PX0019

prime_field_polynomial_trim_left_pad

Actual trimming identifies its input as the retained coefficient prefix with exactly the removed leading zeros restored.

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) → PolynomialLeftPad(d,e,M,t,b,c)

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

Definition DAG

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

Actual proof prerequisites

prime_field_polynomial_zero_suffix_left_pad
Original expanded first-order statement 
forall p b c L t d e M. ((((L)=(t)+(M)) /\ (((forall fom_index_pfp_trim_to_pad_sourceinput. (exists fom_gap_pfp_trim_to_pad_sourceinput_index_bound. fom_gap_pfp_trim_to_pad_sourceinput_index_bound + S (fom_index_pfp_trim_to_pad_sourceinput) = L) -> exists fom_value_pfp_trim_to_pad_sourceinput. ((((exists fom_beta_height_pfp_trim_to_pad_sourceinput_entry. fom_beta_height_pfp_trim_to_pad_sourceinput_entry + S (fom_value_pfp_trim_to_pad_sourceinput) = S ((S (fom_index_pfp_trim_to_pad_sourceinput)) * c)) /\ exists fom_beta_quotient_pfp_trim_to_pad_sourceinput_entry. b = fom_beta_quotient_pfp_trim_to_pad_sourceinput_entry * S ((S (fom_index_pfp_trim_to_pad_sourceinput)) * c) + (fom_value_pfp_trim_to_pad_sourceinput))) /\ (exists fom_gap_pfp_trim_to_pad_sourceinput_value_bound. fom_gap_pfp_trim_to_pad_sourceinput_value_bound + S (fom_value_pfp_trim_to_pad_sourceinput) = p))) /\ (((forall pfp_repeat_index_trim_to_pad_sourceremoved. (exists pfa_gap_trim_to_pad_sourceremovedindex. pfa_gap_trim_to_pad_sourceremovedindex + S (pfp_repeat_index_trim_to_pad_sourceremoved) = (t)) -> (((exists ff_h_pfp_trim_to_pad_sourceremovedentry. ff_h_pfp_trim_to_pad_sourceremovedentry + S (0) = S ((S (pfp_repeat_index_trim_to_pad_sourceremoved)) * c)) /\ exists ff_q_pfp_trim_to_pad_sourceremovedentry. b = ff_q_pfp_trim_to_pad_sourceremovedentry * S ((S (pfp_repeat_index_trim_to_pad_sourceremoved)) * c) + (0)))) /\ (((forall pftrim_index_trim_to_pad_sourcesuffix pftrim_value_trim_to_pad_sourcesuffix. (exists pfa_gap_trim_to_pad_sourcesuffixbound. pfa_gap_trim_to_pad_sourcesuffixbound + S (pftrim_index_trim_to_pad_sourcesuffix) = (M)) -> (((exists ff_h_pfp_trim_to_pad_sourcesuffixsource. ff_h_pfp_trim_to_pad_sourcesuffixsource + S (pftrim_value_trim_to_pad_sourcesuffix) = S ((S ((t)+pftrim_index_trim_to_pad_sourcesuffix)) * c)) /\ exists ff_q_pfp_trim_to_pad_sourcesuffixsource. b = ff_q_pfp_trim_to_pad_sourcesuffixsource * S ((S ((t)+pftrim_index_trim_to_pad_sourcesuffix)) * c) + (pftrim_value_trim_to_pad_sourcesuffix))) -> (((exists ff_h_pfp_trim_to_pad_sourcesuffixoutput. ff_h_pfp_trim_to_pad_sourcesuffixoutput + S (pftrim_value_trim_to_pad_sourcesuffix) = S ((S (pftrim_index_trim_to_pad_sourcesuffix)) * e)) /\ exists ff_q_pfp_trim_to_pad_sourcesuffixoutput. d = ff_q_pfp_trim_to_pad_sourcesuffixoutput * S ((S (pftrim_index_trim_to_pad_sourcesuffix)) * e) + (pftrim_value_trim_to_pad_sourcesuffix)))) /\ (((M)=0 \/ (exists pftrim_leading_trim_to_pad_sourcenormal. ((((exists ff_h_pfp_trim_to_pad_sourcenormalentry. ff_h_pfp_trim_to_pad_sourcenormalentry + S (pftrim_leading_trim_to_pad_sourcenormal) = S ((S (0)) * e)) /\ exists ff_q_pfp_trim_to_pad_sourcenormalentry. d = ff_q_pfp_trim_to_pad_sourcenormalentry * S ((S (0)) * e) + (pftrim_leading_trim_to_pad_sourcenormal))) /\ ((~(pftrim_leading_trim_to_pad_sourcenormal=0))))))))))))))) -> (((forall pfp_repeat_index_trim_to_pad_resultzeros. (exists pfa_gap_trim_to_pad_resultzerosindex. pfa_gap_trim_to_pad_resultzerosindex + S (pfp_repeat_index_trim_to_pad_resultzeros) = (t)) -> (((exists ff_h_pfp_trim_to_pad_resultzerosentry. ff_h_pfp_trim_to_pad_resultzerosentry + S (0) = S ((S (pfp_repeat_index_trim_to_pad_resultzeros)) * c)) /\ exists ff_q_pfp_trim_to_pad_resultzerosentry. b = ff_q_pfp_trim_to_pad_resultzerosentry * S ((S (pfp_repeat_index_trim_to_pad_resultzeros)) * c) + (0)))) /\ ((forall pfrep_index_trim_to_pad_result pfrep_value_trim_to_pad_result. (exists pfa_gap_trim_to_pad_resultbound. pfa_gap_trim_to_pad_resultbound + S (pfrep_index_trim_to_pad_result) = (M)) -> (((exists ff_h_pfp_trim_to_pad_resultinput. ff_h_pfp_trim_to_pad_resultinput + S (pfrep_value_trim_to_pad_result) = S ((S (pfrep_index_trim_to_pad_result)) * e)) /\ exists ff_q_pfp_trim_to_pad_resultinput. d = ff_q_pfp_trim_to_pad_resultinput * S ((S (pfrep_index_trim_to_pad_result)) * e) + (pfrep_value_trim_to_pad_result))) -> (((exists ff_h_pfp_trim_to_pad_resultoutput. ff_h_pfp_trim_to_pad_resultoutput + S (pfrep_value_trim_to_pad_result) = S ((S ((t)+pfrep_index_trim_to_pad_result)) * c)) /\ exists ff_q_pfp_trim_to_pad_resultoutput. b = ff_q_pfp_trim_to_pad_resultoutput * S ((S ((t)+pfrep_index_trim_to_pad_result)) * c) + (pfrep_value_trim_to_pad_result)))))))

Complete tactic proof in conservative notation

All 22 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

22 script commands · 3 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.

Named ingredients (1)
prime field polynomial zero suffix 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
02Separate the logical casesL10–13

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

  1. L10
    cases h
  2. L11
    cases h_right
  3. L12
    cases h_right_right
  4. L13
    cases h_right_right_right
03Use earlier factsL14–22

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

  1. L14
    specialize prime_field_polynomial_zero_suffix_left_pad (d)
  2. L15
    specialize prime_field_polynomial_zero_suffix_left_pad (e)
  3. L16
    specialize prime_field_polynomial_zero_suffix_left_pad (M)
  4. L17
    specialize prime_field_polynomial_zero_suffix_left_pad (t)
  5. L18
    specialize prime_field_polynomial_zero_suffix_left_pad (b)
  6. L19
    specialize prime_field_polynomial_zero_suffix_left_pad (c)
  7. L20
    apply prime_field_polynomial_zero_suffix_left_pad
  8. L21
    exact h_right_right_left
  9. L22
    exact h_right_right_right_left

Library-wide reading audit

Original defined command ledger · 22 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. 0010cases h
  11. 0011cases h_right
  12. 0012cases h_right_right
  13. 0013cases h_right_right_right
  14. 0014specialize prime_field_polynomial_zero_suffix_left_pad (d)
  15. 0015specialize prime_field_polynomial_zero_suffix_left_pad (e)
  16. 0016specialize prime_field_polynomial_zero_suffix_left_pad (M)
  17. 0017specialize prime_field_polynomial_zero_suffix_left_pad (t)
  18. 0018specialize prime_field_polynomial_zero_suffix_left_pad (b)
  19. 0019specialize prime_field_polynomial_zero_suffix_left_pad (c)
  20. 0020apply prime_field_polynomial_zero_suffix_left_pad
  21. 0021exact h_right_right_left
  22. 0022exact h_right_right_right_left