k,a)

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

Definition DAG

PolynomialPowerCoefficient(b,c,L,k,a) · 1Le(a,b) · 1Lt(a,b) · 1BetaAt(b,c,i,x) · 1

Actual proof prerequisites

le_or_lt · checked external prerequisitebeta_at_exists · checked external prerequisite
Original expanded first-order statement 
forall b c L k. exists a. ((exists pfrep_position_power_exists. ((pfrep_position_power_exists+S (k)=(L)) /\ ((((exists ff_h_pfp_power_existsentry. ff_h_pfp_power_existsentry + S (a) = S ((S (pfrep_position_power_exists)) * c)) /\ exists ff_q_pfp_power_existsentry. b = ff_q_pfp_power_existsentry * S ((S (pfrep_position_power_exists)) * c) + (a)))))) \/ (((exists pfrep_gap_power_existsoutside. pfrep_gap_power_existsoutside+(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 · 15 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 assumptionsL1–4

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
02Establish hoL5–8

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply le or lt.

  1. L5
    have ho : Le(L,k) ∨ Lt(k,L)Definitions: Le(L,k)Lt(k,L)Original native command in the exact edition
  2. L6
    specialize le_or_lt (L)
  3. L7
    specialize le_or_lt (k)
  4. L8
    apply le_or_lt
03Separate the logical casesL9–9

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

  1. L9
    cases ho
04Construct an explicit witnessL10–10

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

  1. L10
    exists 0
05Separate the logical casesL11–12

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

  1. L11
    right
  2. L12
    split
06Use earlier factsL13–13

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

  1. L13
    exact ho_left
07Calculate and transport equalitiesL14–14

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

  1. L14
    refl
08Separate the logical casesL15–15

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

  1. L15
    cases ho_right
09Establish hvL16–20

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply beta at exists.

  1. L16
    have hv : ∃ a. BetaAt(b,c,x,a)Definitions: BetaAt(b,c,x,a)Original native command in the exact edition
  2. L17
    specialize beta_at_exists (b)
  3. L18
    specialize beta_at_exists (c)
  4. L19
    specialize beta_at_exists (x)
  5. L20
    apply beta_at_exists
10Separate the logical casesL21–21

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

  1. L21
    cases hv
11Construct an explicit witnessL22–22

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

  1. L22
    exists x1
12Separate the logical casesL23–23

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

  1. L23
    left
13Construct an explicit witnessL24–24

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

  1. L24
    exists x
14Separate the logical casesL25–25

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

  1. L25
    split
15Use earlier factsL26–27

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

  1. L26
    exact ho_right_witness
  2. L27
    exact hv_witness

Library-wide reading audit

Original defined command ledger · 27 lines
  1. 0001intro b
  2. 0002intro c
  3. 0003intro L
  4. 0004intro k
  5. 0005have ho : Le(L,k) ∨ Lt(k,L)
  6. 0006specialize le_or_lt (L)
  7. 0007specialize le_or_lt (k)
  8. 0008apply le_or_lt
  9. 0009cases ho
  10. 0010exists 0
  11. 0011right
  12. 0012split
  13. 0013exact ho_left
  14. 0014refl
  15. 0015cases ho_right
  16. 0016have hv : ∃ a. BetaAt(b,c,x,a)
  17. 0017specialize beta_at_exists (b)
  18. 0018specialize beta_at_exists (c)
  19. 0019specialize beta_at_exists (x)
  20. 0020apply beta_at_exists
  21. 0021cases hv
  22. 0022exists x1
  23. 0023left
  24. 0024exists x
  25. 0025split
  26. 0026exact ho_right_witness
  27. 0027exact hv_witness