PG0004

prime_field_polynomial_shift_zero_prefix

The actual shift of an all-zero prefix is again all zero, including the length-one shift of an empty input.

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

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

Definition DAG

PolynomialShift(b,c,L,d,e) · 1Lt(a,b) · 1Repeat(b,c,a,l) · 2

Actual proof prerequisites

finite_lt_succ_eq_or_lt · checked external prerequisite
Original expanded first-order statement 
forall b c L d e. (forall pfp_repeat_index_shift_zero_source. (exists pfa_gap_shift_zero_sourceindex. pfa_gap_shift_zero_sourceindex + S (pfp_repeat_index_shift_zero_source) = (L)) -> (((exists ff_h_pfp_shift_zero_sourceentry. ff_h_pfp_shift_zero_sourceentry + S (0) = S ((S (pfp_repeat_index_shift_zero_source)) * c)) /\ exists ff_q_pfp_shift_zero_sourceentry. b = ff_q_pfp_shift_zero_sourceentry * S ((S (pfp_repeat_index_shift_zero_source)) * c) + (0)))) -> (((forall mdr_i_pfp_shift_zero_relationprefix mdr_a_pfp_shift_zero_relationprefix. (exists mdr_gap_pfp_shift_zero_relationprefixb. mdr_gap_pfp_shift_zero_relationprefixb + S (mdr_i_pfp_shift_zero_relationprefix) = (L)) -> (((exists ff_h_mdr_pfp_shift_zero_relationprefixo. ff_h_mdr_pfp_shift_zero_relationprefixo + S (mdr_a_pfp_shift_zero_relationprefix) = S ((S (mdr_i_pfp_shift_zero_relationprefix)) * c)) /\ exists ff_q_mdr_pfp_shift_zero_relationprefixo. b = ff_q_mdr_pfp_shift_zero_relationprefixo * S ((S (mdr_i_pfp_shift_zero_relationprefix)) * c) + (mdr_a_pfp_shift_zero_relationprefix))) -> (((exists ff_h_mdr_pfp_shift_zero_relationprefixn. ff_h_mdr_pfp_shift_zero_relationprefixn + S (mdr_a_pfp_shift_zero_relationprefix) = S ((S (mdr_i_pfp_shift_zero_relationprefix)) * e)) /\ exists ff_q_mdr_pfp_shift_zero_relationprefixn. d = ff_q_mdr_pfp_shift_zero_relationprefixn * S ((S (mdr_i_pfp_shift_zero_relationprefix)) * e) + (mdr_a_pfp_shift_zero_relationprefix)))) /\ ((((exists ff_h_pfp_shift_zero_relationlast. ff_h_pfp_shift_zero_relationlast + S (0) = S ((S (L)) * e)) /\ exists ff_q_pfp_shift_zero_relationlast. d = ff_q_pfp_shift_zero_relationlast * S ((S (L)) * e) + (0)))))) -> (forall pfp_repeat_index_shift_zero_result. (exists pfa_gap_shift_zero_resultindex. pfa_gap_shift_zero_resultindex + S (pfp_repeat_index_shift_zero_result) = (S L)) -> (((exists ff_h_pfp_shift_zero_resultentry. ff_h_pfp_shift_zero_resultentry + S (0) = S ((S (pfp_repeat_index_shift_zero_result)) * e)) /\ exists ff_q_pfp_shift_zero_resultentry. d = ff_q_pfp_shift_zero_resultentry * S ((S (pfp_repeat_index_shift_zero_result)) * e) + (0))))

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 · 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.

01Fix variables and assumptionsL1–7

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 hz
  7. L7
    intro hs
02Separate the logical casesL8–8

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

  1. L8
    cases hs
03Fix variables and assumptionsL9–10

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

  1. L9
    intro i
  2. L10
    intro hi
04Establish hoL11–15

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply finite lt succ eq or lt.

  1. L11
    have ho : i = L ∨ Lt(i,L)Definitions: Lt(i,L)Original native command in the exact edition
  2. L12
    specialize finite_lt_succ_eq_or_lt (L)
  3. L13
    specialize finite_lt_succ_eq_or_lt (i)
  4. L14
    apply finite_lt_succ_eq_or_lt
  5. L15
    exact hi
05Separate the logical casesL16–16

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

  1. L16
    cases ho
06Calculate and transport equalitiesL17–18

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

  1. L17
    rewrite ho_left
  2. L18
    rewrite ho_left
07Use earlier factsL19–26

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

  1. L19
    exact hs_right
  2. L20
    specialize hs_left (i)
  3. L21
    specialize hs_left (0)
  4. L22
    apply hs_left
  5. L23
    exact ho_right
  6. L24
    specialize hz (i)
  7. L25
    apply hz
  8. L26
    exact ho_right

Library-wide reading audit

Original defined command ledger · 26 lines
  1. 0001intro b
  2. 0002intro c
  3. 0003intro L
  4. 0004intro d
  5. 0005intro e
  6. 0006intro hz
  7. 0007intro hs
  8. 0008cases hs
  9. 0009intro i
  10. 0010intro hi
  11. 0011have ho : i = L ∨ Lt(i,L)
  12. 0012specialize finite_lt_succ_eq_or_lt (L)
  13. 0013specialize finite_lt_succ_eq_or_lt (i)
  14. 0014apply finite_lt_succ_eq_or_lt
  15. 0015exact hi
  16. 0016cases ho
  17. 0017rewrite ho_left
  18. 0018rewrite ho_left
  19. 0019exact hs_right
  20. 0020specialize hs_left (i)
  21. 0021specialize hs_left (0)
  22. 0022apply hs_left
  23. 0023exact ho_right
  24. 0024specialize hz (i)
  25. 0025apply hz
  26. 0026exact ho_right