PX001C

prime_field_polynomial_trim_equivalent

The actually constructed trimmed representation has exactly the same formal coefficients as its original input.

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

∀ p. ∀ b. ∀ c. ∀ L. ∀ t. ∀ d. ∀ e. ∀ M. FpPolynomialTrim(p,b,c,L,t,d,e,M) → PolynomialEquivalent(b,c,L,d,e,M)

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

Definition DAG

FpPolynomialTrim(p,b,c,L,t,d,e,M) · 2PolynomialEquivalent(b,c,L,d,e,M) · 1

Actual proof prerequisites

prime_field_polynomial_equivalent_symmetricprime_field_polynomial_left_pad_equivalentprime_field_polynomial_trim_left_pad
Original expanded first-order statement 
forall p b c L t d e M. ((((L)=(t)+(M)) /\ (((forall fom_index_pfp_trim_equivalence_sourceinput. (exists fom_gap_pfp_trim_equivalence_sourceinput_index_bound. fom_gap_pfp_trim_equivalence_sourceinput_index_bound + S (fom_index_pfp_trim_equivalence_sourceinput) = L) -> exists fom_value_pfp_trim_equivalence_sourceinput. ((((exists fom_beta_height_pfp_trim_equivalence_sourceinput_entry. fom_beta_height_pfp_trim_equivalence_sourceinput_entry + S (fom_value_pfp_trim_equivalence_sourceinput) = S ((S (fom_index_pfp_trim_equivalence_sourceinput)) * c)) /\ exists fom_beta_quotient_pfp_trim_equivalence_sourceinput_entry. b = fom_beta_quotient_pfp_trim_equivalence_sourceinput_entry * S ((S (fom_index_pfp_trim_equivalence_sourceinput)) * c) + (fom_value_pfp_trim_equivalence_sourceinput))) /\ (exists fom_gap_pfp_trim_equivalence_sourceinput_value_bound. fom_gap_pfp_trim_equivalence_sourceinput_value_bound + S (fom_value_pfp_trim_equivalence_sourceinput) = p))) /\ (((forall pfp_repeat_index_trim_equivalence_sourceremoved. (exists pfa_gap_trim_equivalence_sourceremovedindex. pfa_gap_trim_equivalence_sourceremovedindex + S (pfp_repeat_index_trim_equivalence_sourceremoved) = (t)) -> (((exists ff_h_pfp_trim_equivalence_sourceremovedentry. ff_h_pfp_trim_equivalence_sourceremovedentry + S (0) = S ((S (pfp_repeat_index_trim_equivalence_sourceremoved)) * c)) /\ exists ff_q_pfp_trim_equivalence_sourceremovedentry. b = ff_q_pfp_trim_equivalence_sourceremovedentry * S ((S (pfp_repeat_index_trim_equivalence_sourceremoved)) * c) + (0)))) /\ (((forall pftrim_index_trim_equivalence_sourcesuffix pftrim_value_trim_equivalence_sourcesuffix. (exists pfa_gap_trim_equivalence_sourcesuffixbound. pfa_gap_trim_equivalence_sourcesuffixbound + S (pftrim_index_trim_equivalence_sourcesuffix) = (M)) -> (((exists ff_h_pfp_trim_equivalence_sourcesuffixsource. ff_h_pfp_trim_equivalence_sourcesuffixsource + S (pftrim_value_trim_equivalence_sourcesuffix) = S ((S ((t)+pftrim_index_trim_equivalence_sourcesuffix)) * c)) /\ exists ff_q_pfp_trim_equivalence_sourcesuffixsource. b = ff_q_pfp_trim_equivalence_sourcesuffixsource * S ((S ((t)+pftrim_index_trim_equivalence_sourcesuffix)) * c) + (pftrim_value_trim_equivalence_sourcesuffix))) -> (((exists ff_h_pfp_trim_equivalence_sourcesuffixoutput. ff_h_pfp_trim_equivalence_sourcesuffixoutput + S (pftrim_value_trim_equivalence_sourcesuffix) = S ((S (pftrim_index_trim_equivalence_sourcesuffix)) * e)) /\ exists ff_q_pfp_trim_equivalence_sourcesuffixoutput. d = ff_q_pfp_trim_equivalence_sourcesuffixoutput * S ((S (pftrim_index_trim_equivalence_sourcesuffix)) * e) + (pftrim_value_trim_equivalence_sourcesuffix)))) /\ (((M)=0 \/ (exists pftrim_leading_trim_equivalence_sourcenormal. ((((exists ff_h_pfp_trim_equivalence_sourcenormalentry. ff_h_pfp_trim_equivalence_sourcenormalentry + S (pftrim_leading_trim_equivalence_sourcenormal) = S ((S (0)) * e)) /\ exists ff_q_pfp_trim_equivalence_sourcenormalentry. d = ff_q_pfp_trim_equivalence_sourcenormalentry * S ((S (0)) * e) + (pftrim_leading_trim_equivalence_sourcenormal))) /\ ((~(pftrim_leading_trim_equivalence_sourcenormal=0))))))))))))))) -> (forall pfrep_power_trim_equivalence_result pfrep_left_trim_equivalence_result pfrep_right_trim_equivalence_result. ((exists pfrep_position_trim_equivalence_resultfirst. ((pfrep_position_trim_equivalence_resultfirst+S (pfrep_power_trim_equivalence_result)=(L)) /\ ((((exists ff_h_pfp_trim_equivalence_resultfirstentry. ff_h_pfp_trim_equivalence_resultfirstentry + S (pfrep_left_trim_equivalence_result) = S ((S (pfrep_position_trim_equivalence_resultfirst)) * c)) /\ exists ff_q_pfp_trim_equivalence_resultfirstentry. b = ff_q_pfp_trim_equivalence_resultfirstentry * S ((S (pfrep_position_trim_equivalence_resultfirst)) * c) + (pfrep_left_trim_equivalence_result)))))) \/ (((exists pfrep_gap_trim_equivalence_resultfirstoutside. pfrep_gap_trim_equivalence_resultfirstoutside+(L)=(pfrep_power_trim_equivalence_result)) /\ (((pfrep_left_trim_equivalence_result)=0))))) -> ((exists pfrep_position_trim_equivalence_resultsecond. ((pfrep_position_trim_equivalence_resultsecond+S (pfrep_power_trim_equivalence_result)=(M)) /\ ((((exists ff_h_pfp_trim_equivalence_resultsecondentry. ff_h_pfp_trim_equivalence_resultsecondentry + S (pfrep_right_trim_equivalence_result) = S ((S (pfrep_position_trim_equivalence_resultsecond)) * e)) /\ exists ff_q_pfp_trim_equivalence_resultsecondentry. d = ff_q_pfp_trim_equivalence_resultsecondentry * S ((S (pfrep_position_trim_equivalence_resultsecond)) * e) + (pfrep_right_trim_equivalence_result)))))) \/ (((exists pfrep_gap_trim_equivalence_resultsecondoutside. pfrep_gap_trim_equivalence_resultsecondoutside+(M)=(pfrep_power_trim_equivalence_result)) /\ (((pfrep_right_trim_equivalence_result)=0))))) -> pfrep_left_trim_equivalence_result=pfrep_right_trim_equivalence_result)

Complete tactic proof in conservative notation

All 41 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

41 script commands · 7 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 (3)
prime field polynomial equivalent symmetricprime field polynomial left pad equivalentprime field polynomial trim left pad
01Fix variables and assumptionsL1–9

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

  1. L1
    intro p
  2. L2
    intro b
  3. L3
    intro c
  4. L4
    intro L
  5. L5
    intro t
  6. L6
    intro d
  7. L7
    intro e
  8. L8
    intro M
  9. L9
    intro h
02Establish hcL10–11

Establish this local claim before using it. It is not an additional assumption.

  1. L10
    have hc : FpPolynomialTrim(p,b,c,L,t,d,e,M)Definitions: FpPolynomialTrim(p,b,c,L,t,d,e,M)Original native command in the exact edition
  2. L11
    exact h
03Separate the logical casesL12–15

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

  1. L12
    cases hc
  2. L13
    cases hc_right
  3. L14
    cases hc_right_right
  4. L15
    cases hc_right_right_right
04Calculate and transport equalitiesL16–17

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

  1. L16
    rewrite hc_left
  2. L17
    rewrite hc_left
05Use earlier factsL18–27

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

  1. L18
    specialize prime_field_polynomial_equivalent_symmetric (d)
  2. L19
    specialize prime_field_polynomial_equivalent_symmetric (e)
  3. L20
    specialize prime_field_polynomial_equivalent_symmetric (M)
  4. L21
    specialize prime_field_polynomial_equivalent_symmetric (b)
  5. L22
    specialize prime_field_polynomial_equivalent_symmetric (c)
  6. L23
    specialize prime_field_polynomial_equivalent_symmetric (t+M)
  7. L24
    apply prime_field_polynomial_equivalent_symmetric
  8. L25
    specialize prime_field_polynomial_left_pad_equivalent (d)
  9. L26
    specialize prime_field_polynomial_left_pad_equivalent (e)
  10. L27
    specialize prime_field_polynomial_left_pad_equivalent (M)
06Use earlier factsL28–37

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

  1. L28
    specialize prime_field_polynomial_left_pad_equivalent (t)
  2. L29
    specialize prime_field_polynomial_left_pad_equivalent (b)
  3. L30
    specialize prime_field_polynomial_left_pad_equivalent (c)
  4. L31
    apply prime_field_polynomial_left_pad_equivalent
  5. L32
    specialize prime_field_polynomial_trim_left_pad (p)
  6. L33
    specialize prime_field_polynomial_trim_left_pad (b)
  7. L34
    specialize prime_field_polynomial_trim_left_pad (c)
  8. L35
    specialize prime_field_polynomial_trim_left_pad (L)
  9. L36
    specialize prime_field_polynomial_trim_left_pad (t)
  10. L37
    specialize prime_field_polynomial_trim_left_pad (d)
07Use earlier factsL38–41

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

  1. L38
    specialize prime_field_polynomial_trim_left_pad (e)
  2. L39
    specialize prime_field_polynomial_trim_left_pad (M)
  3. L40
    apply prime_field_polynomial_trim_left_pad
  4. L41
    exact h

Library-wide reading audit

Original defined command ledger · 41 lines
  1. 0001intro p
  2. 0002intro b
  3. 0003intro c
  4. 0004intro L
  5. 0005intro t
  6. 0006intro d
  7. 0007intro e
  8. 0008intro M
  9. 0009intro h
  10. 0010have hc : FpPolynomialTrim(p,b,c,L,t,d,e,M)
  11. 0011exact h
  12. 0012cases hc
  13. 0013cases hc_right
  14. 0014cases hc_right_right
  15. 0015cases hc_right_right_right
  16. 0016rewrite hc_left
  17. 0017rewrite hc_left
  18. 0018specialize prime_field_polynomial_equivalent_symmetric (d)
  19. 0019specialize prime_field_polynomial_equivalent_symmetric (e)
  20. 0020specialize prime_field_polynomial_equivalent_symmetric (M)
  21. 0021specialize prime_field_polynomial_equivalent_symmetric (b)
  22. 0022specialize prime_field_polynomial_equivalent_symmetric (c)
  23. 0023specialize prime_field_polynomial_equivalent_symmetric (t+M)
  24. 0024apply prime_field_polynomial_equivalent_symmetric
  25. 0025specialize prime_field_polynomial_left_pad_equivalent (d)
  26. 0026specialize prime_field_polynomial_left_pad_equivalent (e)
  27. 0027specialize prime_field_polynomial_left_pad_equivalent (M)
  28. 0028specialize prime_field_polynomial_left_pad_equivalent (t)
  29. 0029specialize prime_field_polynomial_left_pad_equivalent (b)
  30. 0030specialize prime_field_polynomial_left_pad_equivalent (c)
  31. 0031apply prime_field_polynomial_left_pad_equivalent
  32. 0032specialize prime_field_polynomial_trim_left_pad (p)
  33. 0033specialize prime_field_polynomial_trim_left_pad (b)
  34. 0034specialize prime_field_polynomial_trim_left_pad (c)
  35. 0035specialize prime_field_polynomial_trim_left_pad (L)
  36. 0036specialize prime_field_polynomial_trim_left_pad (t)
  37. 0037specialize prime_field_polynomial_trim_left_pad (d)
  38. 0038specialize prime_field_polynomial_trim_left_pad (e)
  39. 0039specialize prime_field_polynomial_trim_left_pad (M)
  40. 0040apply prime_field_polynomial_trim_left_pad
  41. 0041exact h