PX0063

polynomial_diagonal_term_left_padding_zero_right

A genuine antidiagonal term is zero when its right factor index lies in the actual leading padding block.

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) → Lt(i,t + j) → PolynomialDiagonalTerm(ab,ac,L,BB,BC,t + M,i,j,z) → z = 0

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

Definition DAG

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

Actual proof prerequisites

polynomial_zero_extended_entry_functional · checked external prerequisitepolynomial_zero_extended_left_pad_beforeadd_right_cancel · checked external prerequisiteadd_assoc · checked external prerequisiteadd_comm · checked external prerequisiteadd_succ_left · 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_zero_term_pad_rightzeros. (exists pfa_gap_zero_term_pad_rightzerosindex. pfa_gap_zero_term_pad_rightzerosindex + S (pfp_repeat_index_zero_term_pad_rightzeros) = (t)) -> (((exists ff_h_pfp_zero_term_pad_rightzerosentry. ff_h_pfp_zero_term_pad_rightzerosentry + S (0) = S ((S (pfp_repeat_index_zero_term_pad_rightzeros)) * BC)) /\ exists ff_q_pfp_zero_term_pad_rightzerosentry. BB = ff_q_pfp_zero_term_pad_rightzerosentry * S ((S (pfp_repeat_index_zero_term_pad_rightzeros)) * BC) + (0)))) /\ ((forall pfrep_index_zero_term_pad_right pfrep_value_zero_term_pad_right. (exists pfa_gap_zero_term_pad_rightbound. pfa_gap_zero_term_pad_rightbound + S (pfrep_index_zero_term_pad_right) = (M)) -> (((exists ff_h_pfp_zero_term_pad_rightinput. ff_h_pfp_zero_term_pad_rightinput + S (pfrep_value_zero_term_pad_right) = S ((S (pfrep_index_zero_term_pad_right)) * bc)) /\ exists ff_q_pfp_zero_term_pad_rightinput. bb = ff_q_pfp_zero_term_pad_rightinput * S ((S (pfrep_index_zero_term_pad_right)) * bc) + (pfrep_value_zero_term_pad_right))) -> (((exists ff_h_pfp_zero_term_pad_rightoutput. ff_h_pfp_zero_term_pad_rightoutput + S (pfrep_value_zero_term_pad_right) = S ((S ((t)+pfrep_index_zero_term_pad_right)) * BC)) /\ exists ff_q_pfp_zero_term_pad_rightoutput. BB = ff_q_pfp_zero_term_pad_rightoutput * S ((S ((t)+pfrep_index_zero_term_pad_right)) * BC) + (pfrep_value_zero_term_pad_right))))))) -> (exists pfa_gap_term_zero_right_bound. pfa_gap_term_zero_right_bound + S (i) = (t+j)) -> (exists pfc_complement_zero_term_actual_right pfc_left_zero_term_actual_right pfc_right_zero_term_actual_right. (((j)+pfc_complement_zero_term_actual_right=(i)) /\ ((((((exists pfa_gap_zero_term_actual_rightleftinside. pfa_gap_zero_term_actual_rightleftinside + S (j) = (L)) /\ ((((exists ff_h_pfp_zero_term_actual_rightleftentry. ff_h_pfp_zero_term_actual_rightleftentry + S (pfc_left_zero_term_actual_right) = S ((S (j)) * ac)) /\ exists ff_q_pfp_zero_term_actual_rightleftentry. ab = ff_q_pfp_zero_term_actual_rightleftentry * S ((S (j)) * ac) + (pfc_left_zero_term_actual_right)))))) \/ (((exists pfc_gap_zero_term_actual_rightleftoutside. pfc_gap_zero_term_actual_rightleftoutside+(L)=(j)) /\ (((pfc_left_zero_term_actual_right)=0))))) /\ ((((((exists pfa_gap_zero_term_actual_rightrightinside. pfa_gap_zero_term_actual_rightrightinside + S (pfc_complement_zero_term_actual_right) = (t+M)) /\ ((((exists ff_h_pfp_zero_term_actual_rightrightentry. ff_h_pfp_zero_term_actual_rightrightentry + S (pfc_right_zero_term_actual_right) = S ((S (pfc_complement_zero_term_actual_right)) * BC)) /\ exists ff_q_pfp_zero_term_actual_rightrightentry. BB = ff_q_pfp_zero_term_actual_rightrightentry * S ((S (pfc_complement_zero_term_actual_right)) * BC) + (pfc_right_zero_term_actual_right)))))) \/ (((exists pfc_gap_zero_term_actual_rightrightoutside. pfc_gap_zero_term_actual_rightrightoutside+(t+M)=(pfc_complement_zero_term_actual_right)) /\ (((pfc_right_zero_term_actual_right)=0))))) /\ (((z)=pfc_left_zero_term_actual_right*pfc_right_zero_term_actual_right)))))))) -> (z=0)

Complete tactic proof in conservative notation

All 59 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

59 script commands · 14 reading checkpoints · 2 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 before
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 assumptionsL11–15

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

  1. L11
    intro j
  2. L12
    intro z
  3. L13
    intro hpad
  4. L14
    intro hbound
  5. L15
    intro ht
03Separate the logical casesL16–21

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

  1. L16
    cases ht
  2. L17
    cases ht_witness
  3. L18
    cases ht_witness_witness
  4. L19
    cases ht_witness_witness_witness
  5. L20
    cases ht_witness_witness_witness_right
  6. L21
    cases ht_witness_witness_witness_right_right
04Establish hindexL22–22

Establish this local claim before using it. It is not an additional assumption.

  1. L22
    have hindex : Lt(x,t)Definitions: Lt(x,t)Original native command in the exact edition
05Separate the logical casesL23–23

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

  1. L23
    cases hbound
06Construct an explicit witnessL24–24

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

  1. L24
    exists x3
07Use earlier factsL25–28

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

  1. L25
    specialize add_right_cancel (x3+S x)
  2. L26
    specialize add_right_cancel (t)
  3. L27
    specialize add_right_cancel (j)
  4. L28
    apply add_right_cancel
08Calculate and transport equalitiesL29–34

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

  1. L29
    trans x3+S (j+x)
  2. L30
    simp [add_assoc,add_comm,add_succ_left]
  3. L31
    trans x3+S i
  4. L32
    congr
  5. L33
    refl
  6. L34
    congr
09Use earlier factsL35–36

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

  1. L35
    exact ht_witness_witness_witness_left
  2. L36
    exact hbound_witness
10Establish hzeroL37–46

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply polynomial zero extended entry functional.

  1. L37
    have hzero : x2=0
  2. L38
    specialize polynomial_zero_extended_entry_functional (BB)
  3. L39
    specialize polynomial_zero_extended_entry_functional (BC)
  4. L40
    specialize polynomial_zero_extended_entry_functional (t+M)
  5. L41
    specialize polynomial_zero_extended_entry_functional (x)
  6. L42
    specialize polynomial_zero_extended_entry_functional (x2)
  7. L43
    specialize polynomial_zero_extended_entry_functional (0)
  8. L44
    apply polynomial_zero_extended_entry_functional
  9. L45
    exact ht_witness_witness_witness_right_right_left
  10. L46
    specialize polynomial_zero_extended_left_pad_before (bb)
11Use earlier factsL47–55

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

  1. L47
    specialize polynomial_zero_extended_left_pad_before (bc)
  2. L48
    specialize polynomial_zero_extended_left_pad_before (M)
  3. L49
    specialize polynomial_zero_extended_left_pad_before (t)
  4. L50
    specialize polynomial_zero_extended_left_pad_before (BB)
  5. L51
    specialize polynomial_zero_extended_left_pad_before (BC)
  6. L52
    specialize polynomial_zero_extended_left_pad_before (x)
  7. L53
    apply polynomial_zero_extended_left_pad_before
  8. L54
    exact hpad
  9. L55
    exact hindex
12Calculate and transport equalitiesL56–56

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

  1. L56
    trans x1*x2
13Use earlier factsL57–57

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

  1. L57
    exact ht_witness_witness_witness_right_right_right
14Calculate and transport equalitiesL58–59

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

  1. L58
    rewrite hzero
  2. L59
    simp

Library-wide reading audit

Original defined command ledger · 59 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 hpad
  14. 0014intro hbound
  15. 0015intro ht
  16. 0016cases ht
  17. 0017cases ht_witness
  18. 0018cases ht_witness_witness
  19. 0019cases ht_witness_witness_witness
  20. 0020cases ht_witness_witness_witness_right
  21. 0021cases ht_witness_witness_witness_right_right
  22. 0022have hindex : Lt(x,t)
  23. 0023cases hbound
  24. 0024exists x3
  25. 0025specialize add_right_cancel (x3+S x)
  26. 0026specialize add_right_cancel (t)
  27. 0027specialize add_right_cancel (j)
  28. 0028apply add_right_cancel
  29. 0029trans x3+S (j+x)
  30. 0030simp [add_assoc,add_comm,add_succ_left]
  31. 0031trans x3+S i
  32. 0032congr
  33. 0033refl
  34. 0034congr
  35. 0035exact ht_witness_witness_witness_left
  36. 0036exact hbound_witness
  37. 0037have hzero : x2=0
  38. 0038specialize polynomial_zero_extended_entry_functional (BB)
  39. 0039specialize polynomial_zero_extended_entry_functional (BC)
  40. 0040specialize polynomial_zero_extended_entry_functional (t+M)
  41. 0041specialize polynomial_zero_extended_entry_functional (x)
  42. 0042specialize polynomial_zero_extended_entry_functional (x2)
  43. 0043specialize polynomial_zero_extended_entry_functional (0)
  44. 0044apply polynomial_zero_extended_entry_functional
  45. 0045exact ht_witness_witness_witness_right_right_left
  46. 0046specialize polynomial_zero_extended_left_pad_before (bb)
  47. 0047specialize polynomial_zero_extended_left_pad_before (bc)
  48. 0048specialize polynomial_zero_extended_left_pad_before (M)
  49. 0049specialize polynomial_zero_extended_left_pad_before (t)
  50. 0050specialize polynomial_zero_extended_left_pad_before (BB)
  51. 0051specialize polynomial_zero_extended_left_pad_before (BC)
  52. 0052specialize polynomial_zero_extended_left_pad_before (x)
  53. 0053apply polynomial_zero_extended_left_pad_before
  54. 0054exact hpad
  55. 0055exact hindex
  56. 0056trans x1*x2
  57. 0057exact ht_witness_witness_witness_right_right_right
  58. 0058rewrite hzero
  59. 0059simp