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. PowerValuation(p,a,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
1 occurrences
In local proof propositions
none
0 occurrences
Exact expanded native-PA statement
forall p a. exists e. (((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))Proof neighborhood
Direct theorem prerequisites
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.
Named ingredients (1)
01Fix variables and assumptionsL1–2
Original defined command ledger · 6 lines
- 0001
intro p - 0002
intro a - 0003
specialize bounded_power_valuation_exists p - 0004
specialize bounded_power_valuation_exists a - 0005
specialize bounded_power_valuation_exists a - 0006
exact bounded_power_valuation_exists