EU0022

euler_theorem_for_units

Full exact G014: m>1 and a genuinely witnessed modular unit and independently counted Phi imply an actual Pow witness congruent to one.

Alpha v34 checked-use · first admitted v31 · 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.

The exact G014 theorem covers m>1 and genuinely invertible a. Phi counts coprime residues independently of the conclusion. The broader coprime theorem handles m=1 by congruence, not by asserting that one is a canonical remainder. Multiplicative-order and RSA statements are not claimed.

Exact theorem in conservative defined notation

∀ a. ∀ m. ∀ t. Lt(1,m) ∧ (Unit(a,m) ∧ Phi(m,t)) → ∃ x. Pow(a,t,x) ∧ ModEq(m,x,1)

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

Definition DAG

Phi(n,t) · 1Unit(a,m) · 1Lt(a,b) · 1ModEq(m,a,b) · 1Pow(a,e,z) · 1

Actual proof prerequisites

euler_modular_unit_totient_power
Original expanded first-order statement 
forall a m t. ((exists eut_gap_eu_G014_domain. eut_gap_eu_G014_domain + S (1) = (m)) /\ (((exists eut_gap_eu_G014_unit_domain. eut_gap_eu_G014_unit_domain + S (1) = (m)) /\ exists eu_inverse_G014_unit. (exists eut_gap_eu_G014_unit_bound. eut_gap_eu_G014_unit_bound + S (eu_inverse_G014_unit) = (m)) /\ (exists eu_mod_left_G014_unit_inverse eu_mod_right_G014_unit_inverse. ((a)*eu_inverse_G014_unit) + (m) * eu_mod_left_G014_unit_inverse = (1) + (m) * eu_mod_right_G014_unit_inverse)) /\ ((~((m)=0) /\ (exists eut_code_eu_G014_phi_count eut_scale_eu_G014_phi_count. (forall eut_index_eu_G014_phi_count_mask. (exists eut_gap_eu_G014_phi_count_mask_bound. eut_gap_eu_G014_phi_count_mask_bound + S (eut_index_eu_G014_phi_count_mask) = (m)) -> exists eut_bit_eu_G014_phi_count_mask. (((exists fs_h_eut_eu_G014_phi_count_mask_entry. fs_h_eut_eu_G014_phi_count_mask_entry + S (eut_bit_eu_G014_phi_count_mask) = S ((S (eut_index_eu_G014_phi_count_mask)) * eut_scale_eu_G014_phi_count)) /\ exists fs_q_eut_eu_G014_phi_count_mask_entry. eut_code_eu_G014_phi_count = fs_q_eut_eu_G014_phi_count_mask_entry * S ((S (eut_index_eu_G014_phi_count_mask)) * eut_scale_eu_G014_phi_count) + (eut_bit_eu_G014_phi_count_mask))) /\ ((((forall eut_divisor_eu_G014_phi_count_mask_choice_coprime. (exists eut_left_eu_G014_phi_count_mask_choice_coprime. (eut_index_eu_G014_phi_count_mask) = eut_divisor_eu_G014_phi_count_mask_choice_coprime * eut_left_eu_G014_phi_count_mask_choice_coprime) -> (exists eut_right_eu_G014_phi_count_mask_choice_coprime. (m) = eut_divisor_eu_G014_phi_count_mask_choice_coprime * eut_right_eu_G014_phi_count_mask_choice_coprime) -> eut_divisor_eu_G014_phi_count_mask_choice_coprime = 1) /\ (eut_bit_eu_G014_phi_count_mask) = 1) \/ (~(forall eut_divisor_eu_G014_phi_count_mask_choice_coprime. (exists eut_left_eu_G014_phi_count_mask_choice_coprime. (eut_index_eu_G014_phi_count_mask) = eut_divisor_eu_G014_phi_count_mask_choice_coprime * eut_left_eu_G014_phi_count_mask_choice_coprime) -> (exists eut_right_eu_G014_phi_count_mask_choice_coprime. (m) = eut_divisor_eu_G014_phi_count_mask_choice_coprime * eut_right_eu_G014_phi_count_mask_choice_coprime) -> eut_divisor_eu_G014_phi_count_mask_choice_coprime = 1) /\ (eut_bit_eu_G014_phi_count_mask) = 0)))) /\ (exists fs_u_eut_eu_G014_phi_count_sum fs_v_eut_eu_G014_phi_count_sum. ((((exists fs_h_eut_eu_G014_phi_count_sum_body_start. fs_h_eut_eu_G014_phi_count_sum_body_start + S (0) = S ((S (0)) * fs_v_eut_eu_G014_phi_count_sum)) /\ exists fs_q_eut_eu_G014_phi_count_sum_body_start. fs_u_eut_eu_G014_phi_count_sum = fs_q_eut_eu_G014_phi_count_sum_body_start * S ((S (0)) * fs_v_eut_eu_G014_phi_count_sum) + (0))) /\ ((((exists fs_h_eut_eu_G014_phi_count_sum_body_terminal. fs_h_eut_eu_G014_phi_count_sum_body_terminal + S (t) = S ((S (m)) * fs_v_eut_eu_G014_phi_count_sum)) /\ exists fs_q_eut_eu_G014_phi_count_sum_body_terminal. fs_u_eut_eu_G014_phi_count_sum = fs_q_eut_eu_G014_phi_count_sum_body_terminal * S ((S (m)) * fs_v_eut_eu_G014_phi_count_sum) + (t))) /\ forall fs_i_eut_eu_G014_phi_count_sum_body_steps. (exists fs_lt_eut_eu_G014_phi_count_sum_body_steps_bound. fs_lt_eut_eu_G014_phi_count_sum_body_steps_bound + S fs_i_eut_eu_G014_phi_count_sum_body_steps = m) -> exists fs_a_eut_eu_G014_phi_count_sum_body_steps fs_r_eut_eu_G014_phi_count_sum_body_steps fs_s_eut_eu_G014_phi_count_sum_body_steps. ((((exists fs_h_eut_eu_G014_phi_count_sum_body_steps_summand. fs_h_eut_eu_G014_phi_count_sum_body_steps_summand + S (fs_a_eut_eu_G014_phi_count_sum_body_steps) = S ((S (fs_i_eut_eu_G014_phi_count_sum_body_steps)) * eut_scale_eu_G014_phi_count)) /\ exists fs_q_eut_eu_G014_phi_count_sum_body_steps_summand. eut_code_eu_G014_phi_count = fs_q_eut_eu_G014_phi_count_sum_body_steps_summand * S ((S (fs_i_eut_eu_G014_phi_count_sum_body_steps)) * eut_scale_eu_G014_phi_count) + (fs_a_eut_eu_G014_phi_count_sum_body_steps))) /\ ((((exists fs_h_eut_eu_G014_phi_count_sum_body_steps_partial. fs_h_eut_eu_G014_phi_count_sum_body_steps_partial + S (fs_r_eut_eu_G014_phi_count_sum_body_steps) = S ((S (fs_i_eut_eu_G014_phi_count_sum_body_steps)) * fs_v_eut_eu_G014_phi_count_sum)) /\ exists fs_q_eut_eu_G014_phi_count_sum_body_steps_partial. fs_u_eut_eu_G014_phi_count_sum = fs_q_eut_eu_G014_phi_count_sum_body_steps_partial * S ((S (fs_i_eut_eu_G014_phi_count_sum_body_steps)) * fs_v_eut_eu_G014_phi_count_sum) + (fs_r_eut_eu_G014_phi_count_sum_body_steps))) /\ ((((exists fs_h_eut_eu_G014_phi_count_sum_body_steps_successor. fs_h_eut_eu_G014_phi_count_sum_body_steps_successor + S (fs_s_eut_eu_G014_phi_count_sum_body_steps) = S ((S (S fs_i_eut_eu_G014_phi_count_sum_body_steps)) * fs_v_eut_eu_G014_phi_count_sum)) /\ exists fs_q_eut_eu_G014_phi_count_sum_body_steps_successor. fs_u_eut_eu_G014_phi_count_sum = fs_q_eut_eu_G014_phi_count_sum_body_steps_successor * S ((S (S fs_i_eut_eu_G014_phi_count_sum_body_steps)) * fs_v_eut_eu_G014_phi_count_sum) + (fs_s_eut_eu_G014_phi_count_sum_body_steps))) /\ fs_s_eut_eu_G014_phi_count_sum_body_steps = fs_r_eut_eu_G014_phi_count_sum_body_steps + fs_a_eut_eu_G014_phi_count_sum_body_steps))))))))))) -> exists w. (exists pa_b_euta_eu_G014_power pa_c_euta_eu_G014_power. ((forall pa_i_euta_eu_G014_power_repeat. (exists pa_lt_euta_eu_G014_power_repeat_bound. pa_lt_euta_eu_G014_power_repeat_bound + S pa_i_euta_eu_G014_power_repeat = t) -> (((exists pa_h_euta_eu_G014_power_repeat_decoded. pa_h_euta_eu_G014_power_repeat_decoded + S (a) = S ((S (pa_i_euta_eu_G014_power_repeat)) * pa_c_euta_eu_G014_power)) /\ exists pa_q_euta_eu_G014_power_repeat_decoded. pa_b_euta_eu_G014_power = pa_q_euta_eu_G014_power_repeat_decoded * S ((S (pa_i_euta_eu_G014_power_repeat)) * pa_c_euta_eu_G014_power) + (a)))) /\ (exists pa_u_euta_eu_G014_power_product pa_v_euta_eu_G014_power_product. ((((exists pa_h_euta_eu_G014_power_product_start. pa_h_euta_eu_G014_power_product_start + S (1) = S ((S (0)) * pa_v_euta_eu_G014_power_product)) /\ exists pa_q_euta_eu_G014_power_product_start. pa_u_euta_eu_G014_power_product = pa_q_euta_eu_G014_power_product_start * S ((S (0)) * pa_v_euta_eu_G014_power_product) + (1))) /\ ((((exists pa_h_euta_eu_G014_power_product_terminal. pa_h_euta_eu_G014_power_product_terminal + S (w) = S ((S (t)) * pa_v_euta_eu_G014_power_product)) /\ exists pa_q_euta_eu_G014_power_product_terminal. pa_u_euta_eu_G014_power_product = pa_q_euta_eu_G014_power_product_terminal * S ((S (t)) * pa_v_euta_eu_G014_power_product) + (w))) /\ forall pa_i_euta_eu_G014_power_product. (exists pa_lt_euta_eu_G014_power_product_bound. pa_lt_euta_eu_G014_power_product_bound + S pa_i_euta_eu_G014_power_product = t) -> exists pa_p_euta_eu_G014_power_product pa_r_euta_eu_G014_power_product pa_s_euta_eu_G014_power_product. ((((exists pa_h_euta_eu_G014_power_product_factor. pa_h_euta_eu_G014_power_product_factor + S (pa_p_euta_eu_G014_power_product) = S ((S (pa_i_euta_eu_G014_power_product)) * pa_c_euta_eu_G014_power)) /\ exists pa_q_euta_eu_G014_power_product_factor. pa_b_euta_eu_G014_power = pa_q_euta_eu_G014_power_product_factor * S ((S (pa_i_euta_eu_G014_power_product)) * pa_c_euta_eu_G014_power) + (pa_p_euta_eu_G014_power_product))) /\ ((((exists pa_h_euta_eu_G014_power_product_partial. pa_h_euta_eu_G014_power_product_partial + S (pa_r_euta_eu_G014_power_product) = S ((S (pa_i_euta_eu_G014_power_product)) * pa_v_euta_eu_G014_power_product)) /\ exists pa_q_euta_eu_G014_power_product_partial. pa_u_euta_eu_G014_power_product = pa_q_euta_eu_G014_power_product_partial * S ((S (pa_i_euta_eu_G014_power_product)) * pa_v_euta_eu_G014_power_product) + (pa_r_euta_eu_G014_power_product))) /\ ((((exists pa_h_euta_eu_G014_power_product_successor. pa_h_euta_eu_G014_power_product_successor + S (pa_s_euta_eu_G014_power_product) = S ((S (S pa_i_euta_eu_G014_power_product)) * pa_v_euta_eu_G014_power_product)) /\ exists pa_q_euta_eu_G014_power_product_successor. pa_u_euta_eu_G014_power_product = pa_q_euta_eu_G014_power_product_successor * S ((S (S pa_i_euta_eu_G014_power_product)) * pa_v_euta_eu_G014_power_product) + (pa_s_euta_eu_G014_power_product))) /\ pa_s_euta_eu_G014_power_product = pa_r_euta_eu_G014_power_product * pa_p_euta_eu_G014_power_product)))))))) /\ (exists eu_mod_left_G014_result eu_mod_right_G014_result. (w) + (m) * eu_mod_left_G014_result = (1) + (m) * eu_mod_right_G014_result)

Complete tactic proof in conservative notation

All 12 original proof lines are preserved. Only local proposition formulas are abbreviated; every abbreviation has an exact binder-safe expansion check. The linked exact edition contains the unchanged replay script.

Read the argument

Proof checkpoints

12 script commands · 3 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 (1)
euler modular unit totient power
01Fix variables and assumptionsL1–4

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

  1. L1
    intro a
  2. L2
    intro m
  3. L3
    intro t
  4. L4
    intro h
02Separate the logical casesL5–6

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

  1. L5
    cases h
  2. L6
    cases h_right
03Use earlier factsL7–12

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

  1. L7
    specialize euler_modular_unit_totient_power (a)
  2. L8
    specialize euler_modular_unit_totient_power (m)
  3. L9
    specialize euler_modular_unit_totient_power (t)
  4. L10
    apply euler_modular_unit_totient_power
  5. L11
    exact h_right_left
  6. L12
    exact h_right_right

Library-wide reading audit

Original defined command ledger · 12 lines
  1. 0001intro a
  2. 0002intro m
  3. 0003intro t
  4. 0004intro h
  5. 0005cases h
  6. 0006cases h_right
  7. 0007specialize euler_modular_unit_totient_power (a)
  8. 0008specialize euler_modular_unit_totient_power (m)
  9. 0009specialize euler_modular_unit_totient_power (t)
  10. 0010apply euler_modular_unit_totient_power
  11. 0011exact h_right_left
  12. 0012exact h_right_right