PX0022

prime_field_polynomial_zero_prefix_equivalent_empty

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

An actual all-zero ambient convolution prefix represents the same formal polynomial as an empty product.

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. (forall pfp_repeat_index_zero_equivalent_source. (exists pfa_gap_zero_equivalent_sourceindex. pfa_gap_zero_equivalent_sourceindex + S (pfp_repeat_index_zero_equivalent_source) = (L)) -> (((exists ff_h_pfp_zero_equivalent_sourceentry. ff_h_pfp_zero_equivalent_sourceentry + S (0) = S ((S (pfp_repeat_index_zero_equivalent_source)) * c)) /\ exists ff_q_pfp_zero_equivalent_sourceentry. b = ff_q_pfp_zero_equivalent_sourceentry * S ((S (pfp_repeat_index_zero_equivalent_source)) * c) + (0)))) -> (forall pfrep_power_zero_equivalent_empty pfrep_left_zero_equivalent_empty pfrep_right_zero_equivalent_empty. ((exists pfrep_position_zero_equivalent_emptyfirst. ((pfrep_position_zero_equivalent_emptyfirst+S (pfrep_power_zero_equivalent_empty)=(L)) /\ ((((exists ff_h_pfp_zero_equivalent_emptyfirstentry. ff_h_pfp_zero_equivalent_emptyfirstentry + S (pfrep_left_zero_equivalent_empty) = S ((S (pfrep_position_zero_equivalent_emptyfirst)) * c)) /\ exists ff_q_pfp_zero_equivalent_emptyfirstentry. b = ff_q_pfp_zero_equivalent_emptyfirstentry * S ((S (pfrep_position_zero_equivalent_emptyfirst)) * c) + (pfrep_left_zero_equivalent_empty)))))) \/ (((exists pfrep_gap_zero_equivalent_emptyfirstoutside. pfrep_gap_zero_equivalent_emptyfirstoutside+(L)=(pfrep_power_zero_equivalent_empty)) /\ (((pfrep_left_zero_equivalent_empty)=0))))) -> ((exists pfrep_position_zero_equivalent_emptysecond. ((pfrep_position_zero_equivalent_emptysecond+S (pfrep_power_zero_equivalent_empty)=(0)) /\ ((((exists ff_h_pfp_zero_equivalent_emptysecondentry. ff_h_pfp_zero_equivalent_emptysecondentry + S (pfrep_right_zero_equivalent_empty) = S ((S (pfrep_position_zero_equivalent_emptysecond)) * 0)) /\ exists ff_q_pfp_zero_equivalent_emptysecondentry. 0 = ff_q_pfp_zero_equivalent_emptysecondentry * S ((S (pfrep_position_zero_equivalent_emptysecond)) * 0) + (pfrep_right_zero_equivalent_empty)))))) \/ (((exists pfrep_gap_zero_equivalent_emptysecondoutside. pfrep_gap_zero_equivalent_emptysecondoutside+(0)=(pfrep_power_zero_equivalent_empty)) /\ (((pfrep_right_zero_equivalent_empty)=0))))) -> pfrep_left_zero_equivalent_empty=pfrep_right_zero_equivalent_empty)

Constructive proof overview

Generated structural guide

An actual all-zero ambient convolution prefix represents the same formal polynomial as an empty product.

The unchanged tactic script uses 2 declared prerequisites and contains 34 exact native proof lines.

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

Proof neighborhood

Direct dependencies

PX0021 prime_field_polynomial_zero_power_coefficient beta_repeat_empty Alpha theorem; checked-use authorized

Direct dependents

PX006E prime_field_polynomial_convolution_left_padding_equivalent_left PX006F prime_field_polynomial_convolution_left_padding_equivalent_right

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

34 script commands · 9 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.

Named ingredients (1)
prime field polynomial zero power coefficient
01Fix variables and assumptionsL1–9

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 hz
  5. L5
    intro k
  6. L6
    intro a
  7. L7
    intro r
  8. L8
    intro ha
  9. L9
    intro hr
02Calculate and transport equalitiesL10–10

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

  1. L10
    trans 0
03Use earlier factsL11–18

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

  1. L11
    specialize prime_field_polynomial_zero_power_coefficient (b)
  2. L12
    specialize prime_field_polynomial_zero_power_coefficient (c)
  3. L13
    specialize prime_field_polynomial_zero_power_coefficient (L)
  4. L14
    specialize prime_field_polynomial_zero_power_coefficient (k)
  5. L15
    specialize prime_field_polynomial_zero_power_coefficient (a)
  6. L16
    apply prime_field_polynomial_zero_power_coefficient
  7. L17
    exact hz
  8. L18
    exact ha
04Establish heqL19–28

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply prime field polynomial zero power coefficient.

  1. L19
    have heq : r=0
  2. L20
    specialize prime_field_polynomial_zero_power_coefficient (0)
  3. L21
    specialize prime_field_polynomial_zero_power_coefficient (0)
  4. L22
    specialize prime_field_polynomial_zero_power_coefficient (0)
  5. L23
    specialize prime_field_polynomial_zero_power_coefficient (k)
  6. L24
    specialize prime_field_polynomial_zero_power_coefficient (r)
  7. L25
    apply prime_field_polynomial_zero_power_coefficient
  8. L26
    specialize beta_repeat_empty (0)
  9. L27
    specialize beta_repeat_empty (0)
  10. L28
    specialize beta_repeat_empty (0)
05Use earlier factsL29–30

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

  1. L29
    specialize beta_repeat_empty (0)
  2. L30
    apply beta_repeat_empty
06Calculate and transport equalitiesL31–31

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

  1. L31
    refl
07Use earlier factsL32–32

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

  1. L32
    exact hr
08Calculate and transport equalitiesL33–33

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

  1. L33
    symm
09Use earlier factsL34–34

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

  1. L34
    exact heq

Library-wide reading audit

Original exact command ledger · 34 lines
  1. 0001intro b
  2. 0002intro c
  3. 0003intro L
  4. 0004intro hz
  5. 0005intro k
  6. 0006intro a
  7. 0007intro r
  8. 0008intro ha
  9. 0009intro hr
  10. 0010trans 0
  11. 0011specialize prime_field_polynomial_zero_power_coefficient (b)
  12. 0012specialize prime_field_polynomial_zero_power_coefficient (c)
  13. 0013specialize prime_field_polynomial_zero_power_coefficient (L)
  14. 0014specialize prime_field_polynomial_zero_power_coefficient (k)
  15. 0015specialize prime_field_polynomial_zero_power_coefficient (a)
  16. 0016apply prime_field_polynomial_zero_power_coefficient
  17. 0017exact hz
  18. 0018exact ha
  19. 0019have heq : r=0
  20. 0020specialize prime_field_polynomial_zero_power_coefficient (0)
  21. 0021specialize prime_field_polynomial_zero_power_coefficient (0)
  22. 0022specialize prime_field_polynomial_zero_power_coefficient (0)
  23. 0023specialize prime_field_polynomial_zero_power_coefficient (k)
  24. 0024specialize prime_field_polynomial_zero_power_coefficient (r)
  25. 0025apply prime_field_polynomial_zero_power_coefficient
  26. 0026specialize beta_repeat_empty (0)
  27. 0027specialize beta_repeat_empty (0)
  28. 0028specialize beta_repeat_empty (0)
  29. 0029specialize beta_repeat_empty (0)
  30. 0030apply beta_repeat_empty
  31. 0031refl
  32. 0032exact hr
  33. 0033symm
  34. 0034exact heq