LU0007

lucas_base_p_digit_functional

Dependency-curried candidate body; not Alpha-enrolled; no checked-use authority

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.

Exact expanded first-order arithmetic 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))

Constructive proof overview

Generated structural guide

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

The unchanged tactic script uses 1 declared prerequisite and contains 21 exact native proof lines.

dependency-curried kernel-checked theorem body; Alpha enrollment and checked-use authority follow separately sealed release evidence; Stable membership remains unchanged

Proof neighborhood

Direct dependencies

division_remainder_unique Stable theorem; checked-use authorized

Direct dependents

none

Formal native tactic body

Dependencies are introduced as named hypotheses before line 1. Local theorem links identify exact declared prerequisites. This dependency-curried candidate body does not grant checked theorem use or Stable membership.

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.

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 exact 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