LU000V · theorem body

lucas_predecessor_digit_below_base

Alpha v34 checked-use · independently kernel and Lean verified; not Stable

The predecessor of a nonzero digit strictly below the base remains strictly below the base.

historical independently replay-verified empty-context experiment; the experiment itself persisted no certificate and granted no release authority; current checked use follows separately sealed proof bundles; no Stable promotion.

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.

Statement with defined notation

∀ p. ∀ k. Lt(S k,p) → Lt(k,p)

Every purple notation token opens its conservative definition. This reading surface never changes the unchanged intuitionistic kernel or confers checked-use authority.

Definitions used by this theorem

In the theorem statement

PD0002 Lt

In local proof propositions

none
Exact expanded first-order statement
forall p k. (exists lcv_gap_predecessor_source. lcv_gap_predecessor_source + S (S k) = (p)) -> (exists lcv_gap_predecessor_result. lcv_gap_predecessor_result + S (k) = (p))

Proof neighborhood

Direct theorem prerequisites

le_succ_self · Stable closed le_trans · Stable closed

Direct theorem dependents

LU000W lucas_prime_shift_below_base

Definition-aware tactic body

Only propositions whose conservative expansion has been checked for exact first-order equivalence are compacted. Every changed line retains its immutable exact replay command.

Read the argument

Proof checkpoints

10 script commands · 2 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 assumptionsL1–3

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

  1. L1
    intro p
  2. L2
    intro k
  3. L3
    intro hbound
02Use earlier factsL4–10

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

  1. L4
    specialize le_trans (S k)
  2. L5
    specialize le_trans (S (S k))
  3. L6
    specialize le_trans p
  4. L7
    apply le_trans
  5. L8
    specialize le_succ_self (S k)
  6. L9
    exact le_succ_self
  7. L10
    exact hbound

Library-wide reading audit

Original defined command ledger · 10 lines
  1. 0001intro p
  2. 0002intro k
  3. 0003intro hbound
  4. 0004specialize le_trans (S k)
  5. 0005specialize le_trans (S (S k))
  6. 0006specialize le_trans p
  7. 0007apply le_trans
  8. 0008specialize le_succ_self (S k)
  9. 0009exact le_succ_self
  10. 0010exact hbound