LU0019 · theorem body

lucas_digit_chain_exists

Alpha v34 checked-use · independently kernel and Lean verified; not Stable

Every nonzero base and natural input possess a coherent successive quotient/digit chain of every finite requested length.

historical independently replay-verified empty-context experiment; the experiment itself persisted no certificate and granted no release authority; current checked use follows separately sealed proof bundles; no Stable promotion.

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. ∀ l. ¬p = 0 → ∃ x. ∃ y. ∃ z. ∃ m. BetaAt(x,y,0,n) ∧ (∀ k. Lt(k,l) → ∃ i. ∃ j. ∃ u. BetaAt(x,y,k,i) ∧ (BetaAt(x,y,S k,j) ∧ (BetaAt(z,m,k,u) ∧ DivRem(i,p,j,u))))

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 PD0013 BetaAt

In local proof propositions

PD0002 Lt PD0007 DivRem PD0013 BetaAt
Exact expanded first-order statement
forall p n l. ~(p = 0) -> exists qb qc db dc. (((((exists ff_h_lmd_total_result_initial. ff_h_lmd_total_result_initial + S (n) = S ((S (0)) * qc)) /\ exists ff_q_lmd_total_result_initial. qb = ff_q_lmd_total_result_initial * S ((S (0)) * qc) + (n))) /\ forall lmd_index_total_result. (exists lmd_gap_total_result_index. lmd_gap_total_result_index + S (lmd_index_total_result) = (l)) -> exists lmd_current_total_result lmd_successor_total_result lmd_digit_total_result. ((((exists ff_h_lmd_total_result_current. ff_h_lmd_total_result_current + S (lmd_current_total_result) = S ((S (lmd_index_total_result)) * qc)) /\ exists ff_q_lmd_total_result_current. qb = ff_q_lmd_total_result_current * S ((S (lmd_index_total_result)) * qc) + (lmd_current_total_result))) /\ ((((exists ff_h_lmd_total_result_successor. ff_h_lmd_total_result_successor + S (lmd_successor_total_result) = S ((S (S lmd_index_total_result)) * qc)) /\ exists ff_q_lmd_total_result_successor. qb = ff_q_lmd_total_result_successor * S ((S (S lmd_index_total_result)) * qc) + (lmd_successor_total_result))) /\ ((((exists ff_h_lmd_total_result_digit. ff_h_lmd_total_result_digit + S (lmd_digit_total_result) = S ((S (lmd_index_total_result)) * dc)) /\ exists ff_q_lmd_total_result_digit. db = ff_q_lmd_total_result_digit * S ((S (lmd_index_total_result)) * dc) + (lmd_digit_total_result))) /\ ((lmd_current_total_result = (p) * (lmd_successor_total_result) + (lmd_digit_total_result)) /\ (exists lmd_gap_total_result_digit_bound. lmd_gap_total_result_digit_bound + S (lmd_digit_total_result) = (p))))))))

Proof neighborhood

Direct theorem prerequisites

LU0017 lucas_digit_chain_empty_exists LU0018 lucas_digit_chain_extend

Direct theorem dependents

LU001A lucas_prime_digit_chain_exists

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 · 5 reading checkpoints · 1 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 (2)
lucas digit chain empty existslucas digit chain extend
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 l
02Induction on lL4–7

Split the argument into the base and successor obligations. The induction hypothesis is available only in the successor branch.

  1. L4
    induction l
  2. L5
    intro hnonzero
  3. L6
    apply lucas_digit_chain_empty_exists
  4. L7
    intro hnonzero
03Establish hpreviousL8–10

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply IH.

  1. L8
    have hprevious : ∃ qb. ∃ qc. ∃ db. ∃ dc. BetaAt(qb,qc,0,n) ∧ (∀ x. Lt(x,l) → ∃ y. ∃ z. ∃ m. BetaAt(qb,qc,x,y) ∧ (BetaAt(qb,qc,S x,z) ∧ (BetaAt(db,dc,x,m) ∧ DivRem(y,p,z,m))))Definitions: BetaAt(qb,qc,0,n)Lt(x,l)BetaAt(qb,qc,x,y)BetaAt(qb,qc,S x,z)BetaAt(db,dc,x,m)DivRem(y,p,z,m)Original native command in the exact edition
  2. L9
    apply IH
  3. L10
    exact hnonzero
04Separate the logical casesL11–14

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

  1. L11
    cases hprevious
  2. L12
    cases hprevious_witness
  3. L13
    cases hprevious_witness_witness
  4. L14
    cases hprevious_witness_witness_witness
05Use earlier factsL15–24

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

  1. L15
    specialize lucas_digit_chain_extend p
  2. L16
    specialize lucas_digit_chain_extend n
  3. L17
    specialize lucas_digit_chain_extend x
  4. L18
    specialize lucas_digit_chain_extend x1
  5. L19
    specialize lucas_digit_chain_extend x2
  6. L20
    specialize lucas_digit_chain_extend x3
  7. L21
    specialize lucas_digit_chain_extend l
  8. L22
    apply lucas_digit_chain_extend
  9. L23
    exact hnonzero
  10. L24
    exact hprevious_witness_witness_witness_witness

Library-wide reading audit

Original defined command ledger · 24 lines
  1. 0001intro p
  2. 0002intro n
  3. 0003intro l
  4. 0004induction l
  5. 0005intro hnonzero
  6. 0006apply lucas_digit_chain_empty_exists
  7. 0007intro hnonzero
  8. 0008have hprevious : ∃ qb. ∃ qc. ∃ db. ∃ dc. BetaAt(qb,qc,0,n) ∧ (∀ x. Lt(x,l) → ∃ y. ∃ z. ∃ m. BetaAt(qb,qc,x,y) ∧ (BetaAt(qb,qc,S x,z) ∧ (BetaAt(db,dc,x,m) ∧ DivRem(y,p,z,m))))
    Exact native replay linehave hprevious : exists qb qc db dc. (((((exists ff_h_lmd_total_previous_initial. ff_h_lmd_total_previous_initial + S (n) = S ((S (0)) * qc)) /\ exists ff_q_lmd_total_previous_initial. qb = ff_q_lmd_total_previous_initial * S ((S (0)) * qc) + (n))) /\ forall lmd_index_total_previous. (exists lmd_gap_total_previous_index. lmd_gap_total_previous_index + S (lmd_index_total_previous) = (l)) -> exists lmd_current_total_previous lmd_successor_total_previous lmd_digit_total_previous. ((((exists ff_h_lmd_total_previous_current. ff_h_lmd_total_previous_current + S (lmd_current_total_previous) = S ((S (lmd_index_total_previous)) * qc)) /\ exists ff_q_lmd_total_previous_current. qb = ff_q_lmd_total_previous_current * S ((S (lmd_index_total_previous)) * qc) + (lmd_current_total_previous))) /\ ((((exists ff_h_lmd_total_previous_successor. ff_h_lmd_total_previous_successor + S (lmd_successor_total_previous) = S ((S (S lmd_index_total_previous)) * qc)) /\ exists ff_q_lmd_total_previous_successor. qb = ff_q_lmd_total_previous_successor * S ((S (S lmd_index_total_previous)) * qc) + (lmd_successor_total_previous))) /\ ((((exists ff_h_lmd_total_previous_digit. ff_h_lmd_total_previous_digit + S (lmd_digit_total_previous) = S ((S (lmd_index_total_previous)) * dc)) /\ exists ff_q_lmd_total_previous_digit. db = ff_q_lmd_total_previous_digit * S ((S (lmd_index_total_previous)) * dc) + (lmd_digit_total_previous))) /\ ((lmd_current_total_previous = (p) * (lmd_successor_total_previous) + (lmd_digit_total_previous)) /\ (exists lmd_gap_total_previous_digit_bound. lmd_gap_total_previous_digit_bound + S (lmd_digit_total_previous) = (p))))))))
  9. 0009apply IH
  10. 0010exact hnonzero
  11. 0011cases hprevious
  12. 0012cases hprevious_witness
  13. 0013cases hprevious_witness_witness
  14. 0014cases hprevious_witness_witness_witness
  15. 0015specialize lucas_digit_chain_extend p
  16. 0016specialize lucas_digit_chain_extend n
  17. 0017specialize lucas_digit_chain_extend x
  18. 0018specialize lucas_digit_chain_extend x1
  19. 0019specialize lucas_digit_chain_extend x2
  20. 0020specialize lucas_digit_chain_extend x3
  21. 0021specialize lucas_digit_chain_extend l
  22. 0022apply lucas_digit_chain_extend
  23. 0023exact hnonzero
  24. 0024exact hprevious_witness_witness_witness_witness