LU000S · theorem body

lucas_positive_digit_has_bounded_complement

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

Every nonzero digit strictly below its prime base has a complementary digit strictly below that 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. ¬k = 0 → Lt(k,p) → ∃ x. k + x = p ∧ Lt(x,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. ~(k = 0) -> (exists lcv_gap_complement_source. lcv_gap_complement_source + S (k) = (p)) -> exists j. ((k + j = p) /\ (exists lcv_gap_complement_result. lcv_gap_complement_result + S (j) = (p)))

Proof neighborhood

Direct theorem prerequisites

nonzero_is_succ · Stable closed add_comm · Stable closed add_succ_left · Stable closed

Direct theorem dependents

LU000T lucas_prime_row_interior_zero_mod

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

22 script commands · 10 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 assumptionsL1–4

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

  1. L1
    intro p
  2. L2
    intro k
  3. L3
    intro hnonzero
  4. L4
    intro hbound
02Establish hsuccessorL5–8

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply nonzero is succ.

  1. L5
    have hsuccessor : exists a. k = S a
  2. L6
    specialize nonzero_is_succ k
  3. L7
    apply nonzero_is_succ
  4. L8
    exact hnonzero
03Separate the logical casesL9–10

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

  1. L9
    cases hsuccessor
  2. L10
    cases hbound
04Construct an explicit witnessL11–11

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

  1. L11
    exists S x1
05Separate the logical casesL12–12

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

  1. L12
    split
06Calculate and transport equalitiesL13–16

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

  1. L13
    rewrite hsuccessor_witness
  2. L14
    trans x1 + S (S x)
  3. L15
    simp [add_comm, add_succ_left]
  4. L16
    rewrite hsuccessor_witness at hbound_witness
07Use earlier factsL17–17

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

  1. L17
    exact hbound_witness
08Construct an explicit witnessL18–18

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

  1. L18
    exists x
09Calculate and transport equalitiesL19–21

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

  1. L19
    trans x1 + S (S x)
  2. L20
    simp [add_comm, add_succ_left]
  3. L21
    rewrite hsuccessor_witness at hbound_witness
10Use earlier factsL22–22

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

  1. L22
    exact hbound_witness

Library-wide reading audit

Original defined command ledger · 22 lines
  1. 0001intro p
  2. 0002intro k
  3. 0003intro hnonzero
  4. 0004intro hbound
  5. 0005have hsuccessor : exists a. k = S a
  6. 0006specialize nonzero_is_succ k
  7. 0007apply nonzero_is_succ
  8. 0008exact hnonzero
  9. 0009cases hsuccessor
  10. 0010cases hbound
  11. 0011exists S x1
  12. 0012split
  13. 0013rewrite hsuccessor_witness
  14. 0014trans x1 + S (S x)
  15. 0015simp [add_comm, add_succ_left]
  16. 0016rewrite hsuccessor_witness at hbound_witness
  17. 0017exact hbound_witness
  18. 0018exists x
  19. 0019trans x1 + S (S x)
  20. 0020simp [add_comm, add_succ_left]
  21. 0021rewrite hsuccessor_witness at hbound_witness
  22. 0022exact hbound_witness