PG000E

prime_field_polynomial_shift_power_zero

The actual constant coefficient of a trailing-zero shift is zero, even when the original representation is empty.

Alpha v34 checked-use · first admitted v34 · 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 products and sums use actual beta-coded coefficients; equality is formal coefficient equivalence, not equality of evaluations or raw codes. The zero gcd is included. Uniqueness is up to formal equivalence, not unique Bézout coefficients. This proves the polynomial gcd component, not G091 irreducible-polynomial existence or arbitrary prime-power-field construction.

Exact theorem in conservative defined notation

∀ b. ∀ c. ∀ L. ∀ d. ∀ e. PolynomialShift(b,c,L,d,e) → PolynomialPowerCoefficient(d,e,S L,0,0)

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

Definition DAG

PolynomialPowerCoefficient(b,c,L,k,a) · 1PolynomialShift(b,c,L,d,e) · 1

Actual proof prerequisites

none
Original expanded first-order statement 
forall b c L d e. (((forall mdr_i_pfp_shift_power_zero_relationprefix mdr_a_pfp_shift_power_zero_relationprefix. (exists mdr_gap_pfp_shift_power_zero_relationprefixb. mdr_gap_pfp_shift_power_zero_relationprefixb + S (mdr_i_pfp_shift_power_zero_relationprefix) = (L)) -> (((exists ff_h_mdr_pfp_shift_power_zero_relationprefixo. ff_h_mdr_pfp_shift_power_zero_relationprefixo + S (mdr_a_pfp_shift_power_zero_relationprefix) = S ((S (mdr_i_pfp_shift_power_zero_relationprefix)) * c)) /\ exists ff_q_mdr_pfp_shift_power_zero_relationprefixo. b = ff_q_mdr_pfp_shift_power_zero_relationprefixo * S ((S (mdr_i_pfp_shift_power_zero_relationprefix)) * c) + (mdr_a_pfp_shift_power_zero_relationprefix))) -> (((exists ff_h_mdr_pfp_shift_power_zero_relationprefixn. ff_h_mdr_pfp_shift_power_zero_relationprefixn + S (mdr_a_pfp_shift_power_zero_relationprefix) = S ((S (mdr_i_pfp_shift_power_zero_relationprefix)) * e)) /\ exists ff_q_mdr_pfp_shift_power_zero_relationprefixn. d = ff_q_mdr_pfp_shift_power_zero_relationprefixn * S ((S (mdr_i_pfp_shift_power_zero_relationprefix)) * e) + (mdr_a_pfp_shift_power_zero_relationprefix)))) /\ ((((exists ff_h_pfp_shift_power_zero_relationlast. ff_h_pfp_shift_power_zero_relationlast + S (0) = S ((S (L)) * e)) /\ exists ff_q_pfp_shift_power_zero_relationlast. d = ff_q_pfp_shift_power_zero_relationlast * S ((S (L)) * e) + (0)))))) -> ((exists pfrep_position_shift_power_zero_result. ((pfrep_position_shift_power_zero_result+S (0)=(S L)) /\ ((((exists ff_h_pfp_shift_power_zero_resultentry. ff_h_pfp_shift_power_zero_resultentry + S (0) = S ((S (pfrep_position_shift_power_zero_result)) * e)) /\ exists ff_q_pfp_shift_power_zero_resultentry. d = ff_q_pfp_shift_power_zero_resultentry * S ((S (pfrep_position_shift_power_zero_result)) * e) + (0)))))) \/ (((exists pfrep_gap_shift_power_zero_resultoutside. pfrep_gap_shift_power_zero_resultoutside+(S L)=(0)) /\ (((0)=0)))))

Complete tactic proof in conservative notation

All 12 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

12 script commands · 6 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 assumptionsL1–6

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 d
  5. L5
    intro e
  6. L6
    intro hs
02Separate the logical casesL7–8

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

  1. L7
    cases hs
  2. L8
    left
03Construct an explicit witnessL9–9

Supply the displayed value, then prove that it has the required property.

  1. L9
    exists L
04Separate the logical casesL10–10

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

  1. L10
    split
05Calculate and transport equalitiesL11–11

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

  1. L11
    simp
06Use earlier factsL12–12

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

  1. L12
    exact hs_right

Library-wide reading audit

Original defined command ledger · 12 lines
  1. 0001intro b
  2. 0002intro c
  3. 0003intro L
  4. 0004intro d
  5. 0005intro e
  6. 0006intro hs
  7. 0007cases hs
  8. 0008left
  9. 0009exists L
  10. 0010split
  11. 0011simp
  12. 0012exact hs_right