PX0018

prime_field_polynomial_zero_suffix_left_pad

A real suffix after an actual zero block gives the reverse decoding needed by genuine left padding.

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. Repeat(d,e,0,t) → PolynomialSuffix(d,e,t,b,c,L) → PolynomialLeftPad(b,c,L,t,d,e)

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

Definition DAG

PolynomialSuffix(b,c,t,d,e,M) · 1PolynomialLeftPad(b,c,L,t,d,e) · 1BetaAt(b,c,i,x) · 1Repeat(b,c,a,l) · 1

Actual proof prerequisites

beta_at_exists · checked external prerequisitebeta_at_unique · checked external prerequisite
Original expanded first-order statement 
forall b c L t d e. (forall pfp_repeat_index_suffix_to_pad_zero. (exists pfa_gap_suffix_to_pad_zeroindex. pfa_gap_suffix_to_pad_zeroindex + S (pfp_repeat_index_suffix_to_pad_zero) = (t)) -> (((exists ff_h_pfp_suffix_to_pad_zeroentry. ff_h_pfp_suffix_to_pad_zeroentry + S (0) = S ((S (pfp_repeat_index_suffix_to_pad_zero)) * e)) /\ exists ff_q_pfp_suffix_to_pad_zeroentry. d = ff_q_pfp_suffix_to_pad_zeroentry * S ((S (pfp_repeat_index_suffix_to_pad_zero)) * e) + (0)))) -> (forall pftrim_index_suffix_to_pad_suffix pftrim_value_suffix_to_pad_suffix. (exists pfa_gap_suffix_to_pad_suffixbound. pfa_gap_suffix_to_pad_suffixbound + S (pftrim_index_suffix_to_pad_suffix) = (L)) -> (((exists ff_h_pfp_suffix_to_pad_suffixsource. ff_h_pfp_suffix_to_pad_suffixsource + S (pftrim_value_suffix_to_pad_suffix) = S ((S ((t)+pftrim_index_suffix_to_pad_suffix)) * e)) /\ exists ff_q_pfp_suffix_to_pad_suffixsource. d = ff_q_pfp_suffix_to_pad_suffixsource * S ((S ((t)+pftrim_index_suffix_to_pad_suffix)) * e) + (pftrim_value_suffix_to_pad_suffix))) -> (((exists ff_h_pfp_suffix_to_pad_suffixoutput. ff_h_pfp_suffix_to_pad_suffixoutput + S (pftrim_value_suffix_to_pad_suffix) = S ((S (pftrim_index_suffix_to_pad_suffix)) * c)) /\ exists ff_q_pfp_suffix_to_pad_suffixoutput. b = ff_q_pfp_suffix_to_pad_suffixoutput * S ((S (pftrim_index_suffix_to_pad_suffix)) * c) + (pftrim_value_suffix_to_pad_suffix)))) -> (((forall pfp_repeat_index_suffix_to_pad_resultzeros. (exists pfa_gap_suffix_to_pad_resultzerosindex. pfa_gap_suffix_to_pad_resultzerosindex + S (pfp_repeat_index_suffix_to_pad_resultzeros) = (t)) -> (((exists ff_h_pfp_suffix_to_pad_resultzerosentry. ff_h_pfp_suffix_to_pad_resultzerosentry + S (0) = S ((S (pfp_repeat_index_suffix_to_pad_resultzeros)) * e)) /\ exists ff_q_pfp_suffix_to_pad_resultzerosentry. d = ff_q_pfp_suffix_to_pad_resultzerosentry * S ((S (pfp_repeat_index_suffix_to_pad_resultzeros)) * e) + (0)))) /\ ((forall pfrep_index_suffix_to_pad_result pfrep_value_suffix_to_pad_result. (exists pfa_gap_suffix_to_pad_resultbound. pfa_gap_suffix_to_pad_resultbound + S (pfrep_index_suffix_to_pad_result) = (L)) -> (((exists ff_h_pfp_suffix_to_pad_resultinput. ff_h_pfp_suffix_to_pad_resultinput + S (pfrep_value_suffix_to_pad_result) = S ((S (pfrep_index_suffix_to_pad_result)) * c)) /\ exists ff_q_pfp_suffix_to_pad_resultinput. b = ff_q_pfp_suffix_to_pad_resultinput * S ((S (pfrep_index_suffix_to_pad_result)) * c) + (pfrep_value_suffix_to_pad_result))) -> (((exists ff_h_pfp_suffix_to_pad_resultoutput. ff_h_pfp_suffix_to_pad_resultoutput + S (pfrep_value_suffix_to_pad_result) = S ((S ((t)+pfrep_index_suffix_to_pad_result)) * e)) /\ exists ff_q_pfp_suffix_to_pad_resultoutput. d = ff_q_pfp_suffix_to_pad_resultoutput * S ((S ((t)+pfrep_index_suffix_to_pad_result)) * e) + (pfrep_value_suffix_to_pad_result)))))))

Complete tactic proof in conservative notation

All 36 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

36 script commands · 10 reading checkpoints · 2 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 assumptionsL1–8

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 hz
  8. L8
    intro hs
02Separate the logical casesL9–9

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

  1. L9
    split
03Use earlier factsL10–10

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

  1. L10
    exact hz
04Fix variables and assumptionsL11–14

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

  1. L11
    intro i
  2. L12
    intro a
  3. L13
    intro hi
  4. L14
    intro ha
05Establish hvL15–19

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply beta at exists.

  1. L15
    have hv : ∃ r. BetaAt(d,e,t + i,r)Definitions: BetaAt(d,e,t + i,r)Original native command in the exact edition
  2. L16
    specialize beta_at_exists (d)
  3. L17
    specialize beta_at_exists (e)
  4. L18
    specialize beta_at_exists (t+i)
  5. L19
    apply beta_at_exists
06Separate the logical casesL20–20

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

  1. L20
    cases hv
07Establish heqL21–30

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply beta at unique.

  1. L21
    have heq : x=a
  2. L22
    specialize beta_at_unique (b)
  3. L23
    specialize beta_at_unique (c)
  4. L24
    specialize beta_at_unique (i)
  5. L25
    specialize beta_at_unique (x)
  6. L26
    specialize beta_at_unique (a)
  7. L27
    apply beta_at_unique
  8. L28
    specialize hs (i)
  9. L29
    specialize hs (x)
  10. L30
    apply hs
08Use earlier factsL31–33

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

  1. L31
    exact hi
  2. L32
    exact hv_witness
  3. L33
    exact ha
09Calculate and transport equalitiesL34–35

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

  1. L34
    rewrite heq at hv_witness
  2. L35
    rewrite heq at hv_witness
10Use earlier factsL36–36

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

  1. L36
    exact hv_witness

Library-wide reading audit

Original defined command ledger · 36 lines
  1. 0001intro b
  2. 0002intro c
  3. 0003intro L
  4. 0004intro t
  5. 0005intro d
  6. 0006intro e
  7. 0007intro hz
  8. 0008intro hs
  9. 0009split
  10. 0010exact hz
  11. 0011intro i
  12. 0012intro a
  13. 0013intro hi
  14. 0014intro ha
  15. 0015have hv : ∃ r. BetaAt(d,e,t + i,r)
  16. 0016specialize beta_at_exists (d)
  17. 0017specialize beta_at_exists (e)
  18. 0018specialize beta_at_exists (t+i)
  19. 0019apply beta_at_exists
  20. 0020cases hv
  21. 0021have heq : x=a
  22. 0022specialize beta_at_unique (b)
  23. 0023specialize beta_at_unique (c)
  24. 0024specialize beta_at_unique (i)
  25. 0025specialize beta_at_unique (x)
  26. 0026specialize beta_at_unique (a)
  27. 0027apply beta_at_unique
  28. 0028specialize hs (i)
  29. 0029specialize hs (x)
  30. 0030apply hs
  31. 0031exact hi
  32. 0032exact hv_witness
  33. 0033exact ha
  34. 0034rewrite heq at hv_witness
  35. 0035rewrite heq at hv_witness
  36. 0036exact hv_witness