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.
Statement with defined notation
∀ p. ∀ a. ∀ e. ∀ f. PowerValuation(p,a,e) → Le(f,a) → PowerDivides(p,f,a) → Le(f,e)Every purple notation token opens its conservative definition. Expanding the displayed statement recovers the exact first-order Peano-arithmetic formula checked by the unchanged kernel.
Definitions used by this theorem
In the theorem statement
4 occurrences
In local proof propositions
none
0 occurrences
Exact expanded native-PA statement
forall p a e f. (((exists bpv_gap_canonical_exponent_bound. bpv_gap_canonical_exponent_bound + e = a) /\ (exists bpv_result_canonical_selected. ((exists ff_b_canonical_selected_power ff_c_canonical_selected_power. ((forall ff_i_canonical_selected_power_repeat. (exists ff_lt_canonical_selected_power_repeat_bound. ff_lt_canonical_selected_power_repeat_bound + S ff_i_canonical_selected_power_repeat = e) -> (((exists ff_h_canonical_selected_power_repeat_decoded. ff_h_canonical_selected_power_repeat_decoded + S (p) = S ((S (ff_i_canonical_selected_power_repeat)) * ff_c_canonical_selected_power)) /\ exists ff_q_canonical_selected_power_repeat_decoded. ff_b_canonical_selected_power = ff_q_canonical_selected_power_repeat_decoded * S ((S (ff_i_canonical_selected_power_repeat)) * ff_c_canonical_selected_power) + (p)))) /\ (exists ff_u_canonical_selected_power_product ff_v_canonical_selected_power_product. ((((exists ff_h_canonical_selected_power_product_start. ff_h_canonical_selected_power_product_start + S (1) = S ((S (0)) * ff_v_canonical_selected_power_product)) /\ exists ff_q_canonical_selected_power_product_start. ff_u_canonical_selected_power_product = ff_q_canonical_selected_power_product_start * S ((S (0)) * ff_v_canonical_selected_power_product) + (1))) /\ ((((exists ff_h_canonical_selected_power_product_terminal. ff_h_canonical_selected_power_product_terminal + S (bpv_result_canonical_selected) = S ((S (e)) * ff_v_canonical_selected_power_product)) /\ exists ff_q_canonical_selected_power_product_terminal. ff_u_canonical_selected_power_product = ff_q_canonical_selected_power_product_terminal * S ((S (e)) * ff_v_canonical_selected_power_product) + (bpv_result_canonical_selected))) /\ forall ff_i_canonical_selected_power_product. (exists ff_lt_canonical_selected_power_product_bound. ff_lt_canonical_selected_power_product_bound + S ff_i_canonical_selected_power_product = e) -> exists ff_p_canonical_selected_power_product ff_r_canonical_selected_power_product ff_s_canonical_selected_power_product. ((((exists ff_h_canonical_selected_power_product_factor. ff_h_canonical_selected_power_product_factor + S (ff_p_canonical_selected_power_product) = S ((S (ff_i_canonical_selected_power_product)) * ff_c_canonical_selected_power)) /\ exists ff_q_canonical_selected_power_product_factor. ff_b_canonical_selected_power = ff_q_canonical_selected_power_product_factor * S ((S (ff_i_canonical_selected_power_product)) * ff_c_canonical_selected_power) + (ff_p_canonical_selected_power_product))) /\ ((((exists ff_h_canonical_selected_power_product_partial. ff_h_canonical_selected_power_product_partial + S (ff_r_canonical_selected_power_product) = S ((S (ff_i_canonical_selected_power_product)) * ff_v_canonical_selected_power_product)) /\ exists ff_q_canonical_selected_power_product_partial. ff_u_canonical_selected_power_product = ff_q_canonical_selected_power_product_partial * S ((S (ff_i_canonical_selected_power_product)) * ff_v_canonical_selected_power_product) + (ff_r_canonical_selected_power_product))) /\ ((((exists ff_h_canonical_selected_power_product_successor. ff_h_canonical_selected_power_product_successor + S (ff_s_canonical_selected_power_product) = S ((S (S ff_i_canonical_selected_power_product)) * ff_v_canonical_selected_power_product)) /\ exists ff_q_canonical_selected_power_product_successor. ff_u_canonical_selected_power_product = ff_q_canonical_selected_power_product_successor * S ((S (S ff_i_canonical_selected_power_product)) * ff_v_canonical_selected_power_product) + (ff_s_canonical_selected_power_product))) /\ ff_s_canonical_selected_power_product = ff_r_canonical_selected_power_product * ff_p_canonical_selected_power_product)))))))) /\ (exists bpv_factor_canonical_selected_divides. a = bpv_result_canonical_selected * bpv_factor_canonical_selected_divides)))) /\ forall bpv_candidate_canonical. (exists bpv_gap_canonical_candidate_bound. bpv_gap_canonical_candidate_bound + bpv_candidate_canonical = a) -> (exists bpv_result_canonical_candidate. ((exists ff_b_canonical_candidate_power ff_c_canonical_candidate_power. ((forall ff_i_canonical_candidate_power_repeat. (exists ff_lt_canonical_candidate_power_repeat_bound. ff_lt_canonical_candidate_power_repeat_bound + S ff_i_canonical_candidate_power_repeat = bpv_candidate_canonical) -> (((exists ff_h_canonical_candidate_power_repeat_decoded. ff_h_canonical_candidate_power_repeat_decoded + S (p) = S ((S (ff_i_canonical_candidate_power_repeat)) * ff_c_canonical_candidate_power)) /\ exists ff_q_canonical_candidate_power_repeat_decoded. ff_b_canonical_candidate_power = ff_q_canonical_candidate_power_repeat_decoded * S ((S (ff_i_canonical_candidate_power_repeat)) * ff_c_canonical_candidate_power) + (p)))) /\ (exists ff_u_canonical_candidate_power_product ff_v_canonical_candidate_power_product. ((((exists ff_h_canonical_candidate_power_product_start. ff_h_canonical_candidate_power_product_start + S (1) = S ((S (0)) * ff_v_canonical_candidate_power_product)) /\ exists ff_q_canonical_candidate_power_product_start. ff_u_canonical_candidate_power_product = ff_q_canonical_candidate_power_product_start * S ((S (0)) * ff_v_canonical_candidate_power_product) + (1))) /\ ((((exists ff_h_canonical_candidate_power_product_terminal. ff_h_canonical_candidate_power_product_terminal + S (bpv_result_canonical_candidate) = S ((S (bpv_candidate_canonical)) * ff_v_canonical_candidate_power_product)) /\ exists ff_q_canonical_candidate_power_product_terminal. ff_u_canonical_candidate_power_product = ff_q_canonical_candidate_power_product_terminal * S ((S (bpv_candidate_canonical)) * ff_v_canonical_candidate_power_product) + (bpv_result_canonical_candidate))) /\ forall ff_i_canonical_candidate_power_product. (exists ff_lt_canonical_candidate_power_product_bound. ff_lt_canonical_candidate_power_product_bound + S ff_i_canonical_candidate_power_product = bpv_candidate_canonical) -> exists ff_p_canonical_candidate_power_product ff_r_canonical_candidate_power_product ff_s_canonical_candidate_power_product. ((((exists ff_h_canonical_candidate_power_product_factor. ff_h_canonical_candidate_power_product_factor + S (ff_p_canonical_candidate_power_product) = S ((S (ff_i_canonical_candidate_power_product)) * ff_c_canonical_candidate_power)) /\ exists ff_q_canonical_candidate_power_product_factor. ff_b_canonical_candidate_power = ff_q_canonical_candidate_power_product_factor * S ((S (ff_i_canonical_candidate_power_product)) * ff_c_canonical_candidate_power) + (ff_p_canonical_candidate_power_product))) /\ ((((exists ff_h_canonical_candidate_power_product_partial. ff_h_canonical_candidate_power_product_partial + S (ff_r_canonical_candidate_power_product) = S ((S (ff_i_canonical_candidate_power_product)) * ff_v_canonical_candidate_power_product)) /\ exists ff_q_canonical_candidate_power_product_partial. ff_u_canonical_candidate_power_product = ff_q_canonical_candidate_power_product_partial * S ((S (ff_i_canonical_candidate_power_product)) * ff_v_canonical_candidate_power_product) + (ff_r_canonical_candidate_power_product))) /\ ((((exists ff_h_canonical_candidate_power_product_successor. ff_h_canonical_candidate_power_product_successor + S (ff_s_canonical_candidate_power_product) = S ((S (S ff_i_canonical_candidate_power_product)) * ff_v_canonical_candidate_power_product)) /\ exists ff_q_canonical_candidate_power_product_successor. ff_u_canonical_candidate_power_product = ff_q_canonical_candidate_power_product_successor * S ((S (S ff_i_canonical_candidate_power_product)) * ff_v_canonical_candidate_power_product) + (ff_s_canonical_candidate_power_product))) /\ ff_s_canonical_candidate_power_product = ff_r_canonical_candidate_power_product * ff_p_canonical_candidate_power_product)))))))) /\ (exists bpv_factor_canonical_candidate_divides. a = bpv_result_canonical_candidate * bpv_factor_canonical_candidate_divides))) -> (exists bpv_gap_canonical_maximal. bpv_gap_canonical_maximal + bpv_candidate_canonical = e)) -> (exists bpv_gap_dominates_bound. bpv_gap_dominates_bound + f = a) -> (exists bpv_result_dominates_candidate. ((exists ff_b_dominates_candidate_power ff_c_dominates_candidate_power. ((forall ff_i_dominates_candidate_power_repeat. (exists ff_lt_dominates_candidate_power_repeat_bound. ff_lt_dominates_candidate_power_repeat_bound + S ff_i_dominates_candidate_power_repeat = f) -> (((exists ff_h_dominates_candidate_power_repeat_decoded. ff_h_dominates_candidate_power_repeat_decoded + S (p) = S ((S (ff_i_dominates_candidate_power_repeat)) * ff_c_dominates_candidate_power)) /\ exists ff_q_dominates_candidate_power_repeat_decoded. ff_b_dominates_candidate_power = ff_q_dominates_candidate_power_repeat_decoded * S ((S (ff_i_dominates_candidate_power_repeat)) * ff_c_dominates_candidate_power) + (p)))) /\ (exists ff_u_dominates_candidate_power_product ff_v_dominates_candidate_power_product. ((((exists ff_h_dominates_candidate_power_product_start. ff_h_dominates_candidate_power_product_start + S (1) = S ((S (0)) * ff_v_dominates_candidate_power_product)) /\ exists ff_q_dominates_candidate_power_product_start. ff_u_dominates_candidate_power_product = ff_q_dominates_candidate_power_product_start * S ((S (0)) * ff_v_dominates_candidate_power_product) + (1))) /\ ((((exists ff_h_dominates_candidate_power_product_terminal. ff_h_dominates_candidate_power_product_terminal + S (bpv_result_dominates_candidate) = S ((S (f)) * ff_v_dominates_candidate_power_product)) /\ exists ff_q_dominates_candidate_power_product_terminal. ff_u_dominates_candidate_power_product = ff_q_dominates_candidate_power_product_terminal * S ((S (f)) * ff_v_dominates_candidate_power_product) + (bpv_result_dominates_candidate))) /\ forall ff_i_dominates_candidate_power_product. (exists ff_lt_dominates_candidate_power_product_bound. ff_lt_dominates_candidate_power_product_bound + S ff_i_dominates_candidate_power_product = f) -> exists ff_p_dominates_candidate_power_product ff_r_dominates_candidate_power_product ff_s_dominates_candidate_power_product. ((((exists ff_h_dominates_candidate_power_product_factor. ff_h_dominates_candidate_power_product_factor + S (ff_p_dominates_candidate_power_product) = S ((S (ff_i_dominates_candidate_power_product)) * ff_c_dominates_candidate_power)) /\ exists ff_q_dominates_candidate_power_product_factor. ff_b_dominates_candidate_power = ff_q_dominates_candidate_power_product_factor * S ((S (ff_i_dominates_candidate_power_product)) * ff_c_dominates_candidate_power) + (ff_p_dominates_candidate_power_product))) /\ ((((exists ff_h_dominates_candidate_power_product_partial. ff_h_dominates_candidate_power_product_partial + S (ff_r_dominates_candidate_power_product) = S ((S (ff_i_dominates_candidate_power_product)) * ff_v_dominates_candidate_power_product)) /\ exists ff_q_dominates_candidate_power_product_partial. ff_u_dominates_candidate_power_product = ff_q_dominates_candidate_power_product_partial * S ((S (ff_i_dominates_candidate_power_product)) * ff_v_dominates_candidate_power_product) + (ff_r_dominates_candidate_power_product))) /\ ((((exists ff_h_dominates_candidate_power_product_successor. ff_h_dominates_candidate_power_product_successor + S (ff_s_dominates_candidate_power_product) = S ((S (S ff_i_dominates_candidate_power_product)) * ff_v_dominates_candidate_power_product)) /\ exists ff_q_dominates_candidate_power_product_successor. ff_u_dominates_candidate_power_product = ff_q_dominates_candidate_power_product_successor * S ((S (S ff_i_dominates_candidate_power_product)) * ff_v_dominates_candidate_power_product) + (ff_s_dominates_candidate_power_product))) /\ ff_s_dominates_candidate_power_product = ff_r_dominates_candidate_power_product * ff_p_dominates_candidate_power_product)))))))) /\ (exists bpv_factor_dominates_candidate_divides. a = bpv_result_dominates_candidate * bpv_factor_dominates_candidate_divides))) -> (exists bpv_gap_dominates_result. bpv_gap_dominates_result + f = e)Proof neighborhood
Direct theorem prerequisites
none
Direct theorem dependents
Definition-aware tactic body
Only local propositions introduced by have or suffices are compacted. Every changed line has an exact-AST conservative-expansion receipt; the kernel still receives the immutable original tactic script.
Read the argument
Proof checkpoints
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 assumptionsL1–7
02Separate the logical casesL8–8
Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.
- L8
cases hvaluation