LU0009 · theorem body

lucas_base_p_zero_digit_iff_divides

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

The canonical least-significant base-p digit is zero exactly when its value is divisible by p.

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. DivRem(n,p,q,d) → (d = 0 → Dvd(p,n)) ∧ (Dvd(p,n) → d = 0)

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

PD0003 Dvd PD0007 DivRem

In local proof propositions

none
Exact expanded first-order statement
forall p n q d. (((n) = (p) * (q) + (d)) /\ (exists ldc_lt_native_bound. ldc_lt_native_bound + S (d) = p)) -> ((((d = 0) -> (exists ldc_quotient_native_zero. n = p * ldc_quotient_native_zero)) /\ ((exists ldc_quotient_native_zero. n = p * ldc_quotient_native_zero) -> (d = 0))))

Proof neighborhood

Direct theorem prerequisites

zero_remainder_implies_multiple · Stable closed eq_decidable · Stable closed nonzero_remainder_not_multiple · Alpha 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

29 script commands · 11 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 assumptionsL1–5

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 hdigit
02Separate the logical casesL6–7

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

  1. L6
    cases hdigit
  2. L7
    split
03Fix variables and assumptionsL8–8

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

  1. L8
    intro hzero
04Calculate and transport equalitiesL9–9

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

  1. L9
    rewrite hzero at hdigit_left
05Use earlier factsL10–14

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

  1. L10
    specialize zero_remainder_implies_multiple p
  2. L11
    specialize zero_remainder_implies_multiple n
  3. L12
    specialize zero_remainder_implies_multiple q
  4. L13
    apply zero_remainder_implies_multiple
  5. L14
    exact hdigit_left
06Fix variables and assumptionsL15–15

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

  1. L15
    intro hdivides
07Use earlier factsL16–17

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

  1. L16
    specialize eq_decidable d
  2. L17
    specialize eq_decidable 0
08Separate the logical casesL18–18

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

  1. L18
    cases eq_decidable
09Use earlier factsL19–19

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

  1. L19
    exact eq_decidable_left
10Separate the logical casesL20–20

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

  1. L20
    exfalso
11Use earlier factsL21–29

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

  1. L21
    specialize nonzero_remainder_not_multiple p
  2. L22
    specialize nonzero_remainder_not_multiple n
  3. L23
    specialize nonzero_remainder_not_multiple q
  4. L24
    specialize nonzero_remainder_not_multiple d
  5. L25
    apply nonzero_remainder_not_multiple
  6. L26
    exact hdigit_left
  7. L27
    exact eq_decidable_right
  8. L28
    exact hdigit_right
  9. L29
    exact hdivides

Library-wide reading audit

Original defined command ledger · 29 lines
  1. 0001intro p
  2. 0002intro n
  3. 0003intro q
  4. 0004intro d
  5. 0005intro hdigit
  6. 0006cases hdigit
  7. 0007split
  8. 0008intro hzero
  9. 0009rewrite hzero at hdigit_left
  10. 0010specialize zero_remainder_implies_multiple p
  11. 0011specialize zero_remainder_implies_multiple n
  12. 0012specialize zero_remainder_implies_multiple q
  13. 0013apply zero_remainder_implies_multiple
  14. 0014exact hdigit_left
  15. 0015intro hdivides
  16. 0016specialize eq_decidable d
  17. 0017specialize eq_decidable 0
  18. 0018cases eq_decidable
  19. 0019exact eq_decidable_left
  20. 0020exfalso
  21. 0021specialize nonzero_remainder_not_multiple p
  22. 0022specialize nonzero_remainder_not_multiple n
  23. 0023specialize nonzero_remainder_not_multiple q
  24. 0024specialize nonzero_remainder_not_multiple d
  25. 0025apply nonzero_remainder_not_multiple
  26. 0026exact hdigit_left
  27. 0027exact eq_decidable_right
  28. 0028exact hdigit_right
  29. 0029exact hdivides