PX001B

prime_field_polynomial_left_pad_equivalent

Alpha v34 independently verified · alpha_closed; checked-use authorized; not Stable

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

Current library: Alpha v34, 4,223 checked-use theorems; Stable remains 432. Historical first admissions, original proof editions, and non-admitted aliases are preserved.

Exact expanded first-order arithmetic 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)

Constructive proof overview

Generated structural guide

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

The unchanged tactic script uses 2 declared prerequisites and contains 31 exact native proof lines.

Alpha v34 checked-use · first admitted v33 · independently kernel and Lean verified; not Stable

Proof neighborhood

Direct dependencies

PX000D prime_field_polynomial_power_coefficient_functional PX001A prime_field_polynomial_left_pad_power_coefficient

Direct dependents

PX001C prime_field_polynomial_trim_equivalent PX006E prime_field_polynomial_convolution_left_padding_equivalent_left PX006F prime_field_polynomial_convolution_left_padding_equivalent_right PX0072 prime_field_polynomial_equivalent_implies_left_pad PX0075 prime_field_polynomial_add_equivalent_congruent PX0076 prime_field_polynomial_subtract_equivalent_congruent

Formal native tactic body

Dependencies are introduced as named hypotheses before line 1. Local theorem links identify exact declared prerequisites. This exact body belongs to a complete independently kernel-checked constructive proof bundle and has Alpha checked-use authority; it does not imply Stable membership.

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.

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 exact 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