EL0006

lte_twice_correction_successor

Twice the correction has the exact triangular coefficient and an actual next remainder.

Alpha v34 checked-use · first admitted v29 · 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. Exact original first-admission records.

All displayed hypotheses are required. Powers and positive differences are actual existential outputs. The proof constructs second-order correction identities and iterates the prime step; no binomial expansion or LTE oracle is assumed. The 2-adic variants remain separate open targets.

Exact theorem in conservative defined notation

∀ b. ∀ d. ∀ n. ∀ T. ∀ C. ∀ Q. ∀ H. Q = S n · (b · T) + d · C → 2 · C = S n · n · T + d · H → 2 · (b · C + Q) = S S n · S n · (b · T) + d · (b · H + 2 · C)

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

Definition DAG

none

Actual proof prerequisites

lte_twice_correction_polynomiallte_adjacent_coefficient_identity
Original expanded first-order statement 
forall b d n T C Q H. Q = S n * (b * T) + d * C -> 2 * C = (S n * n) * T + d * H -> 2 * (b * C + Q) = (S (S n) * S n) * (b * T) + d * (b * H + 2 * C)

Complete tactic proof in conservative notation

All 26 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

26 script commands · 5 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.

Named ingredients (2)
lte twice correction polynomiallte adjacent coefficient identity
01Fix variables and assumptionsL1–9

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

  1. L1
    intro b
  2. L2
    intro d
  3. L3
    intro n
  4. L4
    intro T
  5. L5
    intro C
  6. L6
    intro Q
  7. L7
    intro H
  8. L8
    intro hQ
  9. L9
    intro hC
02Calculate and transport equalitiesL10–10

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

  1. L10
    trans (S n * n + 2 * S n) * (b * T) + d * (b * H + 2 * C)
03Use earlier factsL11–20

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

  1. L11
    specialize lte_twice_correction_polynomial (b)
  2. L12
    specialize lte_twice_correction_polynomial (d)
  3. L13
    specialize lte_twice_correction_polynomial (S n)
  4. L14
    specialize lte_twice_correction_polynomial (S n * n)
  5. L15
    specialize lte_twice_correction_polynomial (T)
  6. L16
    specialize lte_twice_correction_polynomial (C)
  7. L17
    specialize lte_twice_correction_polynomial (Q)
  8. L18
    specialize lte_twice_correction_polynomial (H)
  9. L19
    apply lte_twice_correction_polynomial
  10. L20
    exact hQ
04Use earlier factsL21–21

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

  1. L21
    exact hC
05Establish hcoefficientL22–26

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply lte adjacent coefficient identity.

  1. L22
    have hcoefficient : S n * n + 2 * S n = S (S n) * S n
  2. L23
    specialize lte_adjacent_coefficient_identity (n)
  3. L24
    apply lte_adjacent_coefficient_identity
  4. L25
    rewrite hcoefficient
  5. L26
    refl

Library-wide reading audit

Original defined command ledger · 26 lines
  1. 0001intro b
  2. 0002intro d
  3. 0003intro n
  4. 0004intro T
  5. 0005intro C
  6. 0006intro Q
  7. 0007intro H
  8. 0008intro hQ
  9. 0009intro hC
  10. 0010trans (S n * n + 2 * S n) * (b * T) + d * (b * H + 2 * C)
  11. 0011specialize lte_twice_correction_polynomial (b)
  12. 0012specialize lte_twice_correction_polynomial (d)
  13. 0013specialize lte_twice_correction_polynomial (S n)
  14. 0014specialize lte_twice_correction_polynomial (S n * n)
  15. 0015specialize lte_twice_correction_polynomial (T)
  16. 0016specialize lte_twice_correction_polynomial (C)
  17. 0017specialize lte_twice_correction_polynomial (Q)
  18. 0018specialize lte_twice_correction_polynomial (H)
  19. 0019apply lte_twice_correction_polynomial
  20. 0020exact hQ
  21. 0021exact hC
  22. 0022have hcoefficient : S n * n + 2 * S n = S (S n) * S n
  23. 0023specialize lte_adjacent_coefficient_identity (n)
  24. 0024apply lte_adjacent_coefficient_identity
  25. 0025rewrite hcoefficient
  26. 0026refl