PX0015

prime_field_polynomial_left_pad_entry

Each copied coefficient is identified with the actual source value, without identifying codes.

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

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

Definition DAG

PolynomialLeftPad(b,c,L,t,d,e) · 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 L t d e i a r. (((forall pfp_repeat_index_left_pad_entry_graphzeros. (exists pfa_gap_left_pad_entry_graphzerosindex. pfa_gap_left_pad_entry_graphzerosindex + S (pfp_repeat_index_left_pad_entry_graphzeros) = (t)) -> (((exists ff_h_pfp_left_pad_entry_graphzerosentry. ff_h_pfp_left_pad_entry_graphzerosentry + S (0) = S ((S (pfp_repeat_index_left_pad_entry_graphzeros)) * e)) /\ exists ff_q_pfp_left_pad_entry_graphzerosentry. d = ff_q_pfp_left_pad_entry_graphzerosentry * S ((S (pfp_repeat_index_left_pad_entry_graphzeros)) * e) + (0)))) /\ ((forall pfrep_index_left_pad_entry_graph pfrep_value_left_pad_entry_graph. (exists pfa_gap_left_pad_entry_graphbound. pfa_gap_left_pad_entry_graphbound + S (pfrep_index_left_pad_entry_graph) = (L)) -> (((exists ff_h_pfp_left_pad_entry_graphinput. ff_h_pfp_left_pad_entry_graphinput + S (pfrep_value_left_pad_entry_graph) = S ((S (pfrep_index_left_pad_entry_graph)) * c)) /\ exists ff_q_pfp_left_pad_entry_graphinput. b = ff_q_pfp_left_pad_entry_graphinput * S ((S (pfrep_index_left_pad_entry_graph)) * c) + (pfrep_value_left_pad_entry_graph))) -> (((exists ff_h_pfp_left_pad_entry_graphoutput. ff_h_pfp_left_pad_entry_graphoutput + S (pfrep_value_left_pad_entry_graph) = S ((S ((t)+pfrep_index_left_pad_entry_graph)) * e)) /\ exists ff_q_pfp_left_pad_entry_graphoutput. d = ff_q_pfp_left_pad_entry_graphoutput * S ((S ((t)+pfrep_index_left_pad_entry_graph)) * e) + (pfrep_value_left_pad_entry_graph))))))) -> (exists pfa_gap_left_pad_entry_bound. pfa_gap_left_pad_entry_bound + S (i) = (L)) -> (((exists ff_h_pfp_left_pad_entry_source. ff_h_pfp_left_pad_entry_source + S (a) = S ((S (i)) * c)) /\ exists ff_q_pfp_left_pad_entry_source. b = ff_q_pfp_left_pad_entry_source * S ((S (i)) * c) + (a))) -> (((exists ff_h_pfp_left_pad_entry_target. ff_h_pfp_left_pad_entry_target + S (r) = S ((S (t+i)) * e)) /\ exists ff_q_pfp_left_pad_entry_target. d = ff_q_pfp_left_pad_entry_target * S ((S (t+i)) * e) + (r))) -> r=a

Complete tactic proof in conservative notation

All 26 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

26 script commands · 5 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 L
  4. L4
    intro t
  5. L5
    intro d
  6. L6
    intro e
  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
03Separate the logical casesL14–14

Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.

  1. L14
    cases h
04Use earlier factsL15–24

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

  1. L15
    specialize beta_at_unique (d)
  2. L16
    specialize beta_at_unique (e)
  3. L17
    specialize beta_at_unique (t+i)
  4. L18
    specialize beta_at_unique (r)
  5. L19
    specialize beta_at_unique (a)
  6. L20
    apply beta_at_unique
  7. L21
    exact hr
  8. L22
    specialize h_right (i)
  9. L23
    specialize h_right (a)
  10. L24
    apply h_right
05Use earlier factsL25–26

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

  1. L25
    exact hi
  2. L26
    exact ha

Library-wide reading audit

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