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.
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Pow(a,e,z)Exact expansion
exists ff_b_defined_power ff_c_defined_power. ((forall ff_i_defined_power_repeat. (exists ff_lt_defined_power_repeat_bound. ff_lt_defined_power_repeat_bound + S ff_i_defined_power_repeat = e) -> (((exists ff_h_defined_power_repeat_decoded. ff_h_defined_power_repeat_decoded + S (a) = S ((S (ff_i_defined_power_repeat)) * ff_c_defined_power)) /\ exists ff_q_defined_power_repeat_decoded. ff_b_defined_power = ff_q_defined_power_repeat_decoded * S ((S (ff_i_defined_power_repeat)) * ff_c_defined_power) + (a)))) /\ (exists ff_u_defined_power_product ff_v_defined_power_product. ((((exists ff_h_defined_power_product_start. ff_h_defined_power_product_start + S (1) = S ((S (0)) * ff_v_defined_power_product)) /\ exists ff_q_defined_power_product_start. ff_u_defined_power_product = ff_q_defined_power_product_start * S ((S (0)) * ff_v_defined_power_product) + (1))) /\ ((((exists ff_h_defined_power_product_terminal. ff_h_defined_power_product_terminal + S (z) = S ((S (e)) * ff_v_defined_power_product)) /\ exists ff_q_defined_power_product_terminal. ff_u_defined_power_product = ff_q_defined_power_product_terminal * S ((S (e)) * ff_v_defined_power_product) + (z))) /\ forall ff_i_defined_power_product. (exists ff_lt_defined_power_product_bound. ff_lt_defined_power_product_bound + S ff_i_defined_power_product = e) -> exists ff_p_defined_power_product ff_r_defined_power_product ff_s_defined_power_product. ((((exists ff_h_defined_power_product_factor. ff_h_defined_power_product_factor + S (ff_p_defined_power_product) = S ((S (ff_i_defined_power_product)) * ff_c_defined_power)) /\ exists ff_q_defined_power_product_factor. ff_b_defined_power = ff_q_defined_power_product_factor * S ((S (ff_i_defined_power_product)) * ff_c_defined_power) + (ff_p_defined_power_product))) /\ ((((exists ff_h_defined_power_product_partial. ff_h_defined_power_product_partial + S (ff_r_defined_power_product) = S ((S (ff_i_defined_power_product)) * ff_v_defined_power_product)) /\ exists ff_q_defined_power_product_partial. ff_u_defined_power_product = ff_q_defined_power_product_partial * S ((S (ff_i_defined_power_product)) * ff_v_defined_power_product) + (ff_r_defined_power_product))) /\ ((((exists ff_h_defined_power_product_successor. ff_h_defined_power_product_successor + S (ff_s_defined_power_product) = S ((S (S ff_i_defined_power_product)) * ff_v_defined_power_product)) /\ exists ff_q_defined_power_product_successor. ff_u_defined_power_product = ff_q_defined_power_product_successor * S ((S (S ff_i_defined_power_product)) * ff_v_defined_power_product) + (ff_s_defined_power_product))) /\ ff_s_defined_power_product = ff_r_defined_power_product * ff_p_defined_power_product)))))))This node is notation, not a theorem, axiom, predicate constant, or kernel rule. The elaboration layer must expand it before proof checking.
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Used by theorem statements or local proof propositions
PA0046 pow_exists PA004B pow_zero PA004D pow_successor_decompose PA005E pow_predecessor_parity_mod PA005G pow_functional PA005H pow_successor_pair_mul PA005I pow_mod_congruent PA005T pow_one_from_zero_successor PA005U pow_one PA005V pow_two_from_one_successor PA005W pow_two PA005X pow_add PA005Y pow_mul_exp PA007I beta_sign_factor_product_power PA007J beta_sign_factor_product_power_exists PA007Q beta_product_pointwise_scale_mod PA007R gauss_signed_pointwise_mul_product_mod PA0080 gauss_signed_products_balance_mod PA0084 gauss_signed_products_cancel_mod PA0085 gauss_lemma_power_congruence_exists PA0088 pow_congruent_base_witness PA008J prime_mul_residue_product_balance PA008L fermat_predecessor_exponent_mod_one PA008M quadratic_residue_half_power_mod_one PA00A0 beta_adjacent_target_pairs_product_power PA00BK scaled_pair_order_terminal_power_mod_predecessor PA00BL scaled_inverse_nonresidue_half_power_mod_predecessor PA00BM quadratic_nonresidue_half_power_mod_predecessor PA00BN bounded_euler_criterion_dichotomy PA00BQ bounded_euler_criterion_residue_iff PA00BR arbitrary_euler_criterion_residue_iff PA00BS bounded_euler_criterion_nonresidue_iff PA00BT arbitrary_euler_criterion_nonresidue_iff PA00BU arbitrary_euler_criterion_complete PA00BV arbitrary_gauss_lemma_complete