BX000E

binary_modular_exponentiation_result_exists

Every a,e and every modulus m>1 have an exact bounded residue of the relational power a^e.

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.

G102 was OPEN when this family was first admitted in Alpha v21. It is now CLOSED in Alpha v23: every arbitrary exponent has actual canonical beta-coded digits, a complete modular execution, and exact counted bound operations≤3*BitLen(e)+2.

Exact theorem in conservative defined notation

∀ a. ∀ e. ∀ m. BinaryModulus(m) → ∃ x. BinaryModularPower(a,e,m,x)

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

Definition DAG

BinaryModulus(m) · 1BinaryModularPower(a,e,m,r) · 1

Actual proof prerequisites

pow_exists · checked external prerequisitebinary_canonical_residue_exists
Original expanded first-order statement 
forall a e m. (exists ff_modulus_gap_binary_guard. ff_modulus_gap_binary_guard + S 1 = m) -> exists r. (exists ff_power_binary_result. ((exists ff_b_binary_result_value ff_c_binary_result_value. ((forall ff_i_binary_result_value_repeat. (exists ff_lt_binary_result_value_repeat_bound. ff_lt_binary_result_value_repeat_bound + S ff_i_binary_result_value_repeat = e) -> (((exists ff_h_binary_result_value_repeat_decoded. ff_h_binary_result_value_repeat_decoded + S (a) = S ((S (ff_i_binary_result_value_repeat)) * ff_c_binary_result_value)) /\ exists ff_q_binary_result_value_repeat_decoded. ff_b_binary_result_value = ff_q_binary_result_value_repeat_decoded * S ((S (ff_i_binary_result_value_repeat)) * ff_c_binary_result_value) + (a)))) /\ (exists ff_u_binary_result_value_product ff_v_binary_result_value_product. ((((exists ff_h_binary_result_value_product_start. ff_h_binary_result_value_product_start + S (1) = S ((S (0)) * ff_v_binary_result_value_product)) /\ exists ff_q_binary_result_value_product_start. ff_u_binary_result_value_product = ff_q_binary_result_value_product_start * S ((S (0)) * ff_v_binary_result_value_product) + (1))) /\ ((((exists ff_h_binary_result_value_product_terminal. ff_h_binary_result_value_product_terminal + S (ff_power_binary_result) = S ((S (e)) * ff_v_binary_result_value_product)) /\ exists ff_q_binary_result_value_product_terminal. ff_u_binary_result_value_product = ff_q_binary_result_value_product_terminal * S ((S (e)) * ff_v_binary_result_value_product) + (ff_power_binary_result))) /\ forall ff_i_binary_result_value_product. (exists ff_lt_binary_result_value_product_bound. ff_lt_binary_result_value_product_bound + S ff_i_binary_result_value_product = e) -> exists ff_p_binary_result_value_product ff_r_binary_result_value_product ff_s_binary_result_value_product. ((((exists ff_h_binary_result_value_product_factor. ff_h_binary_result_value_product_factor + S (ff_p_binary_result_value_product) = S ((S (ff_i_binary_result_value_product)) * ff_c_binary_result_value)) /\ exists ff_q_binary_result_value_product_factor. ff_b_binary_result_value = ff_q_binary_result_value_product_factor * S ((S (ff_i_binary_result_value_product)) * ff_c_binary_result_value) + (ff_p_binary_result_value_product))) /\ ((((exists ff_h_binary_result_value_product_partial. ff_h_binary_result_value_product_partial + S (ff_r_binary_result_value_product) = S ((S (ff_i_binary_result_value_product)) * ff_v_binary_result_value_product)) /\ exists ff_q_binary_result_value_product_partial. ff_u_binary_result_value_product = ff_q_binary_result_value_product_partial * S ((S (ff_i_binary_result_value_product)) * ff_v_binary_result_value_product) + (ff_r_binary_result_value_product))) /\ ((((exists ff_h_binary_result_value_product_successor. ff_h_binary_result_value_product_successor + S (ff_s_binary_result_value_product) = S ((S (S ff_i_binary_result_value_product)) * ff_v_binary_result_value_product)) /\ exists ff_q_binary_result_value_product_successor. ff_u_binary_result_value_product = ff_q_binary_result_value_product_successor * S ((S (S ff_i_binary_result_value_product)) * ff_v_binary_result_value_product) + (ff_s_binary_result_value_product))) /\ ff_s_binary_result_value_product = ff_r_binary_result_value_product * ff_p_binary_result_value_product)))))))) /\ (((exists ff_gap_binary_result_residue. ff_gap_binary_result_residue + S (r) = m) /\ (exists ff_left_binary_result_residue_congruence ff_right_binary_result_residue_congruence. (ff_power_binary_result) + m * ff_left_binary_result_residue_congruence = (r) + m * ff_right_binary_result_residue_congruence)))))

Complete unchanged native tactic proof

All 18 lines are the exact independently kernel-checked original script.

Read the argument

Proof checkpoints

18 script commands · 8 reading checkpoints · 1 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 (1)
binary canonical residue exists
01Fix variables and assumptionsL1–4

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

  1. L1
    intro a
  2. L2
    intro e
  3. L3
    intro m
  4. L4
    intro hmodulus
02Use earlier factsL5–6

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

  1. L5
    specialize pow_exists a
  2. L6
    specialize pow_exists e
03Separate the logical casesL7–7

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

  1. L7
    cases pow_exists
04Establish hresidueL8–12

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply binary canonical residue exists.

  1. L8
    have hresidue : exists r. (((exists ff_gap_binary_power_exists. ff_gap_binary_power_exists + S (r) = m) /\ (exists ff_left_binary_power_exists_congruence ff_right_binary_power_exists_congruence. (x) + m * ff_left_binary_power_exists_congruence = (r) + m * ff_right_binary_power_exists_congruence)))
  2. L9
    specialize binary_canonical_residue_exists m
  3. L10
    specialize binary_canonical_residue_exists x
  4. L11
    apply binary_canonical_residue_exists
  5. L12
    exact hmodulus
05Separate the logical casesL13–13

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

  1. L13
    cases hresidue
06Construct an explicit witnessL14–15

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

  1. L14
    exists x1
  2. L15
    exists x
07Separate the logical casesL16–16

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

  1. L16
    split
08Use earlier factsL17–18

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

  1. L17
    exact pow_exists_witness
  2. L18
    exact hresidue_witness

Library-wide reading audit

Original defined command ledger · 18 lines
  1. 0001intro a
  2. 0002intro e
  3. 0003intro m
  4. 0004intro hmodulus
  5. 0005specialize pow_exists a
  6. 0006specialize pow_exists e
  7. 0007cases pow_exists
  8. 0008have hresidue : exists r. (((exists ff_gap_binary_power_exists. ff_gap_binary_power_exists + S (r) = m) /\ (exists ff_left_binary_power_exists_congruence ff_right_binary_power_exists_congruence. (x) + m * ff_left_binary_power_exists_congruence = (r) + m * ff_right_binary_power_exists_congruence)))
  9. 0009specialize binary_canonical_residue_exists m
  10. 0010specialize binary_canonical_residue_exists x
  11. 0011apply binary_canonical_residue_exists
  12. 0012exact hmodulus
  13. 0013cases hresidue
  14. 0014exists x1
  15. 0015exists x
  16. 0016split
  17. 0017exact pow_exists_witness
  18. 0018exact hresidue_witness