LU0008 · theorem body

lucas_base_p_digit_of_small_value

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

A value strictly below its base has quotient zero and itself as its unique canonical digit.

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

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 PD0007 DivRem

In local proof propositions

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

Proof neighborhood

Direct theorem prerequisites

division_remainder_unique · Stable closed zero_add · 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

18 script commands · 5 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 assumptionsL1–6

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 hsmall
  6. L6
    intro hdigit
02Separate the logical casesL7–7

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

  1. L7
    cases hdigit
03Use earlier factsL8–16

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

  1. L8
    specialize division_remainder_unique p
  2. L9
    specialize division_remainder_unique n
  3. L10
    specialize division_remainder_unique q
  4. L11
    specialize division_remainder_unique d
  5. L12
    specialize division_remainder_unique 0
  6. L13
    specialize division_remainder_unique n
  7. L14
    apply division_remainder_unique
  8. L15
    exact hdigit_left
  9. L16
    exact hdigit_right
04Calculate and transport equalitiesL17–17

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

  1. L17
    simp [zero_add]
05Use earlier factsL18–18

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

  1. L18
    exact hsmall

Library-wide reading audit

Original defined command ledger · 18 lines
  1. 0001intro p
  2. 0002intro n
  3. 0003intro q
  4. 0004intro d
  5. 0005intro hsmall
  6. 0006intro hdigit
  7. 0007cases hdigit
  8. 0008specialize division_remainder_unique p
  9. 0009specialize division_remainder_unique n
  10. 0010specialize division_remainder_unique q
  11. 0011specialize division_remainder_unique d
  12. 0012specialize division_remainder_unique 0
  13. 0013specialize division_remainder_unique n
  14. 0014apply division_remainder_unique
  15. 0015exact hdigit_left
  16. 0016exact hdigit_right
  17. 0017simp [zero_add]
  18. 0018exact hsmall