LU000D

lucas_base_p_two_digit_total

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

Every nonzero base constructively extracts two genuinely successive digits: the second source is exactly the first quotient.

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. ~(p = 0) -> exists q0 d0 q1 d1. (((((n) = (p) * (q0) + (d0)) /\ (exists ldc_lt_first_bound. ldc_lt_first_bound + S (d0) = p)) /\ (((q0) = (p) * (q1) + (d1)) /\ (exists ldc_lt_second_bound. ldc_lt_second_bound + S (d1) = p))))

Constructive proof overview

Generated structural guide

Every nonzero base constructively extracts two genuinely successive digits: the second source is exactly the first quotient.

The unchanged tactic script uses 1 declared prerequisite and contains 24 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

LU0005 lucas_base_p_digit_total

Direct dependents

LU000E lucas_prime_base_two_digit_total

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

24 script commands · 8 reading checkpoints · 2 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.

Named ingredients (1)
lucas base p digit total
01Fix variables and assumptionsL1–3

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

  1. L1
    intro p
  2. L2
    intro n
  3. L3
    intro hnonzero
02Establish hfirstL4–8

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply lucas base p digit total.

  1. L4
    have hfirst : exists q0 d0. (((n) = (p) * (q0) + (d0)) /\ (exists ldc_lt_first_bound. ldc_lt_first_bound + S (d0) = p))
  2. L5
    specialize lucas_base_p_digit_total p
  3. L6
    specialize lucas_base_p_digit_total n
  4. L7
    apply lucas_base_p_digit_total
  5. L8
    exact hnonzero
03Separate the logical casesL9–10

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

  1. L9
    cases hfirst
  2. L10
    cases hfirst_witness
04Establish hsecondL11–15

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply lucas base p digit total.

  1. L11
    have hsecond : exists q1 d1. (((x) = (p) * (q1) + (d1)) /\ (exists ldc_lt_second_exists_bound. ldc_lt_second_exists_bound + S (d1) = p))
  2. L12
    specialize lucas_base_p_digit_total p
  3. L13
    specialize lucas_base_p_digit_total x
  4. L14
    apply lucas_base_p_digit_total
  5. L15
    exact hnonzero
05Separate the logical casesL16–17

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

  1. L16
    cases hsecond
  2. L17
    cases hsecond_witness
06Construct an explicit witnessL18–21

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

  1. L18
    exists x
  2. L19
    exists x1
  3. L20
    exists x2
  4. L21
    exists x3
07Separate the logical casesL22–22

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

  1. L22
    split
08Use earlier factsL23–24

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

  1. L23
    exact hfirst_witness_witness
  2. L24
    exact hsecond_witness_witness

Library-wide reading audit

Original exact command ledger · 24 lines
  1. 0001intro p
  2. 0002intro n
  3. 0003intro hnonzero
  4. 0004have hfirst : exists q0 d0. (((n) = (p) * (q0) + (d0)) /\ (exists ldc_lt_first_bound. ldc_lt_first_bound + S (d0) = p))
  5. 0005specialize lucas_base_p_digit_total p
  6. 0006specialize lucas_base_p_digit_total n
  7. 0007apply lucas_base_p_digit_total
  8. 0008exact hnonzero
  9. 0009cases hfirst
  10. 0010cases hfirst_witness
  11. 0011have hsecond : exists q1 d1. (((x) = (p) * (q1) + (d1)) /\ (exists ldc_lt_second_exists_bound. ldc_lt_second_exists_bound + S (d1) = p))
  12. 0012specialize lucas_base_p_digit_total p
  13. 0013specialize lucas_base_p_digit_total x
  14. 0014apply lucas_base_p_digit_total
  15. 0015exact hnonzero
  16. 0016cases hsecond
  17. 0017cases hsecond_witness
  18. 0018exists x
  19. 0019exists x1
  20. 0020exists x2
  21. 0021exists x3
  22. 0022split
  23. 0023exact hfirst_witness_witness
  24. 0024exact hsecond_witness_witness