PG002E

polynomial_diagonal_left_unit_tail_term

Every later term is zero because its left input index is outside a genuine length-one prefix. The value of that prefix is irrelevant here.

Alpha v34 checked-use · first admitted v34 · 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.

All products and sums use actual beta-coded coefficients; equality is formal coefficient equivalence, not equality of evaluations or raw codes. The zero gcd is included. Uniqueness is up to formal equivalence, not unique Bézout coefficients. This proves the polynomial gcd component, not G091 irreducible-polynomial existence or arbitrary prime-power-field construction.

Exact theorem in conservative defined notation

∀ ub. ∀ uc. ∀ ab. ∀ ac. ∀ L. ∀ i. ∀ j. ∀ t. Lt(0,j) → PolynomialDiagonalTerm(ub,uc,1,ab,ac,L,i,j,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) · 1Lt(a,b) · 1

Actual proof prerequisites

polynomial_zero_extended_entry_functional · checked external prerequisitemul_zero_left · checked external prerequisite
Original expanded first-order statement 
forall ub uc ab ac L i j t. (exists pfc_gap_unit_tail_index. pfc_gap_unit_tail_index+(1)=(j)) -> (exists pfc_complement_unit_tail_actual pfc_left_unit_tail_actual pfc_right_unit_tail_actual. (((j)+pfc_complement_unit_tail_actual=(i)) /\ ((((((exists pfa_gap_unit_tail_actualleftinside. pfa_gap_unit_tail_actualleftinside + S (j) = (1)) /\ ((((exists ff_h_pfp_unit_tail_actualleftentry. ff_h_pfp_unit_tail_actualleftentry + S (pfc_left_unit_tail_actual) = S ((S (j)) * uc)) /\ exists ff_q_pfp_unit_tail_actualleftentry. ub = ff_q_pfp_unit_tail_actualleftentry * S ((S (j)) * uc) + (pfc_left_unit_tail_actual)))))) \/ (((exists pfc_gap_unit_tail_actualleftoutside. pfc_gap_unit_tail_actualleftoutside+(1)=(j)) /\ (((pfc_left_unit_tail_actual)=0))))) /\ ((((((exists pfa_gap_unit_tail_actualrightinside. pfa_gap_unit_tail_actualrightinside + S (pfc_complement_unit_tail_actual) = (L)) /\ ((((exists ff_h_pfp_unit_tail_actualrightentry. ff_h_pfp_unit_tail_actualrightentry + S (pfc_right_unit_tail_actual) = S ((S (pfc_complement_unit_tail_actual)) * ac)) /\ exists ff_q_pfp_unit_tail_actualrightentry. ab = ff_q_pfp_unit_tail_actualrightentry * S ((S (pfc_complement_unit_tail_actual)) * ac) + (pfc_right_unit_tail_actual)))))) \/ (((exists pfc_gap_unit_tail_actualrightoutside. pfc_gap_unit_tail_actualrightoutside+(L)=(pfc_complement_unit_tail_actual)) /\ (((pfc_right_unit_tail_actual)=0))))) /\ (((t)=pfc_left_unit_tail_actual*pfc_right_unit_tail_actual)))))))) -> (t=0)

Complete tactic proof in conservative notation

All 33 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

33 script commands · 9 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.

01Fix variables and assumptionsL1–10

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

  1. L1
    intro ub
  2. L2
    intro uc
  3. L3
    intro ab
  4. L4
    intro ac
  5. L5
    intro L
  6. L6
    intro i
  7. L7
    intro j
  8. L8
    intro t
  9. L9
    intro hj
  10. L10
    intro ht
02Separate the logical casesL11–16

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

  1. L11
    cases ht
  2. L12
    cases ht_witness
  3. L13
    cases ht_witness_witness
  4. L14
    cases ht_witness_witness_witness
  5. L15
    cases ht_witness_witness_witness_right
  6. L16
    cases ht_witness_witness_witness_right_right
03Establish hzL17–25

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

  1. L17
    have hz : x1=0
  2. L18
    specialize polynomial_zero_extended_entry_functional (ub)
  3. L19
    specialize polynomial_zero_extended_entry_functional (uc)
  4. L20
    specialize polynomial_zero_extended_entry_functional (1)
  5. L21
    specialize polynomial_zero_extended_entry_functional (j)
  6. L22
    specialize polynomial_zero_extended_entry_functional (x1)
  7. L23
    specialize polynomial_zero_extended_entry_functional (0)
  8. L24
    apply polynomial_zero_extended_entry_functional
  9. L25
    exact ht_witness_witness_witness_right_left
04Separate the logical casesL26–27

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

  1. L26
    right
  2. L27
    split
05Use earlier factsL28–28

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

  1. L28
    exact hj
06Calculate and transport equalitiesL29–30

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

  1. L29
    refl
  2. L30
    trans x1*x2
07Use earlier factsL31–31

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

  1. L31
    exact ht_witness_witness_witness_right_right_right
08Calculate and transport equalitiesL32–32

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

  1. L32
    rewrite hz
09Use earlier factsL33–33

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

  1. L33
    apply mul_zero_left

Library-wide reading audit

Original defined command ledger · 33 lines
  1. 0001intro ub
  2. 0002intro uc
  3. 0003intro ab
  4. 0004intro ac
  5. 0005intro L
  6. 0006intro i
  7. 0007intro j
  8. 0008intro t
  9. 0009intro hj
  10. 0010intro ht
  11. 0011cases ht
  12. 0012cases ht_witness
  13. 0013cases ht_witness_witness
  14. 0014cases ht_witness_witness_witness
  15. 0015cases ht_witness_witness_witness_right
  16. 0016cases ht_witness_witness_witness_right_right
  17. 0017have hz : x1=0
  18. 0018specialize polynomial_zero_extended_entry_functional (ub)
  19. 0019specialize polynomial_zero_extended_entry_functional (uc)
  20. 0020specialize polynomial_zero_extended_entry_functional (1)
  21. 0021specialize polynomial_zero_extended_entry_functional (j)
  22. 0022specialize polynomial_zero_extended_entry_functional (x1)
  23. 0023specialize polynomial_zero_extended_entry_functional (0)
  24. 0024apply polynomial_zero_extended_entry_functional
  25. 0025exact ht_witness_witness_witness_right_left
  26. 0026right
  27. 0027split
  28. 0028exact hj
  29. 0029refl
  30. 0030trans x1*x2
  31. 0031exact ht_witness_witness_witness_right_right_right
  32. 0032rewrite hz
  33. 0033apply mul_zero_left