CF0003

continued_fraction_empty_trace_exists

For every dividend, a fully witnessed empty reverse Euclidean history exists at divisor zero.

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. ContinuedFractionTrace(a,0,0,z,c,0)

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

Definition DAG

ContinuedFractionTrace(a,b,s,u,v,ell) · 1

Actual proof prerequisites

continued_fraction_initial_state_existscontinued_fraction_empty_trace
Original expanded first-order statement 
forall a. exists z c. (exists cf_gcd_empty. ((((exists ff_h_cf_empty_initial_state. ff_h_cf_empty_initial_state + S (((cf_gcd_empty) + (((0) + (0)) * S ((0) + (0)) + ((0) + (0)))) * S ((cf_gcd_empty) + (((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_empty_initial_state. z = ff_q_cf_empty_initial_state * S ((S (0)) * c) + (((cf_gcd_empty) + (((0) + (0)) * S ((0) + (0)) + ((0) + (0)))) * S ((cf_gcd_empty) + (((0) + (0)) * S ((0) + (0)) + ((0) + (0)))) + ((((0) + (0)) * S ((0) + (0)) + ((0) + (0))) + (((0) + (0)) * S ((0) + (0)) + ((0) + (0))))))) /\ ((((exists ff_h_cf_empty_terminal_state. ff_h_cf_empty_terminal_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_empty_terminal_state. z = ff_q_cf_empty_terminal_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))))))) /\ forall cf_index_empty. (exists ff_lt_cf_empty_index. ff_lt_cf_empty_index + S cf_index_empty = 0) -> exists cf_old_a_empty cf_old_b_empty cf_tail_empty cf_new_a_empty cf_new_b_empty cf_head_empty cf_quotient_empty. ((((exists ff_h_cf_empty_previous_state. ff_h_cf_empty_previous_state + S (((cf_old_a_empty) + (((cf_old_b_empty) + (cf_tail_empty)) * S ((cf_old_b_empty) + (cf_tail_empty)) + ((cf_tail_empty) + (cf_tail_empty)))) * S ((cf_old_a_empty) + (((cf_old_b_empty) + (cf_tail_empty)) * S ((cf_old_b_empty) + (cf_tail_empty)) + ((cf_tail_empty) + (cf_tail_empty)))) + ((((cf_old_b_empty) + (cf_tail_empty)) * S ((cf_old_b_empty) + (cf_tail_empty)) + ((cf_tail_empty) + (cf_tail_empty))) + (((cf_old_b_empty) + (cf_tail_empty)) * S ((cf_old_b_empty) + (cf_tail_empty)) + ((cf_tail_empty) + (cf_tail_empty))))) = S ((S (cf_index_empty)) * c)) /\ exists ff_q_cf_empty_previous_state. z = ff_q_cf_empty_previous_state * S ((S (cf_index_empty)) * c) + (((cf_old_a_empty) + (((cf_old_b_empty) + (cf_tail_empty)) * S ((cf_old_b_empty) + (cf_tail_empty)) + ((cf_tail_empty) + (cf_tail_empty)))) * S ((cf_old_a_empty) + (((cf_old_b_empty) + (cf_tail_empty)) * S ((cf_old_b_empty) + (cf_tail_empty)) + ((cf_tail_empty) + (cf_tail_empty)))) + ((((cf_old_b_empty) + (cf_tail_empty)) * S ((cf_old_b_empty) + (cf_tail_empty)) + ((cf_tail_empty) + (cf_tail_empty))) + (((cf_old_b_empty) + (cf_tail_empty)) * S ((cf_old_b_empty) + (cf_tail_empty)) + ((cf_tail_empty) + (cf_tail_empty))))))) /\ ((((exists ff_h_cf_empty_following_state. ff_h_cf_empty_following_state + S (((cf_new_a_empty) + (((cf_new_b_empty) + (cf_head_empty)) * S ((cf_new_b_empty) + (cf_head_empty)) + ((cf_head_empty) + (cf_head_empty)))) * S ((cf_new_a_empty) + (((cf_new_b_empty) + (cf_head_empty)) * S ((cf_new_b_empty) + (cf_head_empty)) + ((cf_head_empty) + (cf_head_empty)))) + ((((cf_new_b_empty) + (cf_head_empty)) * S ((cf_new_b_empty) + (cf_head_empty)) + ((cf_head_empty) + (cf_head_empty))) + (((cf_new_b_empty) + (cf_head_empty)) * S ((cf_new_b_empty) + (cf_head_empty)) + ((cf_head_empty) + (cf_head_empty))))) = S ((S (S cf_index_empty)) * c)) /\ exists ff_q_cf_empty_following_state. z = ff_q_cf_empty_following_state * S ((S (S cf_index_empty)) * c) + (((cf_new_a_empty) + (((cf_new_b_empty) + (cf_head_empty)) * S ((cf_new_b_empty) + (cf_head_empty)) + ((cf_head_empty) + (cf_head_empty)))) * S ((cf_new_a_empty) + (((cf_new_b_empty) + (cf_head_empty)) * S ((cf_new_b_empty) + (cf_head_empty)) + ((cf_head_empty) + (cf_head_empty)))) + ((((cf_new_b_empty) + (cf_head_empty)) * S ((cf_new_b_empty) + (cf_head_empty)) + ((cf_head_empty) + (cf_head_empty))) + (((cf_new_b_empty) + (cf_head_empty)) * S ((cf_new_b_empty) + (cf_head_empty)) + ((cf_head_empty) + (cf_head_empty))))))) /\ (cf_new_b_empty = cf_old_a_empty /\ (cf_new_a_empty = cf_new_b_empty * cf_quotient_empty + cf_old_b_empty /\ ((exists ff_lt_cf_empty_remainder. ff_lt_cf_empty_remainder + S cf_old_b_empty = cf_new_b_empty) /\ (cf_head_empty = S ((cf_quotient_empty + cf_tail_empty) * S (cf_quotient_empty + cf_tail_empty) + (cf_tail_empty + cf_tail_empty)))))))))))

Complete unchanged native tactic proof

All 8 lines are the exact independently kernel-checked original script.

Read the argument

Proof checkpoints

8 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.

Named ingredients (2)
continued fraction initial state existscontinued fraction empty trace
01Fix variables and assumptionsL1–1

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

  1. L1
    intro a
02Use earlier factsL2–2

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

  1. L2
    specialize continued_fraction_initial_state_exists a
03Separate the logical casesL3–4

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

  1. L3
    cases continued_fraction_initial_state_exists
  2. L4
    cases continued_fraction_initial_state_exists_witness
04Construct an explicit witnessL5–6

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

  1. L5
    exists x
  2. L6
    exists x1
05Use earlier factsL7–8

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

  1. L7
    apply continued_fraction_empty_trace
  2. L8
    exact continued_fraction_initial_state_exists_witness_witness

Library-wide reading audit

Original defined command ledger · 8 lines
  1. 0001intro a
  2. 0002specialize continued_fraction_initial_state_exists a
  3. 0003cases continued_fraction_initial_state_exists
  4. 0004cases continued_fraction_initial_state_exists_witness
  5. 0005exists x
  6. 0006exists x1
  7. 0007apply continued_fraction_empty_trace
  8. 0008exact continued_fraction_initial_state_exists_witness_witness