PX001B

prime_field_polynomial_left_pad_equivalent

Leading-zero padding is harmless for formal polynomial coefficients, unlike right padding by trailing zeros.

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. ∀ d. ∀ e. PolynomialLeftPad(b,c,L,t,d,e) → PolynomialEquivalent(b,c,L,d,e,t + L)

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

Definition DAG

PolynomialLeftPad(b,c,L,t,d,e) · 1PolynomialEquivalent(b,c,L,d,e,M) · 1

Actual proof prerequisites

prime_field_polynomial_power_coefficient_functionalprime_field_polynomial_left_pad_power_coefficient
Original expanded first-order statement 
forall b c L t d e. (((forall pfp_repeat_index_pad_equivalent_sourcezeros. (exists pfa_gap_pad_equivalent_sourcezerosindex. pfa_gap_pad_equivalent_sourcezerosindex + S (pfp_repeat_index_pad_equivalent_sourcezeros) = (t)) -> (((exists ff_h_pfp_pad_equivalent_sourcezerosentry. ff_h_pfp_pad_equivalent_sourcezerosentry + S (0) = S ((S (pfp_repeat_index_pad_equivalent_sourcezeros)) * e)) /\ exists ff_q_pfp_pad_equivalent_sourcezerosentry. d = ff_q_pfp_pad_equivalent_sourcezerosentry * S ((S (pfp_repeat_index_pad_equivalent_sourcezeros)) * e) + (0)))) /\ ((forall pfrep_index_pad_equivalent_source pfrep_value_pad_equivalent_source. (exists pfa_gap_pad_equivalent_sourcebound. pfa_gap_pad_equivalent_sourcebound + S (pfrep_index_pad_equivalent_source) = (L)) -> (((exists ff_h_pfp_pad_equivalent_sourceinput. ff_h_pfp_pad_equivalent_sourceinput + S (pfrep_value_pad_equivalent_source) = S ((S (pfrep_index_pad_equivalent_source)) * c)) /\ exists ff_q_pfp_pad_equivalent_sourceinput. b = ff_q_pfp_pad_equivalent_sourceinput * S ((S (pfrep_index_pad_equivalent_source)) * c) + (pfrep_value_pad_equivalent_source))) -> (((exists ff_h_pfp_pad_equivalent_sourceoutput. ff_h_pfp_pad_equivalent_sourceoutput + S (pfrep_value_pad_equivalent_source) = S ((S ((t)+pfrep_index_pad_equivalent_source)) * e)) /\ exists ff_q_pfp_pad_equivalent_sourceoutput. d = ff_q_pfp_pad_equivalent_sourceoutput * S ((S ((t)+pfrep_index_pad_equivalent_source)) * e) + (pfrep_value_pad_equivalent_source))))))) -> (forall pfrep_power_pad_equivalent_result pfrep_left_pad_equivalent_result pfrep_right_pad_equivalent_result. ((exists pfrep_position_pad_equivalent_resultfirst. ((pfrep_position_pad_equivalent_resultfirst+S (pfrep_power_pad_equivalent_result)=(L)) /\ ((((exists ff_h_pfp_pad_equivalent_resultfirstentry. ff_h_pfp_pad_equivalent_resultfirstentry + S (pfrep_left_pad_equivalent_result) = S ((S (pfrep_position_pad_equivalent_resultfirst)) * c)) /\ exists ff_q_pfp_pad_equivalent_resultfirstentry. b = ff_q_pfp_pad_equivalent_resultfirstentry * S ((S (pfrep_position_pad_equivalent_resultfirst)) * c) + (pfrep_left_pad_equivalent_result)))))) \/ (((exists pfrep_gap_pad_equivalent_resultfirstoutside. pfrep_gap_pad_equivalent_resultfirstoutside+(L)=(pfrep_power_pad_equivalent_result)) /\ (((pfrep_left_pad_equivalent_result)=0))))) -> ((exists pfrep_position_pad_equivalent_resultsecond. ((pfrep_position_pad_equivalent_resultsecond+S (pfrep_power_pad_equivalent_result)=(t+L)) /\ ((((exists ff_h_pfp_pad_equivalent_resultsecondentry. ff_h_pfp_pad_equivalent_resultsecondentry + S (pfrep_right_pad_equivalent_result) = S ((S (pfrep_position_pad_equivalent_resultsecond)) * e)) /\ exists ff_q_pfp_pad_equivalent_resultsecondentry. d = ff_q_pfp_pad_equivalent_resultsecondentry * S ((S (pfrep_position_pad_equivalent_resultsecond)) * e) + (pfrep_right_pad_equivalent_result)))))) \/ (((exists pfrep_gap_pad_equivalent_resultsecondoutside. pfrep_gap_pad_equivalent_resultsecondoutside+(t+L)=(pfrep_power_pad_equivalent_result)) /\ (((pfrep_right_pad_equivalent_result)=0))))) -> pfrep_left_pad_equivalent_result=pfrep_right_pad_equivalent_result)

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 · 4 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 (2)
prime field polynomial power coefficient functionalprime field polynomial left pad power coefficient
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 d
  6. L6
    intro e
  7. L7
    intro h
  8. L8
    intro k
  9. L9
    intro a
  10. L10
    intro r
02Fix variables and assumptionsL11–12

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

  1. L11
    intro ha
  2. L12
    intro hr
03Use earlier factsL13–22

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

  1. L13
    specialize prime_field_polynomial_power_coefficient_functional (d)
  2. L14
    specialize prime_field_polynomial_power_coefficient_functional (e)
  3. L15
    specialize prime_field_polynomial_power_coefficient_functional (t+L)
  4. L16
    specialize prime_field_polynomial_power_coefficient_functional (k)
  5. L17
    specialize prime_field_polynomial_power_coefficient_functional (a)
  6. L18
    specialize prime_field_polynomial_power_coefficient_functional (r)
  7. L19
    apply prime_field_polynomial_power_coefficient_functional
  8. L20
    specialize prime_field_polynomial_left_pad_power_coefficient (b)
  9. L21
    specialize prime_field_polynomial_left_pad_power_coefficient (c)
  10. L22
    specialize prime_field_polynomial_left_pad_power_coefficient (L)
04Use earlier factsL23–31

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

  1. L23
    specialize prime_field_polynomial_left_pad_power_coefficient (t)
  2. L24
    specialize prime_field_polynomial_left_pad_power_coefficient (d)
  3. L25
    specialize prime_field_polynomial_left_pad_power_coefficient (e)
  4. L26
    specialize prime_field_polynomial_left_pad_power_coefficient (k)
  5. L27
    specialize prime_field_polynomial_left_pad_power_coefficient (a)
  6. L28
    apply prime_field_polynomial_left_pad_power_coefficient
  7. L29
    exact h
  8. L30
    exact ha
  9. L31
    exact hr

Library-wide reading audit

Original defined command ledger · 31 lines
  1. 0001intro b
  2. 0002intro c
  3. 0003intro L
  4. 0004intro t
  5. 0005intro d
  6. 0006intro e
  7. 0007intro h
  8. 0008intro k
  9. 0009intro a
  10. 0010intro r
  11. 0011intro ha
  12. 0012intro hr
  13. 0013specialize prime_field_polynomial_power_coefficient_functional (d)
  14. 0014specialize prime_field_polynomial_power_coefficient_functional (e)
  15. 0015specialize prime_field_polynomial_power_coefficient_functional (t+L)
  16. 0016specialize prime_field_polynomial_power_coefficient_functional (k)
  17. 0017specialize prime_field_polynomial_power_coefficient_functional (a)
  18. 0018specialize prime_field_polynomial_power_coefficient_functional (r)
  19. 0019apply prime_field_polynomial_power_coefficient_functional
  20. 0020specialize prime_field_polynomial_left_pad_power_coefficient (b)
  21. 0021specialize prime_field_polynomial_left_pad_power_coefficient (c)
  22. 0022specialize prime_field_polynomial_left_pad_power_coefficient (L)
  23. 0023specialize prime_field_polynomial_left_pad_power_coefficient (t)
  24. 0024specialize prime_field_polynomial_left_pad_power_coefficient (d)
  25. 0025specialize prime_field_polynomial_left_pad_power_coefficient (e)
  26. 0026specialize prime_field_polynomial_left_pad_power_coefficient (k)
  27. 0027specialize prime_field_polynomial_left_pad_power_coefficient (a)
  28. 0028apply prime_field_polynomial_left_pad_power_coefficient
  29. 0029exact h
  30. 0030exact ha
  31. 0031exact hr