PG002D

polynomial_diagonal_left_unit_first_term

The first actual antidiagonal term of a length-one left unit is the chosen right-input coefficient, without any primality or commutativity assumption.

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. ∀ a. ∀ t. BetaAt(ub,uc,0,1) → Lt(i,L) → BetaAt(ab,ac,i,a) → PolynomialDiagonalTerm(ub,uc,1,ab,ac,L,i,0,t) → t = a

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) · 1BetaAt(b,c,i,x) · 2

Actual proof prerequisites

polynomial_diagonal_term_functional · checked external prerequisitezero_add · checked external prerequisiteone_mul · checked external prerequisite
Original expanded first-order statement 
forall ub uc ab ac L i a t. (((exists ff_h_pfp_unit_first_entry. ff_h_pfp_unit_first_entry + S (1) = S ((S (0)) * uc)) /\ exists ff_q_pfp_unit_first_entry. ub = ff_q_pfp_unit_first_entry * S ((S (0)) * uc) + (1))) -> (exists pfa_gap_unit_first_index. pfa_gap_unit_first_index + S (i) = (L)) -> (((exists ff_h_pfp_unit_first_A. ff_h_pfp_unit_first_A + S (a) = S ((S (i)) * ac)) /\ exists ff_q_pfp_unit_first_A. ab = ff_q_pfp_unit_first_A * S ((S (i)) * ac) + (a))) -> (exists pfc_complement_unit_first_actual pfc_left_unit_first_actual pfc_right_unit_first_actual. (((0)+pfc_complement_unit_first_actual=(i)) /\ ((((((exists pfa_gap_unit_first_actualleftinside. pfa_gap_unit_first_actualleftinside + S (0) = (1)) /\ ((((exists ff_h_pfp_unit_first_actualleftentry. ff_h_pfp_unit_first_actualleftentry + S (pfc_left_unit_first_actual) = S ((S (0)) * uc)) /\ exists ff_q_pfp_unit_first_actualleftentry. ub = ff_q_pfp_unit_first_actualleftentry * S ((S (0)) * uc) + (pfc_left_unit_first_actual)))))) \/ (((exists pfc_gap_unit_first_actualleftoutside. pfc_gap_unit_first_actualleftoutside+(1)=(0)) /\ (((pfc_left_unit_first_actual)=0))))) /\ ((((((exists pfa_gap_unit_first_actualrightinside. pfa_gap_unit_first_actualrightinside + S (pfc_complement_unit_first_actual) = (L)) /\ ((((exists ff_h_pfp_unit_first_actualrightentry. ff_h_pfp_unit_first_actualrightentry + S (pfc_right_unit_first_actual) = S ((S (pfc_complement_unit_first_actual)) * ac)) /\ exists ff_q_pfp_unit_first_actualrightentry. ab = ff_q_pfp_unit_first_actualrightentry * S ((S (pfc_complement_unit_first_actual)) * ac) + (pfc_right_unit_first_actual)))))) \/ (((exists pfc_gap_unit_first_actualrightoutside. pfc_gap_unit_first_actualrightoutside+(L)=(pfc_complement_unit_first_actual)) /\ (((pfc_right_unit_first_actual)=0))))) /\ (((t)=pfc_left_unit_first_actual*pfc_right_unit_first_actual)))))))) -> (t=a)

Complete tactic proof in conservative notation

All 42 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

42 script commands · 14 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 assumptionsL1–10

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 a
  8. L8
    intro t
  9. L9
    intro hu
  10. L10
    intro hi
02Fix variables and assumptionsL11–12

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

  1. L11
    intro ha
  2. L12
    intro ht
03Use earlier factsL13–22

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

  1. L13
    specialize polynomial_diagonal_term_functional (ub)
  2. L14
    specialize polynomial_diagonal_term_functional (uc)
  3. L15
    specialize polynomial_diagonal_term_functional (1)
  4. L16
    specialize polynomial_diagonal_term_functional (ab)
  5. L17
    specialize polynomial_diagonal_term_functional (ac)
  6. L18
    specialize polynomial_diagonal_term_functional (L)
  7. L19
    specialize polynomial_diagonal_term_functional (i)
  8. L20
    specialize polynomial_diagonal_term_functional (0)
  9. L21
    specialize polynomial_diagonal_term_functional (t)
  10. L22
    specialize polynomial_diagonal_term_functional (a)
04Use earlier factsL23–24

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

  1. L23
    apply polynomial_diagonal_term_functional
  2. L24
    exact ht
05Construct an explicit witnessL25–27

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

  1. L25
    exists i
  2. L26
    exists 1
  3. L27
    exists a
06Separate the logical casesL28–28

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

  1. L28
    split
07Use earlier factsL29–29

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

  1. L29
    apply zero_add
08Separate the logical casesL30–32

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

  1. L30
    split
  2. L31
    left
  3. L32
    split
09Construct an explicit witnessL33–33

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

  1. L33
    exists 0
10Use earlier factsL34–35

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

  1. L34
    apply zero_add
  2. L35
    exact hu
11Separate the logical casesL36–38

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

  1. L36
    split
  2. L37
    left
  3. L38
    split
12Use earlier factsL39–40

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

  1. L39
    exact hi
  2. L40
    exact ha
13Calculate and transport equalitiesL41–41

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

  1. L41
    symm
14Use earlier factsL42–42

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

  1. L42
    apply one_mul

Library-wide reading audit

Original defined command ledger · 42 lines
  1. 0001intro ub
  2. 0002intro uc
  3. 0003intro ab
  4. 0004intro ac
  5. 0005intro L
  6. 0006intro i
  7. 0007intro a
  8. 0008intro t
  9. 0009intro hu
  10. 0010intro hi
  11. 0011intro ha
  12. 0012intro ht
  13. 0013specialize polynomial_diagonal_term_functional (ub)
  14. 0014specialize polynomial_diagonal_term_functional (uc)
  15. 0015specialize polynomial_diagonal_term_functional (1)
  16. 0016specialize polynomial_diagonal_term_functional (ab)
  17. 0017specialize polynomial_diagonal_term_functional (ac)
  18. 0018specialize polynomial_diagonal_term_functional (L)
  19. 0019specialize polynomial_diagonal_term_functional (i)
  20. 0020specialize polynomial_diagonal_term_functional (0)
  21. 0021specialize polynomial_diagonal_term_functional (t)
  22. 0022specialize polynomial_diagonal_term_functional (a)
  23. 0023apply polynomial_diagonal_term_functional
  24. 0024exact ht
  25. 0025exists i
  26. 0026exists 1
  27. 0027exists a
  28. 0028split
  29. 0029apply zero_add
  30. 0030split
  31. 0031left
  32. 0032split
  33. 0033exists 0
  34. 0034apply zero_add
  35. 0035exact hu
  36. 0036split
  37. 0037left
  38. 0038split
  39. 0039exact hi
  40. 0040exact ha
  41. 0041symm
  42. 0042apply one_mul