PX0002

polynomial_diagonal_prefix_left_transport

The same actual first-N antidiagonal table remains valid after a left input prefix extension.

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. ∀ AB. ∀ AC. ∀ K. ∀ bb. ∀ bc. ∀ M. ∀ N. ∀ i. ∀ db. ∀ dc. Le(N,L) → Le(N,K) → BetaPrefixEqual(ab,ac,AB,AC,N) → PolynomialDiagonalPrefix(ab,ac,L,bb,bc,M,i,db,dc,N) → PolynomialDiagonalPrefix(AB,AC,K,bb,bc,M,i,db,dc,N)

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

Definition DAG

BetaPrefixEqual(b,c,d,e,l) · 1PolynomialDiagonalTerm(ab,ac,L,bb,bc,M,i,j,t) · 1PolynomialDiagonalPrefix(ab,ac,L,bb,bc,M,i,db,dc,l) · 2Le(a,b) · 2BetaAt(b,c,i,x) · 1

Actual proof prerequisites

polynomial_diagonal_left_prefix_transport
Original expanded first-order statement 
forall ab ac L AB AC K bb bc M N i db dc. (exists pfc_gap_tri_prefix_old_length. pfc_gap_tri_prefix_old_length+(N)=(L)) -> (exists pfc_gap_tri_prefix_new_length. pfc_gap_tri_prefix_new_length+(N)=(K)) -> (forall mdr_i_pfp_tri_prefix_equal mdr_a_pfp_tri_prefix_equal. (exists mdr_gap_pfp_tri_prefix_equalb. mdr_gap_pfp_tri_prefix_equalb + S (mdr_i_pfp_tri_prefix_equal) = (N)) -> (((exists ff_h_mdr_pfp_tri_prefix_equalo. ff_h_mdr_pfp_tri_prefix_equalo + S (mdr_a_pfp_tri_prefix_equal) = S ((S (mdr_i_pfp_tri_prefix_equal)) * ac)) /\ exists ff_q_mdr_pfp_tri_prefix_equalo. ab = ff_q_mdr_pfp_tri_prefix_equalo * S ((S (mdr_i_pfp_tri_prefix_equal)) * ac) + (mdr_a_pfp_tri_prefix_equal))) -> (((exists ff_h_mdr_pfp_tri_prefix_equaln. ff_h_mdr_pfp_tri_prefix_equaln + S (mdr_a_pfp_tri_prefix_equal) = S ((S (mdr_i_pfp_tri_prefix_equal)) * AC)) /\ exists ff_q_mdr_pfp_tri_prefix_equaln. AB = ff_q_mdr_pfp_tri_prefix_equaln * S ((S (mdr_i_pfp_tri_prefix_equal)) * AC) + (mdr_a_pfp_tri_prefix_equal)))) -> (forall pfc_index_tri_prefix_old. (exists pfa_gap_tri_prefix_oldbound. pfa_gap_tri_prefix_oldbound + S (pfc_index_tri_prefix_old) = (N)) -> exists pfc_value_tri_prefix_old. ((((exists ff_h_pfp_tri_prefix_oldentry. ff_h_pfp_tri_prefix_oldentry + S (pfc_value_tri_prefix_old) = S ((S (pfc_index_tri_prefix_old)) * dc)) /\ exists ff_q_pfp_tri_prefix_oldentry. db = ff_q_pfp_tri_prefix_oldentry * S ((S (pfc_index_tri_prefix_old)) * dc) + (pfc_value_tri_prefix_old))) /\ ((exists pfc_complement_tri_prefix_oldterm pfc_left_tri_prefix_oldterm pfc_right_tri_prefix_oldterm. (((pfc_index_tri_prefix_old)+pfc_complement_tri_prefix_oldterm=(i)) /\ ((((((exists pfa_gap_tri_prefix_oldtermleftinside. pfa_gap_tri_prefix_oldtermleftinside + S (pfc_index_tri_prefix_old) = (L)) /\ ((((exists ff_h_pfp_tri_prefix_oldtermleftentry. ff_h_pfp_tri_prefix_oldtermleftentry + S (pfc_left_tri_prefix_oldterm) = S ((S (pfc_index_tri_prefix_old)) * ac)) /\ exists ff_q_pfp_tri_prefix_oldtermleftentry. ab = ff_q_pfp_tri_prefix_oldtermleftentry * S ((S (pfc_index_tri_prefix_old)) * ac) + (pfc_left_tri_prefix_oldterm)))))) \/ (((exists pfc_gap_tri_prefix_oldtermleftoutside. pfc_gap_tri_prefix_oldtermleftoutside+(L)=(pfc_index_tri_prefix_old)) /\ (((pfc_left_tri_prefix_oldterm)=0))))) /\ ((((((exists pfa_gap_tri_prefix_oldtermrightinside. pfa_gap_tri_prefix_oldtermrightinside + S (pfc_complement_tri_prefix_oldterm) = (M)) /\ ((((exists ff_h_pfp_tri_prefix_oldtermrightentry. ff_h_pfp_tri_prefix_oldtermrightentry + S (pfc_right_tri_prefix_oldterm) = S ((S (pfc_complement_tri_prefix_oldterm)) * bc)) /\ exists ff_q_pfp_tri_prefix_oldtermrightentry. bb = ff_q_pfp_tri_prefix_oldtermrightentry * S ((S (pfc_complement_tri_prefix_oldterm)) * bc) + (pfc_right_tri_prefix_oldterm)))))) \/ (((exists pfc_gap_tri_prefix_oldtermrightoutside. pfc_gap_tri_prefix_oldtermrightoutside+(M)=(pfc_complement_tri_prefix_oldterm)) /\ (((pfc_right_tri_prefix_oldterm)=0))))) /\ (((pfc_value_tri_prefix_old)=pfc_left_tri_prefix_oldterm*pfc_right_tri_prefix_oldterm))))))))))) -> (forall pfc_index_tri_prefix_new. (exists pfa_gap_tri_prefix_newbound. pfa_gap_tri_prefix_newbound + S (pfc_index_tri_prefix_new) = (N)) -> exists pfc_value_tri_prefix_new. ((((exists ff_h_pfp_tri_prefix_newentry. ff_h_pfp_tri_prefix_newentry + S (pfc_value_tri_prefix_new) = S ((S (pfc_index_tri_prefix_new)) * dc)) /\ exists ff_q_pfp_tri_prefix_newentry. db = ff_q_pfp_tri_prefix_newentry * S ((S (pfc_index_tri_prefix_new)) * dc) + (pfc_value_tri_prefix_new))) /\ ((exists pfc_complement_tri_prefix_newterm pfc_left_tri_prefix_newterm pfc_right_tri_prefix_newterm. (((pfc_index_tri_prefix_new)+pfc_complement_tri_prefix_newterm=(i)) /\ ((((((exists pfa_gap_tri_prefix_newtermleftinside. pfa_gap_tri_prefix_newtermleftinside + S (pfc_index_tri_prefix_new) = (K)) /\ ((((exists ff_h_pfp_tri_prefix_newtermleftentry. ff_h_pfp_tri_prefix_newtermleftentry + S (pfc_left_tri_prefix_newterm) = S ((S (pfc_index_tri_prefix_new)) * AC)) /\ exists ff_q_pfp_tri_prefix_newtermleftentry. AB = ff_q_pfp_tri_prefix_newtermleftentry * S ((S (pfc_index_tri_prefix_new)) * AC) + (pfc_left_tri_prefix_newterm)))))) \/ (((exists pfc_gap_tri_prefix_newtermleftoutside. pfc_gap_tri_prefix_newtermleftoutside+(K)=(pfc_index_tri_prefix_new)) /\ (((pfc_left_tri_prefix_newterm)=0))))) /\ ((((((exists pfa_gap_tri_prefix_newtermrightinside. pfa_gap_tri_prefix_newtermrightinside + S (pfc_complement_tri_prefix_newterm) = (M)) /\ ((((exists ff_h_pfp_tri_prefix_newtermrightentry. ff_h_pfp_tri_prefix_newtermrightentry + S (pfc_right_tri_prefix_newterm) = S ((S (pfc_complement_tri_prefix_newterm)) * bc)) /\ exists ff_q_pfp_tri_prefix_newtermrightentry. bb = ff_q_pfp_tri_prefix_newtermrightentry * S ((S (pfc_complement_tri_prefix_newterm)) * bc) + (pfc_right_tri_prefix_newterm)))))) \/ (((exists pfc_gap_tri_prefix_newtermrightoutside. pfc_gap_tri_prefix_newtermrightoutside+(M)=(pfc_complement_tri_prefix_newterm)) /\ (((pfc_right_tri_prefix_newterm)=0))))) /\ (((pfc_value_tri_prefix_new)=pfc_left_tri_prefix_newterm*pfc_right_tri_prefix_newterm)))))))))))

Complete tactic proof in conservative notation

All 47 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

47 script commands · 8 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.

Named ingredients (1)
polynomial diagonal left prefix transport
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 AB
  5. L5
    intro AC
  6. L6
    intro K
  7. L7
    intro bb
  8. L8
    intro bc
  9. L9
    intro M
  10. L10
    intro N
02Fix variables and assumptionsL11–19

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

  1. L11
    intro i
  2. L12
    intro db
  3. L13
    intro dc
  4. L14
    intro hl
  5. L15
    intro hk
  6. L16
    intro he
  7. L17
    intro hd
  8. L18
    intro j
  9. L19
    intro hj
03Establish htL20–23

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply hd.

  1. L20
    have ht : ∃ t. BetaAt(db,dc,j,t) ∧ PolynomialDiagonalTerm(ab,ac,L,bb,bc,M,i,j,t)Definitions: BetaAt(db,dc,j,t)PolynomialDiagonalTerm(ab,ac,L,bb,bc,M,i,j,t)Original native command in the exact edition
  2. L21
    specialize hd (j)
  3. L22
    apply hd
  4. L23
    exact hj
04Separate the logical casesL24–25

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

  1. L24
    cases ht
  2. L25
    cases ht_witness
05Construct an explicit witnessL26–26

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

  1. L26
    exists x
06Separate the logical casesL27–27

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

  1. L27
    split
07Use earlier factsL28–37

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

  1. L28
    exact ht_witness_left
  2. L29
    specialize polynomial_diagonal_left_prefix_transport (ab)
  3. L30
    specialize polynomial_diagonal_left_prefix_transport (ac)
  4. L31
    specialize polynomial_diagonal_left_prefix_transport (L)
  5. L32
    specialize polynomial_diagonal_left_prefix_transport (AB)
  6. L33
    specialize polynomial_diagonal_left_prefix_transport (AC)
  7. L34
    specialize polynomial_diagonal_left_prefix_transport (K)
  8. L35
    specialize polynomial_diagonal_left_prefix_transport (bb)
  9. L36
    specialize polynomial_diagonal_left_prefix_transport (bc)
  10. L37
    specialize polynomial_diagonal_left_prefix_transport (M)
08Use earlier factsL38–47

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

  1. L38
    specialize polynomial_diagonal_left_prefix_transport (N)
  2. L39
    specialize polynomial_diagonal_left_prefix_transport (i)
  3. L40
    specialize polynomial_diagonal_left_prefix_transport (j)
  4. L41
    specialize polynomial_diagonal_left_prefix_transport (x)
  5. L42
    apply polynomial_diagonal_left_prefix_transport
  6. L43
    exact hl
  7. L44
    exact hk
  8. L45
    exact he
  9. L46
    exact hj
  10. L47
    exact ht_witness_right

Library-wide reading audit

Original defined command ledger · 47 lines
  1. 0001intro ab
  2. 0002intro ac
  3. 0003intro L
  4. 0004intro AB
  5. 0005intro AC
  6. 0006intro K
  7. 0007intro bb
  8. 0008intro bc
  9. 0009intro M
  10. 0010intro N
  11. 0011intro i
  12. 0012intro db
  13. 0013intro dc
  14. 0014intro hl
  15. 0015intro hk
  16. 0016intro he
  17. 0017intro hd
  18. 0018intro j
  19. 0019intro hj
  20. 0020have ht : ∃ t. BetaAt(db,dc,j,t) ∧ PolynomialDiagonalTerm(ab,ac,L,bb,bc,M,i,j,t)
  21. 0021specialize hd (j)
  22. 0022apply hd
  23. 0023exact hj
  24. 0024cases ht
  25. 0025cases ht_witness
  26. 0026exists x
  27. 0027split
  28. 0028exact ht_witness_left
  29. 0029specialize polynomial_diagonal_left_prefix_transport (ab)
  30. 0030specialize polynomial_diagonal_left_prefix_transport (ac)
  31. 0031specialize polynomial_diagonal_left_prefix_transport (L)
  32. 0032specialize polynomial_diagonal_left_prefix_transport (AB)
  33. 0033specialize polynomial_diagonal_left_prefix_transport (AC)
  34. 0034specialize polynomial_diagonal_left_prefix_transport (K)
  35. 0035specialize polynomial_diagonal_left_prefix_transport (bb)
  36. 0036specialize polynomial_diagonal_left_prefix_transport (bc)
  37. 0037specialize polynomial_diagonal_left_prefix_transport (M)
  38. 0038specialize polynomial_diagonal_left_prefix_transport (N)
  39. 0039specialize polynomial_diagonal_left_prefix_transport (i)
  40. 0040specialize polynomial_diagonal_left_prefix_transport (j)
  41. 0041specialize polynomial_diagonal_left_prefix_transport (x)
  42. 0042apply polynomial_diagonal_left_prefix_transport
  43. 0043exact hl
  44. 0044exact hk
  45. 0045exact he
  46. 0046exact hj
  47. 0047exact ht_witness_right