PX000A

prime_field_polynomial_left_pad_index_cases

Every index in an actual left-padded window is in its zero block or has an actual bounded source index.

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. Lt(i,t + L) → Lt(i,t) ∨ (∃ x. Lt(x,L) ∧ i = t + x)

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

Definition DAG

Le(a,b) · 1Lt(a,b) · 4

Actual proof prerequisites

le_or_lt · checked external prerequisiteadd_left_cancel · checked external prerequisiteadd_comm · checked external prerequisiteadd_assoc · checked external prerequisite
Original expanded first-order statement 
forall t L i. (exists pfa_gap_index_cases_bound. pfa_gap_index_cases_bound + S (i) = (t+L)) -> (exists pfa_gap_index_cases_zero. pfa_gap_index_cases_zero + S (i) = (t)) \/ exists j. (((exists pfa_gap_index_cases_copy. pfa_gap_index_cases_copy + S (j) = (L)) /\ ((i=t+j))))

Complete tactic proof in conservative notation

All 31 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

31 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 t
  2. L2
    intro L
  3. L3
    intro i
  4. L4
    intro hi
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(t,i) ∨ Lt(i,t)Definitions: Le(t,i)Lt(i,t)Original native command in the exact edition
  2. L6
    specialize le_or_lt (t)
  3. L7
    specialize le_or_lt (i)
  4. L8
    apply le_or_lt
03Separate the logical casesL9–10

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

  1. L9
    cases ho
  2. L10
    cases ho_left
04Establish heqL11–14

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

  1. L11
    have heq : t+x=i
  2. L12
    trans x+t
  3. L13
    apply add_comm
  4. L14
    exact ho_left_witness
05Separate the logical casesL15–15

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

  1. L15
    right
06Construct an explicit witnessL16–16

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

  1. L16
    exists x
07Separate the logical casesL17–18

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

  1. L17
    split
  2. L18
    cases hi
08Construct an explicit witnessL19–19

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

  1. L19
    exists x1
09Use earlier factsL20–23

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

  1. L20
    specialize add_left_cancel (t)
  2. L21
    specialize add_left_cancel (x1+S x)
  3. L22
    specialize add_left_cancel (L)
  4. L23
    apply add_left_cancel
10Calculate and transport equalitiesL24–26

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

  1. L24
    trans x1+S (t+x)
  2. L25
    simp [add_comm,add_assoc]
  3. L26
    rewrite heq
11Use earlier factsL27–27

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

  1. L27
    exact hi_witness
12Calculate and transport equalitiesL28–28

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

  1. L28
    symm
13Use earlier factsL29–29

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

  1. L29
    exact heq
14Separate the logical casesL30–30

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

  1. L30
    left
15Use earlier factsL31–31

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

  1. L31
    exact ho_right

Library-wide reading audit

Original defined command ledger · 31 lines
  1. 0001intro t
  2. 0002intro L
  3. 0003intro i
  4. 0004intro hi
  5. 0005have ho : Le(t,i) ∨ Lt(i,t)
  6. 0006specialize le_or_lt (t)
  7. 0007specialize le_or_lt (i)
  8. 0008apply le_or_lt
  9. 0009cases ho
  10. 0010cases ho_left
  11. 0011have heq : t+x=i
  12. 0012trans x+t
  13. 0013apply add_comm
  14. 0014exact ho_left_witness
  15. 0015right
  16. 0016exists x
  17. 0017split
  18. 0018cases hi
  19. 0019exists x1
  20. 0020specialize add_left_cancel (t)
  21. 0021specialize add_left_cancel (x1+S x)
  22. 0022specialize add_left_cancel (L)
  23. 0023apply add_left_cancel
  24. 0024trans x1+S (t+x)
  25. 0025simp [add_comm,add_assoc]
  26. 0026rewrite heq
  27. 0027exact hi_witness
  28. 0028symm
  29. 0029exact heq
  30. 0030left
  31. 0031exact ho_right