CF0003

continued_fraction_empty_trace_exists

Alpha v34 independently verified · alpha_closed; checked-use authorized; not Stable

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

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 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)))))))))))

Constructive proof overview

Generated structural guide

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

The unchanged tactic script uses 2 declared prerequisites and contains 8 exact native proof lines.

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

Proof neighborhood

Direct dependencies

CF0001 continued_fraction_initial_state_exists CF0002 continued_fraction_empty_trace

Direct dependents

CF0005 continued_fraction_trace_exists_up_to

Formal native tactic body

Dependencies are introduced as named hypotheses before line 1. Local theorem links identify exact declared prerequisites. This exact body belongs to a complete independently kernel-checked constructive proof bundle and has Alpha checked-use authority; it does not imply Stable membership.

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.

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 exact 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