PQ001C

prime_field_polynomial_suffix_entry

Every actual output decoding equals the input coefficient at the supplied shifted index.

Alpha v34 checked-use · first admitted v32 · 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.

All coefficients retain the established highest-degree-first order. Trimming handles empty and all-zero prefixes; monic normalization requires a nonzero leading coefficient. Synthetic division has a nonempty input of length S n and a quotient of length n, unique in decoded values. Its coefficient recurrence, actual evaluation remainder and positive-degree drop are checked. General polynomial Euclidean division, gcd/Bezout, an arbitrary-convolution factor theorem, irreducible-polynomial existence and the full G091 prime-power-field endpoint remain open. These exact theorems are first admitted to Alpha v32; Stable remains unchanged.

Exact theorem in conservative defined notation

∀ b. ∀ c. ∀ t. ∀ d. ∀ e. ∀ M. ∀ i. ∀ a. ∀ r. PolynomialSuffix(b,c,t,d,e,M) → Lt(i,M) → BetaAt(b,c,t + i,a) → BetaAt(d,e,i,r) → r = a

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

Definition DAG

PolynomialSuffix(b,c,t,d,e,M) · 1Lt(a,b) · 1BetaAt(b,c,i,x) · 2

Actual proof prerequisites

beta_at_unique · checked external prerequisite
Original expanded first-order statement 
forall b c t d e M i a r. (forall pftrim_index_suffix_entry pftrim_value_suffix_entry. (exists pfa_gap_suffix_entrybound. pfa_gap_suffix_entrybound + S (pftrim_index_suffix_entry) = (M)) -> (((exists ff_h_pfp_suffix_entrysource. ff_h_pfp_suffix_entrysource + S (pftrim_value_suffix_entry) = S ((S ((t)+pftrim_index_suffix_entry)) * c)) /\ exists ff_q_pfp_suffix_entrysource. b = ff_q_pfp_suffix_entrysource * S ((S ((t)+pftrim_index_suffix_entry)) * c) + (pftrim_value_suffix_entry))) -> (((exists ff_h_pfp_suffix_entryoutput. ff_h_pfp_suffix_entryoutput + S (pftrim_value_suffix_entry) = S ((S (pftrim_index_suffix_entry)) * e)) /\ exists ff_q_pfp_suffix_entryoutput. d = ff_q_pfp_suffix_entryoutput * S ((S (pftrim_index_suffix_entry)) * e) + (pftrim_value_suffix_entry)))) -> (exists pfa_gap_suffix_entry_bound. pfa_gap_suffix_entry_bound + S (i) = (M)) -> (((exists ff_h_pfp_suffix_entry_source. ff_h_pfp_suffix_entry_source + S (a) = S ((S (t+i)) * c)) /\ exists ff_q_pfp_suffix_entry_source. b = ff_q_pfp_suffix_entry_source * S ((S (t+i)) * c) + (a))) -> (((exists ff_h_pfp_suffix_entry_output. ff_h_pfp_suffix_entry_output + S (r) = S ((S (i)) * e)) /\ exists ff_q_pfp_suffix_entry_output. d = ff_q_pfp_suffix_entry_output * S ((S (i)) * e) + (r))) -> r=a

Complete tactic proof in conservative notation

All 25 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

25 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.

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 t
  4. L4
    intro d
  5. L5
    intro e
  6. L6
    intro M
  7. L7
    intro i
  8. L8
    intro a
  9. L9
    intro r
  10. L10
    intro h
02Fix variables and assumptionsL11–13

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

  1. L11
    intro hi
  2. L12
    intro ha
  3. L13
    intro hr
03Use earlier factsL14–23

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

  1. L14
    specialize beta_at_unique (d)
  2. L15
    specialize beta_at_unique (e)
  3. L16
    specialize beta_at_unique (i)
  4. L17
    specialize beta_at_unique (r)
  5. L18
    specialize beta_at_unique (a)
  6. L19
    apply beta_at_unique
  7. L20
    exact hr
  8. L21
    specialize h (i)
  9. L22
    specialize h (a)
  10. L23
    apply h
04Use earlier factsL24–25

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

  1. L24
    exact hi
  2. L25
    exact ha

Library-wide reading audit

Original defined command ledger · 25 lines
  1. 0001intro b
  2. 0002intro c
  3. 0003intro t
  4. 0004intro d
  5. 0005intro e
  6. 0006intro M
  7. 0007intro i
  8. 0008intro a
  9. 0009intro r
  10. 0010intro h
  11. 0011intro hi
  12. 0012intro ha
  13. 0013intro hr
  14. 0014specialize beta_at_unique (d)
  15. 0015specialize beta_at_unique (e)
  16. 0016specialize beta_at_unique (i)
  17. 0017specialize beta_at_unique (r)
  18. 0018specialize beta_at_unique (a)
  19. 0019apply beta_at_unique
  20. 0020exact hr
  21. 0021specialize h (i)
  22. 0022specialize h (a)
  23. 0023apply h
  24. 0024exact hi
  25. 0025exact ha