PX005D

polynomial_left_pad_zero_prefix

An actually zero prefix remains zero after genuine left padding, including an originally empty factor.

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. ∀ t. ∀ B. ∀ C. Repeat(b,c,0,L) → PolynomialLeftPad(b,c,L,t,B,C) → Repeat(B,C,0,t + L)

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

Definition DAG

PolynomialLeftPad(b,c,L,t,d,e) · 1Lt(a,b) · 2Repeat(b,c,a,l) · 2

Actual proof prerequisites

prime_field_polynomial_left_pad_index_cases
Original expanded first-order statement 
forall b c L t B C. (forall pfp_repeat_index_padded_zero_original. (exists pfa_gap_padded_zero_originalindex. pfa_gap_padded_zero_originalindex + S (pfp_repeat_index_padded_zero_original) = (L)) -> (((exists ff_h_pfp_padded_zero_originalentry. ff_h_pfp_padded_zero_originalentry + S (0) = S ((S (pfp_repeat_index_padded_zero_original)) * c)) /\ exists ff_q_pfp_padded_zero_originalentry. b = ff_q_pfp_padded_zero_originalentry * S ((S (pfp_repeat_index_padded_zero_original)) * c) + (0)))) -> (((forall pfp_repeat_index_padded_zero_datazeros. (exists pfa_gap_padded_zero_datazerosindex. pfa_gap_padded_zero_datazerosindex + S (pfp_repeat_index_padded_zero_datazeros) = (t)) -> (((exists ff_h_pfp_padded_zero_datazerosentry. ff_h_pfp_padded_zero_datazerosentry + S (0) = S ((S (pfp_repeat_index_padded_zero_datazeros)) * C)) /\ exists ff_q_pfp_padded_zero_datazerosentry. B = ff_q_pfp_padded_zero_datazerosentry * S ((S (pfp_repeat_index_padded_zero_datazeros)) * C) + (0)))) /\ ((forall pfrep_index_padded_zero_data pfrep_value_padded_zero_data. (exists pfa_gap_padded_zero_databound. pfa_gap_padded_zero_databound + S (pfrep_index_padded_zero_data) = (L)) -> (((exists ff_h_pfp_padded_zero_datainput. ff_h_pfp_padded_zero_datainput + S (pfrep_value_padded_zero_data) = S ((S (pfrep_index_padded_zero_data)) * c)) /\ exists ff_q_pfp_padded_zero_datainput. b = ff_q_pfp_padded_zero_datainput * S ((S (pfrep_index_padded_zero_data)) * c) + (pfrep_value_padded_zero_data))) -> (((exists ff_h_pfp_padded_zero_dataoutput. ff_h_pfp_padded_zero_dataoutput + S (pfrep_value_padded_zero_data) = S ((S ((t)+pfrep_index_padded_zero_data)) * C)) /\ exists ff_q_pfp_padded_zero_dataoutput. B = ff_q_pfp_padded_zero_dataoutput * S ((S ((t)+pfrep_index_padded_zero_data)) * C) + (pfrep_value_padded_zero_data))))))) -> (forall pfp_repeat_index_padded_zero_result. (exists pfa_gap_padded_zero_resultindex. pfa_gap_padded_zero_resultindex + S (pfp_repeat_index_padded_zero_result) = (t+L)) -> (((exists ff_h_pfp_padded_zero_resultentry. ff_h_pfp_padded_zero_resultentry + S (0) = S ((S (pfp_repeat_index_padded_zero_result)) * C)) /\ exists ff_q_pfp_padded_zero_resultentry. B = ff_q_pfp_padded_zero_resultentry * S ((S (pfp_repeat_index_padded_zero_result)) * C) + (0))))

Complete tactic proof in conservative notation

All 32 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

32 script commands · 8 reading checkpoints · 1 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 left pad index cases
01Fix variables and assumptionsL1–10

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 t
  5. L5
    intro B
  6. L6
    intro C
  7. L7
    intro hz
  8. L8
    intro hpad
  9. L9
    intro i
  10. L10
    intro hi
02Separate the logical casesL11–11

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

  1. L11
    cases hpad
03Establish hcL12–17

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply prime field polynomial left pad index cases.

  1. L12
    have hc : Lt(i,t) ∨ (∃ x. Lt(x,L) ∧ i = t + x)Definitions: Lt(i,t)Lt(x,L)Original native command in the exact edition
  2. L13
    specialize prime_field_polynomial_left_pad_index_cases (t)
  3. L14
    specialize prime_field_polynomial_left_pad_index_cases (L)
  4. L15
    specialize prime_field_polynomial_left_pad_index_cases (i)
  5. L16
    apply prime_field_polynomial_left_pad_index_cases
  6. L17
    exact hi
04Separate the logical casesL18–18

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

  1. L18
    cases hc
05Use earlier factsL19–21

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

  1. L19
    specialize hpad_left (i)
  2. L20
    apply hpad_left
  3. L21
    exact hc_left
06Separate the logical casesL22–23

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

  1. L22
    cases hc_right
  2. L23
    cases hc_right_witness
07Calculate and transport equalitiesL24–25

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

  1. L24
    rewrite hc_right_witness_right
  2. L25
    rewrite hc_right_witness_right
08Use earlier factsL26–32

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

  1. L26
    specialize hpad_right (x)
  2. L27
    specialize hpad_right (0)
  3. L28
    apply hpad_right
  4. L29
    exact hc_right_witness_left
  5. L30
    specialize hz (x)
  6. L31
    apply hz
  7. L32
    exact hc_right_witness_left

Library-wide reading audit

Original defined command ledger · 32 lines
  1. 0001intro b
  2. 0002intro c
  3. 0003intro L
  4. 0004intro t
  5. 0005intro B
  6. 0006intro C
  7. 0007intro hz
  8. 0008intro hpad
  9. 0009intro i
  10. 0010intro hi
  11. 0011cases hpad
  12. 0012have hc : Lt(i,t) ∨ (∃ x. Lt(x,L) ∧ i = t + x)
  13. 0013specialize prime_field_polynomial_left_pad_index_cases (t)
  14. 0014specialize prime_field_polynomial_left_pad_index_cases (L)
  15. 0015specialize prime_field_polynomial_left_pad_index_cases (i)
  16. 0016apply prime_field_polynomial_left_pad_index_cases
  17. 0017exact hi
  18. 0018cases hc
  19. 0019specialize hpad_left (i)
  20. 0020apply hpad_left
  21. 0021exact hc_left
  22. 0022cases hc_right
  23. 0023cases hc_right_witness
  24. 0024rewrite hc_right_witness_right
  25. 0025rewrite hc_right_witness_right
  26. 0026specialize hpad_right (x)
  27. 0027specialize hpad_right (0)
  28. 0028apply hpad_right
  29. 0029exact hc_right_witness_left
  30. 0030specialize hz (x)
  31. 0031apply hz
  32. 0032exact hc_right_witness_left