PX000B

prime_field_polynomial_power_index_before_padding

A power beyond the source degree can only access the actual added leading-zero block.

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

∀ t. ∀ L. ∀ i. ∀ k. i + S k = t + L → Le(L,k) → Lt(i,t)

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

Definition DAG

Le(a,b) · 2Lt(a,b) · 3

Actual proof prerequisites

le_or_lt · checked external prerequisiteadd_left_cancel · checked external prerequisiteadd_assoc · checked external prerequisiteadd_comm · checked external prerequisitelt_not_le · checked external prerequisite
Original expanded first-order statement 
forall t L i k. i+S k=t+L -> (exists pfrep_gap_power_before_outside. pfrep_gap_power_before_outside+(L)=(k)) -> (exists pfa_gap_power_before_result. pfa_gap_power_before_result + S (i) = (t))

Complete tactic proof in conservative notation

All 34 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

34 script commands · 11 reading checkpoints · 3 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 assumptionsL1–6

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

  1. L1
    intro t
  2. L2
    intro L
  3. L3
    intro i
  4. L4
    intro k
  5. L5
    intro hi
  6. L6
    intro hk
02Establish hoL7–10

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

  1. L7
    have ho : Le(t,i) ∨ Lt(i,t)Definitions: Le(t,i)Lt(i,t)Original native command in the exact edition
  2. L8
    specialize le_or_lt (t)
  3. L9
    specialize le_or_lt (i)
  4. L10
    apply le_or_lt
03Separate the logical casesL11–12

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

  1. L11
    cases ho
  2. L12
    cases ho_left
04Establish hbadL13–13

Establish this local claim before using it. It is not an additional assumption.

  1. L13
05Construct an explicit witnessL14–14

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

  1. L14
    exists x
06Use earlier factsL15–18

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

  1. L15
    specialize add_left_cancel (t)
  2. L16
    specialize add_left_cancel (x+S k)
  3. L17
    specialize add_left_cancel (L)
  4. L18
    apply add_left_cancel
07Calculate and transport equalitiesL19–20

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

  1. L19
    trans (t+x)+S k
  2. L20
    symm
08Use earlier factsL21–21

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

  1. L21
    apply add_assoc
09Establish heqL22–27

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

  1. L22
    have heq : t+x=i
  2. L23
    trans x+t
  3. L24
    apply add_comm
  4. L25
    exact ho_left_witness
  5. L26
    rewrite heq
  6. L27
    exact hi
10Separate the logical casesL28–28

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

  1. L28
    exfalso
11Use earlier factsL29–34

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

  1. L29
    specialize lt_not_le (k)
  2. L30
    specialize lt_not_le (L)
  3. L31
    apply lt_not_le
  4. L32
    exact hbad
  5. L33
    exact hk
  6. L34
    exact ho_right

Library-wide reading audit

Original defined command ledger · 34 lines
  1. 0001intro t
  2. 0002intro L
  3. 0003intro i
  4. 0004intro k
  5. 0005intro hi
  6. 0006intro hk
  7. 0007have ho : Le(t,i) ∨ Lt(i,t)
  8. 0008specialize le_or_lt (t)
  9. 0009specialize le_or_lt (i)
  10. 0010apply le_or_lt
  11. 0011cases ho
  12. 0012cases ho_left
  13. 0013have hbad : Lt(k,L)
  14. 0014exists x
  15. 0015specialize add_left_cancel (t)
  16. 0016specialize add_left_cancel (x+S k)
  17. 0017specialize add_left_cancel (L)
  18. 0018apply add_left_cancel
  19. 0019trans (t+x)+S k
  20. 0020symm
  21. 0021apply add_assoc
  22. 0022have heq : t+x=i
  23. 0023trans x+t
  24. 0024apply add_comm
  25. 0025exact ho_left_witness
  26. 0026rewrite heq
  27. 0027exact hi
  28. 0028exfalso
  29. 0029specialize lt_not_le (k)
  30. 0030specialize lt_not_le (L)
  31. 0031apply lt_not_le
  32. 0032exact hbad
  33. 0033exact hk
  34. 0034exact ho_right