LU0007 · theorem body

lucas_base_p_digit_functional

dependency-curried kernel-checked candidate body; not enrolled in Alpha or Stable

Both the quotient and the bounded base-p digit are constructively functional.

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. ∀ n. ∀ q. ∀ d. ∀ Q. ∀ D. DivRem(n,p,q,d) → DivRem(n,p,Q,D) → q = Q ∧ d = D

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

PD0007 DivRem

In local proof propositions

none
Exact expanded first-order statement
forall p n q d Q D. (((n) = (p) * (q) + (d)) /\ (exists ldc_lt_native_bound. ldc_lt_native_bound + S (d) = p)) -> (((n) = (p) * (Q) + (D)) /\ (exists ldc_lt_alternate_bound. ldc_lt_alternate_bound + S (D) = p)) -> ((q = Q) /\ (d = D))

Proof neighborhood

Direct theorem prerequisites

division_remainder_unique · Stable closed

Direct theorem dependents

none

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

21 script commands · 4 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 assumptionsL1–8

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

  1. L1
    intro p
  2. L2
    intro n
  3. L3
    intro q
  4. L4
    intro d
  5. L5
    intro Q
  6. L6
    intro D
  7. L7
    intro hfirst
  8. L8
    intro hsecond
02Separate the logical casesL9–10

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

  1. L9
    cases hfirst
  2. L10
    cases hsecond
03Use earlier factsL11–20

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

  1. L11
    specialize division_remainder_unique p
  2. L12
    specialize division_remainder_unique n
  3. L13
    specialize division_remainder_unique q
  4. L14
    specialize division_remainder_unique d
  5. L15
    specialize division_remainder_unique Q
  6. L16
    specialize division_remainder_unique D
  7. L17
    apply division_remainder_unique
  8. L18
    exact hfirst_left
  9. L19
    exact hfirst_right
  10. L20
    exact hsecond_left
04Use earlier factsL21–21

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

  1. L21
    exact hsecond_right

Library-wide reading audit

Original defined command ledger · 21 lines
  1. 0001intro p
  2. 0002intro n
  3. 0003intro q
  4. 0004intro d
  5. 0005intro Q
  6. 0006intro D
  7. 0007intro hfirst
  8. 0008intro hsecond
  9. 0009cases hfirst
  10. 0010cases hsecond
  11. 0011specialize division_remainder_unique p
  12. 0012specialize division_remainder_unique n
  13. 0013specialize division_remainder_unique q
  14. 0014specialize division_remainder_unique d
  15. 0015specialize division_remainder_unique Q
  16. 0016specialize division_remainder_unique D
  17. 0017apply division_remainder_unique
  18. 0018exact hfirst_left
  19. 0019exact hfirst_right
  20. 0020exact hsecond_left
  21. 0021exact hsecond_right