EC000A

euclidean_execution_zero_divisor

The zero-divisor boundary has an actual zero-step beta-history and returns the dividend as its checked relational gcd.

Alpha v34 checked-use · first admitted v21 · 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.

G101 was OPEN when this family was first admitted in Alpha v21. It is now CLOSED in Alpha v23: the actual anchored Euclidean history, terminal gcd, and exact bound steps≤2*BitLen(b)+1 are proved.

Exact theorem in conservative defined notation

∀ a. EuclideanExecution(a,0,a,0)

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

Definition DAG

EuclideanExecution(a,b,g,k) · 1

Actual proof prerequisites

continued_fraction_empty_trace_exists · checked external prerequisiteis_gcd_zero_right · checked external prerequisite
Original expanded first-order statement 
forall a. (exists ec_list_zero ec_history_zero ec_scale_zero. ((exists cf_gcd_ec_zero_trace. ((((exists ff_h_cf_ec_zero_trace_initial_state. ff_h_cf_ec_zero_trace_initial_state + S (((cf_gcd_ec_zero_trace) + (((0) + (0)) * S ((0) + (0)) + ((0) + (0)))) * S ((cf_gcd_ec_zero_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_zero)) /\ exists ff_q_cf_ec_zero_trace_initial_state. ec_history_zero = ff_q_cf_ec_zero_trace_initial_state * S ((S (0)) * ec_scale_zero) + (((cf_gcd_ec_zero_trace) + (((0) + (0)) * S ((0) + (0)) + ((0) + (0)))) * S ((cf_gcd_ec_zero_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_zero_trace_terminal_state. ff_h_cf_ec_zero_trace_terminal_state + S (((a) + (((0) + (ec_list_zero)) * S ((0) + (ec_list_zero)) + ((ec_list_zero) + (ec_list_zero)))) * S ((a) + (((0) + (ec_list_zero)) * S ((0) + (ec_list_zero)) + ((ec_list_zero) + (ec_list_zero)))) + ((((0) + (ec_list_zero)) * S ((0) + (ec_list_zero)) + ((ec_list_zero) + (ec_list_zero))) + (((0) + (ec_list_zero)) * S ((0) + (ec_list_zero)) + ((ec_list_zero) + (ec_list_zero))))) = S ((S (0)) * ec_scale_zero)) /\ exists ff_q_cf_ec_zero_trace_terminal_state. ec_history_zero = ff_q_cf_ec_zero_trace_terminal_state * S ((S (0)) * ec_scale_zero) + (((a) + (((0) + (ec_list_zero)) * S ((0) + (ec_list_zero)) + ((ec_list_zero) + (ec_list_zero)))) * S ((a) + (((0) + (ec_list_zero)) * S ((0) + (ec_list_zero)) + ((ec_list_zero) + (ec_list_zero)))) + ((((0) + (ec_list_zero)) * S ((0) + (ec_list_zero)) + ((ec_list_zero) + (ec_list_zero))) + (((0) + (ec_list_zero)) * S ((0) + (ec_list_zero)) + ((ec_list_zero) + (ec_list_zero))))))) /\ forall cf_index_ec_zero_trace. (exists ff_lt_cf_ec_zero_trace_index. ff_lt_cf_ec_zero_trace_index + S cf_index_ec_zero_trace = 0) -> exists cf_old_a_ec_zero_trace cf_old_b_ec_zero_trace cf_tail_ec_zero_trace cf_new_a_ec_zero_trace cf_new_b_ec_zero_trace cf_head_ec_zero_trace cf_quotient_ec_zero_trace. ((((exists ff_h_cf_ec_zero_trace_previous_state. ff_h_cf_ec_zero_trace_previous_state + S (((cf_old_a_ec_zero_trace) + (((cf_old_b_ec_zero_trace) + (cf_tail_ec_zero_trace)) * S ((cf_old_b_ec_zero_trace) + (cf_tail_ec_zero_trace)) + ((cf_tail_ec_zero_trace) + (cf_tail_ec_zero_trace)))) * S ((cf_old_a_ec_zero_trace) + (((cf_old_b_ec_zero_trace) + (cf_tail_ec_zero_trace)) * S ((cf_old_b_ec_zero_trace) + (cf_tail_ec_zero_trace)) + ((cf_tail_ec_zero_trace) + (cf_tail_ec_zero_trace)))) + ((((cf_old_b_ec_zero_trace) + (cf_tail_ec_zero_trace)) * S ((cf_old_b_ec_zero_trace) + (cf_tail_ec_zero_trace)) + ((cf_tail_ec_zero_trace) + (cf_tail_ec_zero_trace))) + (((cf_old_b_ec_zero_trace) + (cf_tail_ec_zero_trace)) * S ((cf_old_b_ec_zero_trace) + (cf_tail_ec_zero_trace)) + ((cf_tail_ec_zero_trace) + (cf_tail_ec_zero_trace))))) = S ((S (cf_index_ec_zero_trace)) * ec_scale_zero)) /\ exists ff_q_cf_ec_zero_trace_previous_state. ec_history_zero = ff_q_cf_ec_zero_trace_previous_state * S ((S (cf_index_ec_zero_trace)) * ec_scale_zero) + (((cf_old_a_ec_zero_trace) + (((cf_old_b_ec_zero_trace) + (cf_tail_ec_zero_trace)) * S ((cf_old_b_ec_zero_trace) + (cf_tail_ec_zero_trace)) + ((cf_tail_ec_zero_trace) + (cf_tail_ec_zero_trace)))) * S ((cf_old_a_ec_zero_trace) + (((cf_old_b_ec_zero_trace) + (cf_tail_ec_zero_trace)) * S ((cf_old_b_ec_zero_trace) + (cf_tail_ec_zero_trace)) + ((cf_tail_ec_zero_trace) + (cf_tail_ec_zero_trace)))) + ((((cf_old_b_ec_zero_trace) + (cf_tail_ec_zero_trace)) * S ((cf_old_b_ec_zero_trace) + (cf_tail_ec_zero_trace)) + ((cf_tail_ec_zero_trace) + (cf_tail_ec_zero_trace))) + (((cf_old_b_ec_zero_trace) + (cf_tail_ec_zero_trace)) * S ((cf_old_b_ec_zero_trace) + (cf_tail_ec_zero_trace)) + ((cf_tail_ec_zero_trace) + (cf_tail_ec_zero_trace))))))) /\ ((((exists ff_h_cf_ec_zero_trace_following_state. ff_h_cf_ec_zero_trace_following_state + S (((cf_new_a_ec_zero_trace) + (((cf_new_b_ec_zero_trace) + (cf_head_ec_zero_trace)) * S ((cf_new_b_ec_zero_trace) + (cf_head_ec_zero_trace)) + ((cf_head_ec_zero_trace) + (cf_head_ec_zero_trace)))) * S ((cf_new_a_ec_zero_trace) + (((cf_new_b_ec_zero_trace) + (cf_head_ec_zero_trace)) * S ((cf_new_b_ec_zero_trace) + (cf_head_ec_zero_trace)) + ((cf_head_ec_zero_trace) + (cf_head_ec_zero_trace)))) + ((((cf_new_b_ec_zero_trace) + (cf_head_ec_zero_trace)) * S ((cf_new_b_ec_zero_trace) + (cf_head_ec_zero_trace)) + ((cf_head_ec_zero_trace) + (cf_head_ec_zero_trace))) + (((cf_new_b_ec_zero_trace) + (cf_head_ec_zero_trace)) * S ((cf_new_b_ec_zero_trace) + (cf_head_ec_zero_trace)) + ((cf_head_ec_zero_trace) + (cf_head_ec_zero_trace))))) = S ((S (S cf_index_ec_zero_trace)) * ec_scale_zero)) /\ exists ff_q_cf_ec_zero_trace_following_state. ec_history_zero = ff_q_cf_ec_zero_trace_following_state * S ((S (S cf_index_ec_zero_trace)) * ec_scale_zero) + (((cf_new_a_ec_zero_trace) + (((cf_new_b_ec_zero_trace) + (cf_head_ec_zero_trace)) * S ((cf_new_b_ec_zero_trace) + (cf_head_ec_zero_trace)) + ((cf_head_ec_zero_trace) + (cf_head_ec_zero_trace)))) * S ((cf_new_a_ec_zero_trace) + (((cf_new_b_ec_zero_trace) + (cf_head_ec_zero_trace)) * S ((cf_new_b_ec_zero_trace) + (cf_head_ec_zero_trace)) + ((cf_head_ec_zero_trace) + (cf_head_ec_zero_trace)))) + ((((cf_new_b_ec_zero_trace) + (cf_head_ec_zero_trace)) * S ((cf_new_b_ec_zero_trace) + (cf_head_ec_zero_trace)) + ((cf_head_ec_zero_trace) + (cf_head_ec_zero_trace))) + (((cf_new_b_ec_zero_trace) + (cf_head_ec_zero_trace)) * S ((cf_new_b_ec_zero_trace) + (cf_head_ec_zero_trace)) + ((cf_head_ec_zero_trace) + (cf_head_ec_zero_trace))))))) /\ (cf_new_b_ec_zero_trace = cf_old_a_ec_zero_trace /\ (cf_new_a_ec_zero_trace = cf_new_b_ec_zero_trace * cf_quotient_ec_zero_trace + cf_old_b_ec_zero_trace /\ ((exists ff_lt_cf_ec_zero_trace_remainder. ff_lt_cf_ec_zero_trace_remainder + S cf_old_b_ec_zero_trace = cf_new_b_ec_zero_trace) /\ (cf_head_ec_zero_trace = S ((cf_quotient_ec_zero_trace + cf_tail_ec_zero_trace) * S (cf_quotient_ec_zero_trace + cf_tail_ec_zero_trace) + (cf_tail_ec_zero_trace + cf_tail_ec_zero_trace))))))))))) /\ ((((exists ec_gcd_left_zero_result. a = a * ec_gcd_left_zero_result) /\ (exists ec_gcd_right_zero_result. 0 = a * ec_gcd_right_zero_result)) /\ forall ec_gcd_common_zero_result. (exists ec_gcd_common_left_zero_result. a = ec_gcd_common_zero_result * ec_gcd_common_left_zero_result) -> (exists ec_gcd_common_right_zero_result. 0 = ec_gcd_common_zero_result * ec_gcd_common_right_zero_result) -> exists ec_gcd_greatest_zero_result. a = ec_gcd_common_zero_result * ec_gcd_greatest_zero_result))))

Complete unchanged native tactic proof

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

Read the argument

Proof checkpoints

11 script commands · 6 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 factsL2–2

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

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

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

  1. L3
    cases continued_fraction_empty_trace_exists
  2. L4
    cases continued_fraction_empty_trace_exists_witness
04Construct an explicit witnessL5–7

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

  1. L5
    exists 0
  2. L6
    exists x
  3. L7
    exists x1
05Separate the logical casesL8–8

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

  1. L8
    split
06Use earlier factsL9–11

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

  1. L9
    exact continued_fraction_empty_trace_exists_witness_witness
  2. L10
    specialize is_gcd_zero_right a
  3. L11
    exact is_gcd_zero_right

Library-wide reading audit

Original defined command ledger · 11 lines
  1. 0001intro a
  2. 0002specialize continued_fraction_empty_trace_exists a
  3. 0003cases continued_fraction_empty_trace_exists
  4. 0004cases continued_fraction_empty_trace_exists_witness
  5. 0005exists 0
  6. 0006exists x
  7. 0007exists x1
  8. 0008split
  9. 0009exact continued_fraction_empty_trace_exists_witness_witness
  10. 0010specialize is_gcd_zero_right a
  11. 0011exact is_gcd_zero_right