EC000D

euclidean_execution_exists

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

All natural input pairs admit a fully expanded, beta-coded terminating Euclidean execution with a certified relational gcd output.

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 b. exists g l. (exists ec_list_execution ec_history_execution ec_scale_execution. ((exists cf_gcd_ec_execution_trace. ((((exists ff_h_cf_ec_execution_trace_initial_state. ff_h_cf_ec_execution_trace_initial_state + S (((cf_gcd_ec_execution_trace) + (((0) + (0)) * S ((0) + (0)) + ((0) + (0)))) * S ((cf_gcd_ec_execution_trace) + (((0) + (0)) * S ((0) + (0)) + ((0) + (0)))) + ((((0) + (0)) * S ((0) + (0)) + ((0) + (0))) + (((0) + (0)) * S ((0) + (0)) + ((0) + (0))))) = S ((S (0)) * ec_scale_execution)) /\ exists ff_q_cf_ec_execution_trace_initial_state. ec_history_execution = ff_q_cf_ec_execution_trace_initial_state * S ((S (0)) * ec_scale_execution) + (((cf_gcd_ec_execution_trace) + (((0) + (0)) * S ((0) + (0)) + ((0) + (0)))) * S ((cf_gcd_ec_execution_trace) + (((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_ec_execution_trace_terminal_state. ff_h_cf_ec_execution_trace_terminal_state + S (((a) + (((b) + (ec_list_execution)) * S ((b) + (ec_list_execution)) + ((ec_list_execution) + (ec_list_execution)))) * S ((a) + (((b) + (ec_list_execution)) * S ((b) + (ec_list_execution)) + ((ec_list_execution) + (ec_list_execution)))) + ((((b) + (ec_list_execution)) * S ((b) + (ec_list_execution)) + ((ec_list_execution) + (ec_list_execution))) + (((b) + (ec_list_execution)) * S ((b) + (ec_list_execution)) + ((ec_list_execution) + (ec_list_execution))))) = S ((S (l)) * ec_scale_execution)) /\ exists ff_q_cf_ec_execution_trace_terminal_state. ec_history_execution = ff_q_cf_ec_execution_trace_terminal_state * S ((S (l)) * ec_scale_execution) + (((a) + (((b) + (ec_list_execution)) * S ((b) + (ec_list_execution)) + ((ec_list_execution) + (ec_list_execution)))) * S ((a) + (((b) + (ec_list_execution)) * S ((b) + (ec_list_execution)) + ((ec_list_execution) + (ec_list_execution)))) + ((((b) + (ec_list_execution)) * S ((b) + (ec_list_execution)) + ((ec_list_execution) + (ec_list_execution))) + (((b) + (ec_list_execution)) * S ((b) + (ec_list_execution)) + ((ec_list_execution) + (ec_list_execution))))))) /\ forall cf_index_ec_execution_trace. (exists ff_lt_cf_ec_execution_trace_index. ff_lt_cf_ec_execution_trace_index + S cf_index_ec_execution_trace = l) -> exists cf_old_a_ec_execution_trace cf_old_b_ec_execution_trace cf_tail_ec_execution_trace cf_new_a_ec_execution_trace cf_new_b_ec_execution_trace cf_head_ec_execution_trace cf_quotient_ec_execution_trace. ((((exists ff_h_cf_ec_execution_trace_previous_state. ff_h_cf_ec_execution_trace_previous_state + S (((cf_old_a_ec_execution_trace) + (((cf_old_b_ec_execution_trace) + (cf_tail_ec_execution_trace)) * S ((cf_old_b_ec_execution_trace) + (cf_tail_ec_execution_trace)) + ((cf_tail_ec_execution_trace) + (cf_tail_ec_execution_trace)))) * S ((cf_old_a_ec_execution_trace) + (((cf_old_b_ec_execution_trace) + (cf_tail_ec_execution_trace)) * S ((cf_old_b_ec_execution_trace) + (cf_tail_ec_execution_trace)) + ((cf_tail_ec_execution_trace) + (cf_tail_ec_execution_trace)))) + ((((cf_old_b_ec_execution_trace) + (cf_tail_ec_execution_trace)) * S ((cf_old_b_ec_execution_trace) + (cf_tail_ec_execution_trace)) + ((cf_tail_ec_execution_trace) + (cf_tail_ec_execution_trace))) + (((cf_old_b_ec_execution_trace) + (cf_tail_ec_execution_trace)) * S ((cf_old_b_ec_execution_trace) + (cf_tail_ec_execution_trace)) + ((cf_tail_ec_execution_trace) + (cf_tail_ec_execution_trace))))) = S ((S (cf_index_ec_execution_trace)) * ec_scale_execution)) /\ exists ff_q_cf_ec_execution_trace_previous_state. ec_history_execution = ff_q_cf_ec_execution_trace_previous_state * S ((S (cf_index_ec_execution_trace)) * ec_scale_execution) + (((cf_old_a_ec_execution_trace) + (((cf_old_b_ec_execution_trace) + (cf_tail_ec_execution_trace)) * S ((cf_old_b_ec_execution_trace) + (cf_tail_ec_execution_trace)) + ((cf_tail_ec_execution_trace) + (cf_tail_ec_execution_trace)))) * S ((cf_old_a_ec_execution_trace) + (((cf_old_b_ec_execution_trace) + (cf_tail_ec_execution_trace)) * S ((cf_old_b_ec_execution_trace) + (cf_tail_ec_execution_trace)) + ((cf_tail_ec_execution_trace) + (cf_tail_ec_execution_trace)))) + ((((cf_old_b_ec_execution_trace) + (cf_tail_ec_execution_trace)) * S ((cf_old_b_ec_execution_trace) + (cf_tail_ec_execution_trace)) + ((cf_tail_ec_execution_trace) + (cf_tail_ec_execution_trace))) + (((cf_old_b_ec_execution_trace) + (cf_tail_ec_execution_trace)) * S ((cf_old_b_ec_execution_trace) + (cf_tail_ec_execution_trace)) + ((cf_tail_ec_execution_trace) + (cf_tail_ec_execution_trace))))))) /\ ((((exists ff_h_cf_ec_execution_trace_following_state. ff_h_cf_ec_execution_trace_following_state + S (((cf_new_a_ec_execution_trace) + (((cf_new_b_ec_execution_trace) + (cf_head_ec_execution_trace)) * S ((cf_new_b_ec_execution_trace) + (cf_head_ec_execution_trace)) + ((cf_head_ec_execution_trace) + (cf_head_ec_execution_trace)))) * S ((cf_new_a_ec_execution_trace) + (((cf_new_b_ec_execution_trace) + (cf_head_ec_execution_trace)) * S ((cf_new_b_ec_execution_trace) + (cf_head_ec_execution_trace)) + ((cf_head_ec_execution_trace) + (cf_head_ec_execution_trace)))) + ((((cf_new_b_ec_execution_trace) + (cf_head_ec_execution_trace)) * S ((cf_new_b_ec_execution_trace) + (cf_head_ec_execution_trace)) + ((cf_head_ec_execution_trace) + (cf_head_ec_execution_trace))) + (((cf_new_b_ec_execution_trace) + (cf_head_ec_execution_trace)) * S ((cf_new_b_ec_execution_trace) + (cf_head_ec_execution_trace)) + ((cf_head_ec_execution_trace) + (cf_head_ec_execution_trace))))) = S ((S (S cf_index_ec_execution_trace)) * ec_scale_execution)) /\ exists ff_q_cf_ec_execution_trace_following_state. ec_history_execution = ff_q_cf_ec_execution_trace_following_state * S ((S (S cf_index_ec_execution_trace)) * ec_scale_execution) + (((cf_new_a_ec_execution_trace) + (((cf_new_b_ec_execution_trace) + (cf_head_ec_execution_trace)) * S ((cf_new_b_ec_execution_trace) + (cf_head_ec_execution_trace)) + ((cf_head_ec_execution_trace) + (cf_head_ec_execution_trace)))) * S ((cf_new_a_ec_execution_trace) + (((cf_new_b_ec_execution_trace) + (cf_head_ec_execution_trace)) * S ((cf_new_b_ec_execution_trace) + (cf_head_ec_execution_trace)) + ((cf_head_ec_execution_trace) + (cf_head_ec_execution_trace)))) + ((((cf_new_b_ec_execution_trace) + (cf_head_ec_execution_trace)) * S ((cf_new_b_ec_execution_trace) + (cf_head_ec_execution_trace)) + ((cf_head_ec_execution_trace) + (cf_head_ec_execution_trace))) + (((cf_new_b_ec_execution_trace) + (cf_head_ec_execution_trace)) * S ((cf_new_b_ec_execution_trace) + (cf_head_ec_execution_trace)) + ((cf_head_ec_execution_trace) + (cf_head_ec_execution_trace))))))) /\ (cf_new_b_ec_execution_trace = cf_old_a_ec_execution_trace /\ (cf_new_a_ec_execution_trace = cf_new_b_ec_execution_trace * cf_quotient_ec_execution_trace + cf_old_b_ec_execution_trace /\ ((exists ff_lt_cf_ec_execution_trace_remainder. ff_lt_cf_ec_execution_trace_remainder + S cf_old_b_ec_execution_trace = cf_new_b_ec_execution_trace) /\ (cf_head_ec_execution_trace = S ((cf_quotient_ec_execution_trace + cf_tail_ec_execution_trace) * S (cf_quotient_ec_execution_trace + cf_tail_ec_execution_trace) + (cf_tail_ec_execution_trace + cf_tail_ec_execution_trace))))))))))) /\ ((((exists ec_gcd_left_execution_result. a = g * ec_gcd_left_execution_result) /\ (exists ec_gcd_right_execution_result. b = g * ec_gcd_right_execution_result)) /\ forall ec_gcd_common_execution_result. (exists ec_gcd_common_left_execution_result. a = ec_gcd_common_execution_result * ec_gcd_common_left_execution_result) -> (exists ec_gcd_common_right_execution_result. b = ec_gcd_common_execution_result * ec_gcd_common_right_execution_result) -> exists ec_gcd_greatest_execution_result. g = ec_gcd_common_execution_result * ec_gcd_greatest_execution_result))))

Constructive proof overview

Generated structural guide

All natural input pairs admit a fully expanded, beta-coded terminating Euclidean execution with a certified relational gcd output.

The unchanged tactic script uses 2 declared prerequisites and contains 19 exact native proof lines.

Alpha v34 checked-use · first admitted v21 · independently kernel and Lean verified; not Stable

Proof neighborhood

Direct dependencies

continued_fraction_trace_exists Alpha theorem; checked-use authorized gcd_exists_relational Stable theorem; checked-use authorized

Direct dependents

none

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

19 script commands · 8 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.

01Fix variables and assumptionsL1–2

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

  1. L1
    intro a
  2. L2
    intro b
02Use earlier factsL3–4

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

  1. L3
    specialize gcd_exists_relational a
  2. L4
    specialize gcd_exists_relational b
03Separate the logical casesL5–5

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

  1. L5
    cases gcd_exists_relational
04Use earlier factsL6–7

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

  1. L6
    specialize continued_fraction_trace_exists a
  2. L7
    specialize continued_fraction_trace_exists b
05Separate the logical casesL8–11

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

  1. L8
    cases continued_fraction_trace_exists
  2. L9
    cases continued_fraction_trace_exists_witness
  3. L10
    cases continued_fraction_trace_exists_witness_witness
  4. L11
    cases continued_fraction_trace_exists_witness_witness_witness
06Construct an explicit witnessL12–16

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

  1. L12
    exists x
  2. L13
    exists x4
  3. L14
    exists x1
  4. L15
    exists x2
  5. L16
    exists x3
07Separate the logical casesL17–17

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

  1. L17
    split
08Use earlier factsL18–19

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

  1. L18
    exact continued_fraction_trace_exists_witness_witness_witness_witness
  2. L19
    exact gcd_exists_relational_witness

Library-wide reading audit

Original exact command ledger · 19 lines
  1. 0001intro a
  2. 0002intro b
  3. 0003specialize gcd_exists_relational a
  4. 0004specialize gcd_exists_relational b
  5. 0005cases gcd_exists_relational
  6. 0006specialize continued_fraction_trace_exists a
  7. 0007specialize continued_fraction_trace_exists b
  8. 0008cases continued_fraction_trace_exists
  9. 0009cases continued_fraction_trace_exists_witness
  10. 0010cases continued_fraction_trace_exists_witness_witness
  11. 0011cases continued_fraction_trace_exists_witness_witness_witness
  12. 0012exists x
  13. 0013exists x4
  14. 0014exists x1
  15. 0015exists x2
  16. 0016exists x3
  17. 0017split
  18. 0018exact continued_fraction_trace_exists_witness_witness_witness_witness
  19. 0019exact gcd_exists_relational_witness