PX0022

prime_field_polynomial_zero_prefix_equivalent_empty

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

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. Repeat(b,c,0,L) → PolynomialEquivalent(b,c,L,0,0,0)

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

Definition DAG

PolynomialEquivalent(b,c,L,d,e,M) · 1Repeat(b,c,a,l) · 1

Actual proof prerequisites

prime_field_polynomial_zero_power_coefficientbeta_repeat_empty · checked external prerequisite
Original expanded first-order 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)

Complete tactic proof in conservative notation

All 34 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

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.

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 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 defined 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