LU000D · theorem body

lucas_base_p_two_digit_total

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

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.

Statement with defined notation

∀ p. ∀ n. ¬p = 0 → ∃ x. ∃ y. ∃ z. ∃ m. DivRem(n,p,x,y) ∧ DivRem(x,p,z,m)

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

PD0007 DivRem
Exact expanded first-order 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))))

Proof neighborhood

Direct theorem prerequisites

LU0005 lucas_base_p_digit_total

Direct theorem dependents

LU000E lucas_prime_base_two_digit_total

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

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.

Definition notation is shown below. Open the paired exact edition for the original native formulas. Source pairing is not a new equivalence certificate.

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 : ∃ q0. ∃ d0. DivRem(n,p,q0,d0)Definitions: DivRem(n,p,q0,d0)Original native command in the exact edition
  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 : ∃ q1. ∃ d1. DivRem(x,p,q1,d1)Definitions: DivRem(x,p,q1,d1)Original native command in the exact edition
  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 defined command ledger · 24 lines
  1. 0001intro p
  2. 0002intro n
  3. 0003intro hnonzero
  4. 0004have hfirst : ∃ q0. ∃ d0. DivRem(n,p,q0,d0)
    Exact native replay linehave 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 : ∃ q1. ∃ d1. DivRem(x,p,q1,d1)
    Exact native replay linehave 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