PG004D

polynomial_diagonal_left_constant_first_term

The first actual antidiagonal term of a genuine left singleton is the ordered natural product k*a. No modulus, coefficient bound, or commutativity premise is needed.

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

∀ k. ∀ kb. ∀ kc. ∀ ab. ∀ ac. ∀ L. ∀ i. ∀ a. ∀ t. BetaAt(kb,kc,0,k) → Lt(i,L) → BetaAt(ab,ac,i,a) → PolynomialDiagonalTerm(kb,kc,1,ab,ac,L,i,0,t) → t = k · 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 prerequisite
Original expanded first-order statement 
forall k kb kc ab ac L i a t. (((exists ff_h_pfp_left_constant_first_K. ff_h_pfp_left_constant_first_K + S (k) = S ((S (0)) * kc)) /\ exists ff_q_pfp_left_constant_first_K. kb = ff_q_pfp_left_constant_first_K * S ((S (0)) * kc) + (k))) -> (exists pfa_gap_left_constant_first_index. pfa_gap_left_constant_first_index + S (i) = (L)) -> (((exists ff_h_pfp_left_constant_first_A. ff_h_pfp_left_constant_first_A + S (a) = S ((S (i)) * ac)) /\ exists ff_q_pfp_left_constant_first_A. ab = ff_q_pfp_left_constant_first_A * S ((S (i)) * ac) + (a))) -> (exists pfc_complement_left_constant_first_actual pfc_left_left_constant_first_actual pfc_right_left_constant_first_actual. (((0)+pfc_complement_left_constant_first_actual=(i)) /\ ((((((exists pfa_gap_left_constant_first_actualleftinside. pfa_gap_left_constant_first_actualleftinside + S (0) = (1)) /\ ((((exists ff_h_pfp_left_constant_first_actualleftentry. ff_h_pfp_left_constant_first_actualleftentry + S (pfc_left_left_constant_first_actual) = S ((S (0)) * kc)) /\ exists ff_q_pfp_left_constant_first_actualleftentry. kb = ff_q_pfp_left_constant_first_actualleftentry * S ((S (0)) * kc) + (pfc_left_left_constant_first_actual)))))) \/ (((exists pfc_gap_left_constant_first_actualleftoutside. pfc_gap_left_constant_first_actualleftoutside+(1)=(0)) /\ (((pfc_left_left_constant_first_actual)=0))))) /\ ((((((exists pfa_gap_left_constant_first_actualrightinside. pfa_gap_left_constant_first_actualrightinside + S (pfc_complement_left_constant_first_actual) = (L)) /\ ((((exists ff_h_pfp_left_constant_first_actualrightentry. ff_h_pfp_left_constant_first_actualrightentry + S (pfc_right_left_constant_first_actual) = S ((S (pfc_complement_left_constant_first_actual)) * ac)) /\ exists ff_q_pfp_left_constant_first_actualrightentry. ab = ff_q_pfp_left_constant_first_actualrightentry * S ((S (pfc_complement_left_constant_first_actual)) * ac) + (pfc_right_left_constant_first_actual)))))) \/ (((exists pfc_gap_left_constant_first_actualrightoutside. pfc_gap_left_constant_first_actualrightoutside+(L)=(pfc_complement_left_constant_first_actual)) /\ (((pfc_right_left_constant_first_actual)=0))))) /\ (((t)=pfc_left_left_constant_first_actual*pfc_right_left_constant_first_actual)))))))) -> (t=k*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 · 13 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 k
  2. L2
    intro kb
  3. L3
    intro kc
  4. L4
    intro ab
  5. L5
    intro ac
  6. L6
    intro L
  7. L7
    intro i
  8. L8
    intro a
  9. L9
    intro t
  10. L10
    intro hk
02Fix variables and assumptionsL11–13

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

  1. L11
    intro hi
  2. L12
    intro ha
  3. L13
    intro ht
03Use earlier factsL14–23

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

  1. L14
    specialize polynomial_diagonal_term_functional (kb)
  2. L15
    specialize polynomial_diagonal_term_functional (kc)
  3. L16
    specialize polynomial_diagonal_term_functional (1)
  4. L17
    specialize polynomial_diagonal_term_functional (ab)
  5. L18
    specialize polynomial_diagonal_term_functional (ac)
  6. L19
    specialize polynomial_diagonal_term_functional (L)
  7. L20
    specialize polynomial_diagonal_term_functional (i)
  8. L21
    specialize polynomial_diagonal_term_functional (0)
  9. L22
    specialize polynomial_diagonal_term_functional (t)
  10. L23
    specialize polynomial_diagonal_term_functional (k*a)
04Use earlier factsL24–25

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

  1. L24
    apply polynomial_diagonal_term_functional
  2. L25
    exact ht
05Construct an explicit witnessL26–28

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

  1. L26
    exists i
  2. L27
    exists k
  3. L28
    exists a
06Separate the logical casesL29–29

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

  1. L29
    split
07Use earlier factsL30–30

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

  1. L30
    apply zero_add
08Separate the logical casesL31–33

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

  1. L31
    split
  2. L32
    left
  3. L33
    split
09Construct an explicit witnessL34–34

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

  1. L34
    exists 0
10Use earlier factsL35–36

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

  1. L35
    apply zero_add
  2. L36
    exact hk
11Separate the logical casesL37–39

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

  1. L37
    split
  2. L38
    left
  3. L39
    split
12Use earlier factsL40–41

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

  1. L40
    exact hi
  2. L41
    exact ha
13Calculate and transport equalitiesL42–42

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

  1. L42
    refl

Library-wide reading audit

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