CF0001

continued_fraction_initial_state_exists

Every natural initializes an actual beta-coded terminal Euclidean state (a,0,nil).

Alpha v34 checked-use · first admitted v20 · independently kernel and Lean verified; not Stable

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 theorem in conservative defined notation

∀ a. ∃ z. ∃ c. Beta(z,c,0,(a + ((0 + 0) · S (0 + 0) + (0 + 0))) · S (a + ((0 + 0) · S (0 + 0) + (0 + 0))) + ((0 + 0) · S (0 + 0) + (0 + 0) + ((0 + 0) · S (0 + 0) + (0 + 0))))

Every linked abbreviation expands hygienically to the identical original native formula.

Definition DAG

Beta(b,c,i,x) · 1

Actual proof prerequisites

beta_prefix_extend · checked external prerequisite
Original expanded first-order statement 
forall a. exists z c. (((exists ff_h_cf_initial_exists_state. ff_h_cf_initial_exists_state + S (((a) + (((0) + (0)) * S ((0) + (0)) + ((0) + (0)))) * S ((a) + (((0) + (0)) * S ((0) + (0)) + ((0) + (0)))) + ((((0) + (0)) * S ((0) + (0)) + ((0) + (0))) + (((0) + (0)) * S ((0) + (0)) + ((0) + (0))))) = S ((S (0)) * c)) /\ exists ff_q_cf_initial_exists_state. z = ff_q_cf_initial_exists_state * S ((S (0)) * c) + (((a) + (((0) + (0)) * S ((0) + (0)) + ((0) + (0)))) * S ((a) + (((0) + (0)) * S ((0) + (0)) + ((0) + (0)))) + ((((0) + (0)) * S ((0) + (0)) + ((0) + (0))) + (((0) + (0)) * S ((0) + (0)) + ((0) + (0)))))))

Complete unchanged native tactic proof

All 11 lines are the exact independently kernel-checked original script.

Read the argument

Proof checkpoints

11 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 assumptionsL1–1

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

  1. L1
    intro a
02Use earlier factsL2–5

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

  1. L2
    specialize beta_prefix_extend 0
  2. L3
    specialize beta_prefix_extend 0
  3. L4
    specialize beta_prefix_extend 0
  4. L5
    specialize beta_prefix_extend (((a) + (((0) + (0)) * S ((0) + (0)) + ((0) + (0)))) * S ((a) + (((0) + (0)) * S ((0) + (0)) + ((0) + (0)))) + ((((0) + (0)) * S ((0) + (0)) + ((0) + (0))) + (((0) + (0)) * S ((0) + (0)) + ((0) + (0)))))
03Separate the logical casesL6–8

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

  1. L6
    cases beta_prefix_extend
  2. L7
    cases beta_prefix_extend_witness
  3. L8
    cases beta_prefix_extend_witness_witness
04Construct an explicit witnessL9–10

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

  1. L9
    exists x
  2. L10
    exists x1
05Use earlier factsL11–11

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

  1. L11
    exact beta_prefix_extend_witness_witness_left

Library-wide reading audit

Original defined command ledger · 11 lines
  1. 0001intro a
  2. 0002specialize beta_prefix_extend 0
  3. 0003specialize beta_prefix_extend 0
  4. 0004specialize beta_prefix_extend 0
  5. 0005specialize beta_prefix_extend (((a) + (((0) + (0)) * S ((0) + (0)) + ((0) + (0)))) * S ((a) + (((0) + (0)) * S ((0) + (0)) + ((0) + (0)))) + ((((0) + (0)) * S ((0) + (0)) + ((0) + (0))) + (((0) + (0)) * S ((0) + (0)) + ((0) + (0)))))
  6. 0006cases beta_prefix_extend
  7. 0007cases beta_prefix_extend_witness
  8. 0008cases beta_prefix_extend_witness_witness
  9. 0009exists x
  10. 0010exists x1
  11. 0011exact beta_prefix_extend_witness_witness_left