PX0002

polynomial_diagonal_prefix_left_transport

Alpha v34 independently verified · alpha_closed; checked-use authorized; not Stable

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

Current library: Alpha v34, 4,223 checked-use theorems; Stable remains 432. Historical first admissions, original proof editions, and non-admitted aliases are preserved.

Exact expanded first-order arithmetic 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)))))))))))

Constructive proof overview

Generated structural guide

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

The unchanged tactic script uses 1 declared prerequisite and contains 47 exact native proof lines.

Alpha v34 checked-use · first admitted v33 · independently kernel and Lean verified; not Stable

Proof neighborhood

Direct dependencies

PX0001 polynomial_diagonal_left_prefix_transport

Direct dependents

PX0007 polynomial_diagonal_sum_left_append

Formal native tactic body

Dependencies are introduced as named hypotheses before line 1. Local theorem links identify exact declared prerequisites. This exact body belongs to a complete independently kernel-checked constructive proof bundle and has Alpha checked-use authority; it does not imply Stable membership.

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.

Named ingredients (1)
polynomial diagonal left prefix transport

Long local formulas use this family’s existing definitions. Each new abbreviation was expanded back to the identical native formula, including its free-variable context. The original edition is preserved below.

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: PolynomialDiagonalTermBetaAt
  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 exact 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 : exists t. ((((exists ff_h_pfp_tri_prefix_chosen_entry. ff_h_pfp_tri_prefix_chosen_entry + S (t) = S ((S (j)) * dc)) /\ exists ff_q_pfp_tri_prefix_chosen_entry. db = ff_q_pfp_tri_prefix_chosen_entry * S ((S (j)) * dc) + (t))) /\ ((exists pfc_complement_tri_prefix_chosen_term pfc_left_tri_prefix_chosen_term pfc_right_tri_prefix_chosen_term. (((j)+pfc_complement_tri_prefix_chosen_term=(i)) /\ ((((((exists pfa_gap_tri_prefix_chosen_termleftinside. pfa_gap_tri_prefix_chosen_termleftinside + S (j) = (L)) /\ ((((exists ff_h_pfp_tri_prefix_chosen_termleftentry. ff_h_pfp_tri_prefix_chosen_termleftentry + S (pfc_left_tri_prefix_chosen_term) = S ((S (j)) * ac)) /\ exists ff_q_pfp_tri_prefix_chosen_termleftentry. ab = ff_q_pfp_tri_prefix_chosen_termleftentry * S ((S (j)) * ac) + (pfc_left_tri_prefix_chosen_term)))))) \/ (((exists pfc_gap_tri_prefix_chosen_termleftoutside. pfc_gap_tri_prefix_chosen_termleftoutside+(L)=(j)) /\ (((pfc_left_tri_prefix_chosen_term)=0))))) /\ ((((((exists pfa_gap_tri_prefix_chosen_termrightinside. pfa_gap_tri_prefix_chosen_termrightinside + S (pfc_complement_tri_prefix_chosen_term) = (M)) /\ ((((exists ff_h_pfp_tri_prefix_chosen_termrightentry. ff_h_pfp_tri_prefix_chosen_termrightentry + S (pfc_right_tri_prefix_chosen_term) = S ((S (pfc_complement_tri_prefix_chosen_term)) * bc)) /\ exists ff_q_pfp_tri_prefix_chosen_termrightentry. bb = ff_q_pfp_tri_prefix_chosen_termrightentry * S ((S (pfc_complement_tri_prefix_chosen_term)) * bc) + (pfc_right_tri_prefix_chosen_term)))))) \/ (((exists pfc_gap_tri_prefix_chosen_termrightoutside. pfc_gap_tri_prefix_chosen_termrightoutside+(M)=(pfc_complement_tri_prefix_chosen_term)) /\ (((pfc_right_tri_prefix_chosen_term)=0))))) /\ (((t)=pfc_left_tri_prefix_chosen_term*pfc_right_tri_prefix_chosen_term))))))))))
  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