PX0005

polynomial_diagonal_last_term_left_empty

At diagonal index N the absent N-th entry of a length-N left prefix contributes actual zero.

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. ∀ bb. ∀ bc. ∀ M. ∀ N. ∀ t. PolynomialDiagonalTerm(ab,ac,N,bb,bc,M,N,N,t) → t = 0

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

Definition DAG

PolynomialDiagonalTerm(ab,ac,L,bb,bc,M,i,j,t) · 1

Actual proof prerequisites

lt_not_le · checked external prerequisitele_refl · checked external prerequisitemul_zero_left · checked external prerequisite
Original expanded first-order statement 
forall ab ac bb bc M N t. (exists pfc_complement_tri_last_old pfc_left_tri_last_old pfc_right_tri_last_old. (((N)+pfc_complement_tri_last_old=(N)) /\ ((((((exists pfa_gap_tri_last_oldleftinside. pfa_gap_tri_last_oldleftinside + S (N) = (N)) /\ ((((exists ff_h_pfp_tri_last_oldleftentry. ff_h_pfp_tri_last_oldleftentry + S (pfc_left_tri_last_old) = S ((S (N)) * ac)) /\ exists ff_q_pfp_tri_last_oldleftentry. ab = ff_q_pfp_tri_last_oldleftentry * S ((S (N)) * ac) + (pfc_left_tri_last_old)))))) \/ (((exists pfc_gap_tri_last_oldleftoutside. pfc_gap_tri_last_oldleftoutside+(N)=(N)) /\ (((pfc_left_tri_last_old)=0))))) /\ ((((((exists pfa_gap_tri_last_oldrightinside. pfa_gap_tri_last_oldrightinside + S (pfc_complement_tri_last_old) = (M)) /\ ((((exists ff_h_pfp_tri_last_oldrightentry. ff_h_pfp_tri_last_oldrightentry + S (pfc_right_tri_last_old) = S ((S (pfc_complement_tri_last_old)) * bc)) /\ exists ff_q_pfp_tri_last_oldrightentry. bb = ff_q_pfp_tri_last_oldrightentry * S ((S (pfc_complement_tri_last_old)) * bc) + (pfc_right_tri_last_old)))))) \/ (((exists pfc_gap_tri_last_oldrightoutside. pfc_gap_tri_last_oldrightoutside+(M)=(pfc_complement_tri_last_old)) /\ (((pfc_right_tri_last_old)=0))))) /\ (((t)=pfc_left_tri_last_old*pfc_right_tri_last_old)))))))) -> t=0

Complete tactic proof in conservative notation

All 29 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

29 script commands · 8 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 assumptionsL1–8

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

  1. L1
    intro ab
  2. L2
    intro ac
  3. L3
    intro bb
  4. L4
    intro bc
  5. L5
    intro M
  6. L6
    intro N
  7. L7
    intro t
  8. L8
    intro ht
02Separate the logical casesL9–17

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

  1. L9
    cases ht
  2. L10
    cases ht_witness
  3. L11
    cases ht_witness_witness
  4. L12
    cases ht_witness_witness_witness
  5. L13
    cases ht_witness_witness_witness_right
  6. L14
    cases ht_witness_witness_witness_right_right
  7. L15
    cases ht_witness_witness_witness_right_left
  8. L16
    cases ht_witness_witness_witness_right_left_left
  9. L17
    exfalso
03Use earlier factsL18–23

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

  1. L18
    specialize lt_not_le (N)
  2. L19
    specialize lt_not_le (N)
  3. L20
    apply lt_not_le
  4. L21
    exact ht_witness_witness_witness_right_left_left_left
  5. L22
    specialize le_refl (N)
  6. L23
    apply le_refl
04Separate the logical casesL24–24

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

  1. L24
    cases ht_witness_witness_witness_right_left_right
05Calculate and transport equalitiesL25–25

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

  1. L25
    trans x1*x2
06Use earlier factsL26–26

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

  1. L26
    exact ht_witness_witness_witness_right_right_right
07Calculate and transport equalitiesL27–27

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

  1. L27
    rewrite ht_witness_witness_witness_right_left_right_right
08Use earlier factsL28–29

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

  1. L28
    specialize mul_zero_left (x2)
  2. L29
    apply mul_zero_left

Library-wide reading audit

Original defined command ledger · 29 lines
  1. 0001intro ab
  2. 0002intro ac
  3. 0003intro bb
  4. 0004intro bc
  5. 0005intro M
  6. 0006intro N
  7. 0007intro t
  8. 0008intro ht
  9. 0009cases ht
  10. 0010cases ht_witness
  11. 0011cases ht_witness_witness
  12. 0012cases ht_witness_witness_witness
  13. 0013cases ht_witness_witness_witness_right
  14. 0014cases ht_witness_witness_witness_right_right
  15. 0015cases ht_witness_witness_witness_right_left
  16. 0016cases ht_witness_witness_witness_right_left_left
  17. 0017exfalso
  18. 0018specialize lt_not_le (N)
  19. 0019specialize lt_not_le (N)
  20. 0020apply lt_not_le
  21. 0021exact ht_witness_witness_witness_right_left_left_left
  22. 0022specialize le_refl (N)
  23. 0023apply le_refl
  24. 0024cases ht_witness_witness_witness_right_left_right
  25. 0025trans x1*x2
  26. 0026exact ht_witness_witness_witness_right_right_right
  27. 0027rewrite ht_witness_witness_witness_right_left_right_right
  28. 0028specialize mul_zero_left (x2)
  29. 0029apply mul_zero_left