PX005C

polynomial_zero_extended_left_pad_before

Every position in the actual added leading block has zero extended value zero; no source coefficient is read.

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. PolynomialLeftPad(b,c,L,t,B,C) → Lt(i,t) → BetaZeroExtend(B,C,t + L,i,0)

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

Definition DAG

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

Actual proof prerequisites

add_assoc · checked external prerequisiteadd_comm · checked external prerequisiteadd_succ_left · checked external prerequisite
Original expanded first-order statement 
forall b c L t B C i. (((forall pfp_repeat_index_before_paddingzeros. (exists pfa_gap_before_paddingzerosindex. pfa_gap_before_paddingzerosindex + S (pfp_repeat_index_before_paddingzeros) = (t)) -> (((exists ff_h_pfp_before_paddingzerosentry. ff_h_pfp_before_paddingzerosentry + S (0) = S ((S (pfp_repeat_index_before_paddingzeros)) * C)) /\ exists ff_q_pfp_before_paddingzerosentry. B = ff_q_pfp_before_paddingzerosentry * S ((S (pfp_repeat_index_before_paddingzeros)) * C) + (0)))) /\ ((forall pfrep_index_before_padding pfrep_value_before_padding. (exists pfa_gap_before_paddingbound. pfa_gap_before_paddingbound + S (pfrep_index_before_padding) = (L)) -> (((exists ff_h_pfp_before_paddinginput. ff_h_pfp_before_paddinginput + S (pfrep_value_before_padding) = S ((S (pfrep_index_before_padding)) * c)) /\ exists ff_q_pfp_before_paddinginput. b = ff_q_pfp_before_paddinginput * S ((S (pfrep_index_before_padding)) * c) + (pfrep_value_before_padding))) -> (((exists ff_h_pfp_before_paddingoutput. ff_h_pfp_before_paddingoutput + S (pfrep_value_before_padding) = S ((S ((t)+pfrep_index_before_padding)) * C)) /\ exists ff_q_pfp_before_paddingoutput. B = ff_q_pfp_before_paddingoutput * S ((S ((t)+pfrep_index_before_padding)) * C) + (pfrep_value_before_padding))))))) -> (exists pfa_gap_before_bound. pfa_gap_before_bound + S (i) = (t)) -> ((((exists pfa_gap_before_zeroinside. pfa_gap_before_zeroinside + S (i) = (t+L)) /\ ((((exists ff_h_pfp_before_zeroentry. ff_h_pfp_before_zeroentry + S (0) = S ((S (i)) * C)) /\ exists ff_q_pfp_before_zeroentry. B = ff_q_pfp_before_zeroentry * S ((S (i)) * C) + (0)))))) \/ (((exists pfc_gap_before_zerooutside. pfc_gap_before_zerooutside+(t+L)=(i)) /\ (((0)=0)))))

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 · 7 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 assumptionsL1–9

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 hpad
  9. L9
    intro hi
02Separate the logical casesL10–13

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

  1. L10
    cases hpad
  2. L11
    left
  3. L12
    split
  4. L13
    cases hi
03Construct an explicit witnessL14–14

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

  1. L14
    exists x+L
04Calculate and transport equalitiesL15–17

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

  1. L15
    trans (x+S i)+L
  2. L16
    simp [add_assoc,add_comm,add_succ_left]
  3. L17
    congr
05Use earlier factsL18–18

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

  1. L18
    exact hi_witness
06Calculate and transport equalitiesL19–19

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

  1. L19
    refl
07Use earlier factsL20–22

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

  1. L20
    specialize hpad_left (i)
  2. L21
    apply hpad_left
  3. L22
    exact hi

Library-wide reading audit

Original defined command ledger · 22 lines
  1. 0001intro b
  2. 0002intro c
  3. 0003intro L
  4. 0004intro t
  5. 0005intro B
  6. 0006intro C
  7. 0007intro i
  8. 0008intro hpad
  9. 0009intro hi
  10. 0010cases hpad
  11. 0011left
  12. 0012split
  13. 0013cases hi
  14. 0014exists x+L
  15. 0015trans (x+S i)+L
  16. 0016simp [add_assoc,add_comm,add_succ_left]
  17. 0017congr
  18. 0018exact hi_witness
  19. 0019refl
  20. 0020specialize hpad_left (i)
  21. 0021apply hpad_left
  22. 0022exact hi