PX0061

polynomial_diagonal_term_left_padding_right

A genuine right factor leading-zero padding shifts the antidiagonal position while preserving each actual natural product term.

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

∀ ab. ∀ ac. ∀ L. ∀ bb. ∀ bc. ∀ M. ∀ BB. ∀ BC. ∀ t. ∀ i. ∀ j. ∀ z. PolynomialLeftPad(bb,bc,M,t,BB,BC) → PolynomialDiagonalTerm(ab,ac,L,bb,bc,M,i,j,z) → PolynomialDiagonalTerm(ab,ac,L,BB,BC,t + M,t + i,j,z)

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

Definition DAG

PolynomialDiagonalTerm(ab,ac,L,bb,bc,M,i,j,t) · 2PolynomialLeftPad(b,c,L,t,d,e) · 1

Actual proof prerequisites

add_assoc · checked external prerequisitepolynomial_zero_extended_left_pad_shiftadd_comm · checked external prerequisite
Original expanded first-order statement 
forall ab ac L bb bc M BB BC t i j z. (((forall pfp_repeat_index_term_padding_rightzeros. (exists pfa_gap_term_padding_rightzerosindex. pfa_gap_term_padding_rightzerosindex + S (pfp_repeat_index_term_padding_rightzeros) = (t)) -> (((exists ff_h_pfp_term_padding_rightzerosentry. ff_h_pfp_term_padding_rightzerosentry + S (0) = S ((S (pfp_repeat_index_term_padding_rightzeros)) * BC)) /\ exists ff_q_pfp_term_padding_rightzerosentry. BB = ff_q_pfp_term_padding_rightzerosentry * S ((S (pfp_repeat_index_term_padding_rightzeros)) * BC) + (0)))) /\ ((forall pfrep_index_term_padding_right pfrep_value_term_padding_right. (exists pfa_gap_term_padding_rightbound. pfa_gap_term_padding_rightbound + S (pfrep_index_term_padding_right) = (M)) -> (((exists ff_h_pfp_term_padding_rightinput. ff_h_pfp_term_padding_rightinput + S (pfrep_value_term_padding_right) = S ((S (pfrep_index_term_padding_right)) * bc)) /\ exists ff_q_pfp_term_padding_rightinput. bb = ff_q_pfp_term_padding_rightinput * S ((S (pfrep_index_term_padding_right)) * bc) + (pfrep_value_term_padding_right))) -> (((exists ff_h_pfp_term_padding_rightoutput. ff_h_pfp_term_padding_rightoutput + S (pfrep_value_term_padding_right) = S ((S ((t)+pfrep_index_term_padding_right)) * BC)) /\ exists ff_q_pfp_term_padding_rightoutput. BB = ff_q_pfp_term_padding_rightoutput * S ((S ((t)+pfrep_index_term_padding_right)) * BC) + (pfrep_value_term_padding_right))))))) -> (exists pfc_complement_term_original_right pfc_left_term_original_right pfc_right_term_original_right. (((j)+pfc_complement_term_original_right=(i)) /\ ((((((exists pfa_gap_term_original_rightleftinside. pfa_gap_term_original_rightleftinside + S (j) = (L)) /\ ((((exists ff_h_pfp_term_original_rightleftentry. ff_h_pfp_term_original_rightleftentry + S (pfc_left_term_original_right) = S ((S (j)) * ac)) /\ exists ff_q_pfp_term_original_rightleftentry. ab = ff_q_pfp_term_original_rightleftentry * S ((S (j)) * ac) + (pfc_left_term_original_right)))))) \/ (((exists pfc_gap_term_original_rightleftoutside. pfc_gap_term_original_rightleftoutside+(L)=(j)) /\ (((pfc_left_term_original_right)=0))))) /\ ((((((exists pfa_gap_term_original_rightrightinside. pfa_gap_term_original_rightrightinside + S (pfc_complement_term_original_right) = (M)) /\ ((((exists ff_h_pfp_term_original_rightrightentry. ff_h_pfp_term_original_rightrightentry + S (pfc_right_term_original_right) = S ((S (pfc_complement_term_original_right)) * bc)) /\ exists ff_q_pfp_term_original_rightrightentry. bb = ff_q_pfp_term_original_rightrightentry * S ((S (pfc_complement_term_original_right)) * bc) + (pfc_right_term_original_right)))))) \/ (((exists pfc_gap_term_original_rightrightoutside. pfc_gap_term_original_rightrightoutside+(M)=(pfc_complement_term_original_right)) /\ (((pfc_right_term_original_right)=0))))) /\ (((z)=pfc_left_term_original_right*pfc_right_term_original_right)))))))) -> (exists pfc_complement_term_shifted_right pfc_left_term_shifted_right pfc_right_term_shifted_right. (((j)+pfc_complement_term_shifted_right=(t+i)) /\ ((((((exists pfa_gap_term_shifted_rightleftinside. pfa_gap_term_shifted_rightleftinside + S (j) = (L)) /\ ((((exists ff_h_pfp_term_shifted_rightleftentry. ff_h_pfp_term_shifted_rightleftentry + S (pfc_left_term_shifted_right) = S ((S (j)) * ac)) /\ exists ff_q_pfp_term_shifted_rightleftentry. ab = ff_q_pfp_term_shifted_rightleftentry * S ((S (j)) * ac) + (pfc_left_term_shifted_right)))))) \/ (((exists pfc_gap_term_shifted_rightleftoutside. pfc_gap_term_shifted_rightleftoutside+(L)=(j)) /\ (((pfc_left_term_shifted_right)=0))))) /\ ((((((exists pfa_gap_term_shifted_rightrightinside. pfa_gap_term_shifted_rightrightinside + S (pfc_complement_term_shifted_right) = (t+M)) /\ ((((exists ff_h_pfp_term_shifted_rightrightentry. ff_h_pfp_term_shifted_rightrightentry + S (pfc_right_term_shifted_right) = S ((S (pfc_complement_term_shifted_right)) * BC)) /\ exists ff_q_pfp_term_shifted_rightrightentry. BB = ff_q_pfp_term_shifted_rightrightentry * S ((S (pfc_complement_term_shifted_right)) * BC) + (pfc_right_term_shifted_right)))))) \/ (((exists pfc_gap_term_shifted_rightrightoutside. pfc_gap_term_shifted_rightrightoutside+(t+M)=(pfc_complement_term_shifted_right)) /\ (((pfc_right_term_shifted_right)=0))))) /\ (((z)=pfc_left_term_shifted_right*pfc_right_term_shifted_right))))))))

Complete tactic proof in conservative notation

All 44 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

44 script commands · 12 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)
polynomial zero extended left pad shift
01Fix variables and assumptionsL1–10

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

  1. L1
    intro ab
  2. L2
    intro ac
  3. L3
    intro L
  4. L4
    intro bb
  5. L5
    intro bc
  6. L6
    intro M
  7. L7
    intro BB
  8. L8
    intro BC
  9. L9
    intro t
  10. L10
    intro i
02Fix variables and assumptionsL11–14

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

  1. L11
    intro j
  2. L12
    intro z
  3. L13
    intro hp
  4. L14
    intro ht
03Separate the logical casesL15–20

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

  1. L15
    cases ht
  2. L16
    cases ht_witness
  3. L17
    cases ht_witness_witness
  4. L18
    cases ht_witness_witness_witness
  5. L19
    cases ht_witness_witness_witness_right
  6. L20
    cases ht_witness_witness_witness_right_right
04Construct an explicit witnessL21–23

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

  1. L21
    exists t+x
  2. L22
    exists x1
  3. L23
    exists x2
05Separate the logical casesL24–24

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

  1. L24
    split
06Calculate and transport equalitiesL25–28

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

  1. L25
    trans t+(j+x)
  2. L26
    simp [add_assoc,add_comm]
  3. L27
    congr
  4. L28
    refl
07Use earlier factsL29–29

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

  1. L29
    exact ht_witness_witness_witness_left
08Separate the logical casesL30–30

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

  1. L30
    split
09Use earlier factsL31–31

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

  1. L31
    exact ht_witness_witness_witness_right_left
10Separate the logical casesL32–32

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

  1. L32
    split
11Use earlier factsL33–42

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

  1. L33
    specialize polynomial_zero_extended_left_pad_shift (bb)
  2. L34
    specialize polynomial_zero_extended_left_pad_shift (bc)
  3. L35
    specialize polynomial_zero_extended_left_pad_shift (M)
  4. L36
    specialize polynomial_zero_extended_left_pad_shift (t)
  5. L37
    specialize polynomial_zero_extended_left_pad_shift (BB)
  6. L38
    specialize polynomial_zero_extended_left_pad_shift (BC)
  7. L39
    specialize polynomial_zero_extended_left_pad_shift (x)
  8. L40
    specialize polynomial_zero_extended_left_pad_shift (x2)
  9. L41
    apply polynomial_zero_extended_left_pad_shift
  10. L42
    exact hp
12Use earlier factsL43–44

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

  1. L43
    exact ht_witness_witness_witness_right_right_left
  2. L44
    exact ht_witness_witness_witness_right_right_right

Library-wide reading audit

Original defined command ledger · 44 lines
  1. 0001intro ab
  2. 0002intro ac
  3. 0003intro L
  4. 0004intro bb
  5. 0005intro bc
  6. 0006intro M
  7. 0007intro BB
  8. 0008intro BC
  9. 0009intro t
  10. 0010intro i
  11. 0011intro j
  12. 0012intro z
  13. 0013intro hp
  14. 0014intro ht
  15. 0015cases ht
  16. 0016cases ht_witness
  17. 0017cases ht_witness_witness
  18. 0018cases ht_witness_witness_witness
  19. 0019cases ht_witness_witness_witness_right
  20. 0020cases ht_witness_witness_witness_right_right
  21. 0021exists t+x
  22. 0022exists x1
  23. 0023exists x2
  24. 0024split
  25. 0025trans t+(j+x)
  26. 0026simp [add_assoc,add_comm]
  27. 0027congr
  28. 0028refl
  29. 0029exact ht_witness_witness_witness_left
  30. 0030split
  31. 0031exact ht_witness_witness_witness_right_left
  32. 0032split
  33. 0033specialize polynomial_zero_extended_left_pad_shift (bb)
  34. 0034specialize polynomial_zero_extended_left_pad_shift (bc)
  35. 0035specialize polynomial_zero_extended_left_pad_shift (M)
  36. 0036specialize polynomial_zero_extended_left_pad_shift (t)
  37. 0037specialize polynomial_zero_extended_left_pad_shift (BB)
  38. 0038specialize polynomial_zero_extended_left_pad_shift (BC)
  39. 0039specialize polynomial_zero_extended_left_pad_shift (x)
  40. 0040specialize polynomial_zero_extended_left_pad_shift (x2)
  41. 0041apply polynomial_zero_extended_left_pad_shift
  42. 0042exact hp
  43. 0043exact ht_witness_witness_witness_right_right_left
  44. 0044exact ht_witness_witness_witness_right_right_right