PX005B

polynomial_zero_extended_left_pad_shift

Leading-zero padding preserves every shifted zero-extended coefficient, including indices outside the original finite prefix.

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. ∀ B. ∀ C. ∀ i. ∀ a. PolynomialLeftPad(b,c,L,t,B,C) → BetaZeroExtend(b,c,L,i,a) → BetaZeroExtend(B,C,t + L,t + i,a)

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

Definition DAG

BetaZeroExtend(b,c,L,i,a) · 2PolynomialLeftPad(b,c,L,t,d,e) · 1

Actual proof prerequisites

matrix_recursive_lt_add_left · checked external prerequisiteadd_le_add_left · checked external prerequisite
Original expanded first-order statement 
forall b c L t B C i a. (((forall pfp_repeat_index_shift_paddingzeros. (exists pfa_gap_shift_paddingzerosindex. pfa_gap_shift_paddingzerosindex + S (pfp_repeat_index_shift_paddingzeros) = (t)) -> (((exists ff_h_pfp_shift_paddingzerosentry. ff_h_pfp_shift_paddingzerosentry + S (0) = S ((S (pfp_repeat_index_shift_paddingzeros)) * C)) /\ exists ff_q_pfp_shift_paddingzerosentry. B = ff_q_pfp_shift_paddingzerosentry * S ((S (pfp_repeat_index_shift_paddingzeros)) * C) + (0)))) /\ ((forall pfrep_index_shift_padding pfrep_value_shift_padding. (exists pfa_gap_shift_paddingbound. pfa_gap_shift_paddingbound + S (pfrep_index_shift_padding) = (L)) -> (((exists ff_h_pfp_shift_paddinginput. ff_h_pfp_shift_paddinginput + S (pfrep_value_shift_padding) = S ((S (pfrep_index_shift_padding)) * c)) /\ exists ff_q_pfp_shift_paddinginput. b = ff_q_pfp_shift_paddinginput * S ((S (pfrep_index_shift_padding)) * c) + (pfrep_value_shift_padding))) -> (((exists ff_h_pfp_shift_paddingoutput. ff_h_pfp_shift_paddingoutput + S (pfrep_value_shift_padding) = S ((S ((t)+pfrep_index_shift_padding)) * C)) /\ exists ff_q_pfp_shift_paddingoutput. B = ff_q_pfp_shift_paddingoutput * S ((S ((t)+pfrep_index_shift_padding)) * C) + (pfrep_value_shift_padding))))))) -> ((((exists pfa_gap_shift_originalinside. pfa_gap_shift_originalinside + S (i) = (L)) /\ ((((exists ff_h_pfp_shift_originalentry. ff_h_pfp_shift_originalentry + S (a) = S ((S (i)) * c)) /\ exists ff_q_pfp_shift_originalentry. b = ff_q_pfp_shift_originalentry * S ((S (i)) * c) + (a)))))) \/ (((exists pfc_gap_shift_originaloutside. pfc_gap_shift_originaloutside+(L)=(i)) /\ (((a)=0))))) -> ((((exists pfa_gap_shift_resultinside. pfa_gap_shift_resultinside + S (t+i) = (t+L)) /\ ((((exists ff_h_pfp_shift_resultentry. ff_h_pfp_shift_resultentry + S (a) = S ((S (t+i)) * C)) /\ exists ff_q_pfp_shift_resultentry. B = ff_q_pfp_shift_resultentry * S ((S (t+i)) * C) + (a)))))) \/ (((exists pfc_gap_shift_resultoutside. pfc_gap_shift_resultoutside+(t+L)=(t+i)) /\ (((a)=0)))))

Complete tactic proof in conservative notation

All 34 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

34 script commands · 5 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 assumptionsL1–10

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 B
  6. L6
    intro C
  7. L7
    intro i
  8. L8
    intro a
  9. L9
    intro hpad
  10. L10
    intro hvalue
02Separate the logical casesL11–15

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

  1. L11
    cases hpad
  2. L12
    cases hvalue
  3. L13
    cases hvalue_left
  4. L14
    left
  5. L15
    split
03Use earlier factsL16–25

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

  1. L16
    specialize matrix_recursive_lt_add_left (i)
  2. L17
    specialize matrix_recursive_lt_add_left (L)
  3. L18
    specialize matrix_recursive_lt_add_left (t)
  4. L19
    apply matrix_recursive_lt_add_left
  5. L20
    exact hvalue_left_left
  6. L21
    specialize hpad_right (i)
  7. L22
    specialize hpad_right (a)
  8. L23
    apply hpad_right
  9. L24
    exact hvalue_left_left
  10. L25
    exact hvalue_left_right
04Separate the logical casesL26–28

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

  1. L26
    cases hvalue_right
  2. L27
    right
  3. L28
    split
05Use earlier factsL29–34

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

  1. L29
    specialize add_le_add_left (L)
  2. L30
    specialize add_le_add_left (i)
  3. L31
    specialize add_le_add_left (t)
  4. L32
    apply add_le_add_left
  5. L33
    exact hvalue_right_left
  6. L34
    exact hvalue_right_right

Library-wide reading audit

Original defined command ledger · 34 lines
  1. 0001intro b
  2. 0002intro c
  3. 0003intro L
  4. 0004intro t
  5. 0005intro B
  6. 0006intro C
  7. 0007intro i
  8. 0008intro a
  9. 0009intro hpad
  10. 0010intro hvalue
  11. 0011cases hpad
  12. 0012cases hvalue
  13. 0013cases hvalue_left
  14. 0014left
  15. 0015split
  16. 0016specialize matrix_recursive_lt_add_left (i)
  17. 0017specialize matrix_recursive_lt_add_left (L)
  18. 0018specialize matrix_recursive_lt_add_left (t)
  19. 0019apply matrix_recursive_lt_add_left
  20. 0020exact hvalue_left_left
  21. 0021specialize hpad_right (i)
  22. 0022specialize hpad_right (a)
  23. 0023apply hpad_right
  24. 0024exact hvalue_left_left
  25. 0025exact hvalue_left_right
  26. 0026cases hvalue_right
  27. 0027right
  28. 0028split
  29. 0029specialize add_le_add_left (L)
  30. 0030specialize add_le_add_left (i)
  31. 0031specialize add_le_add_left (t)
  32. 0032apply add_le_add_left
  33. 0033exact hvalue_right_left
  34. 0034exact hvalue_right_right