BT00Q9 · Bertrand theorem

power_valuation_power_divides

Alpha v34 checked-use theorem · independently kernel and Lean verified; not Stable

A valuation exponent has a relational power dividing the value.

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) → PowerDivides(p,e,a)

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

PD0044 PowerDivides PD0046 PowerValuation

2 occurrences

In local proof propositions

none

0 occurrences

Exact expanded native-PA statement
forall p a 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)) -> (exists bpv_result_projection. ((exists ff_b_projection_power ff_c_projection_power. ((forall ff_i_projection_power_repeat. (exists ff_lt_projection_power_repeat_bound. ff_lt_projection_power_repeat_bound + S ff_i_projection_power_repeat = e) -> (((exists ff_h_projection_power_repeat_decoded. ff_h_projection_power_repeat_decoded + S (p) = S ((S (ff_i_projection_power_repeat)) * ff_c_projection_power)) /\ exists ff_q_projection_power_repeat_decoded. ff_b_projection_power = ff_q_projection_power_repeat_decoded * S ((S (ff_i_projection_power_repeat)) * ff_c_projection_power) + (p)))) /\ (exists ff_u_projection_power_product ff_v_projection_power_product. ((((exists ff_h_projection_power_product_start. ff_h_projection_power_product_start + S (1) = S ((S (0)) * ff_v_projection_power_product)) /\ exists ff_q_projection_power_product_start. ff_u_projection_power_product = ff_q_projection_power_product_start * S ((S (0)) * ff_v_projection_power_product) + (1))) /\ ((((exists ff_h_projection_power_product_terminal. ff_h_projection_power_product_terminal + S (bpv_result_projection) = S ((S (e)) * ff_v_projection_power_product)) /\ exists ff_q_projection_power_product_terminal. ff_u_projection_power_product = ff_q_projection_power_product_terminal * S ((S (e)) * ff_v_projection_power_product) + (bpv_result_projection))) /\ forall ff_i_projection_power_product. (exists ff_lt_projection_power_product_bound. ff_lt_projection_power_product_bound + S ff_i_projection_power_product = e) -> exists ff_p_projection_power_product ff_r_projection_power_product ff_s_projection_power_product. ((((exists ff_h_projection_power_product_factor. ff_h_projection_power_product_factor + S (ff_p_projection_power_product) = S ((S (ff_i_projection_power_product)) * ff_c_projection_power)) /\ exists ff_q_projection_power_product_factor. ff_b_projection_power = ff_q_projection_power_product_factor * S ((S (ff_i_projection_power_product)) * ff_c_projection_power) + (ff_p_projection_power_product))) /\ ((((exists ff_h_projection_power_product_partial. ff_h_projection_power_product_partial + S (ff_r_projection_power_product) = S ((S (ff_i_projection_power_product)) * ff_v_projection_power_product)) /\ exists ff_q_projection_power_product_partial. ff_u_projection_power_product = ff_q_projection_power_product_partial * S ((S (ff_i_projection_power_product)) * ff_v_projection_power_product) + (ff_r_projection_power_product))) /\ ((((exists ff_h_projection_power_product_successor. ff_h_projection_power_product_successor + S (ff_s_projection_power_product) = S ((S (S ff_i_projection_power_product)) * ff_v_projection_power_product)) /\ exists ff_q_projection_power_product_successor. ff_u_projection_power_product = ff_q_projection_power_product_successor * S ((S (S ff_i_projection_power_product)) * ff_v_projection_power_product) + (ff_s_projection_power_product))) /\ ff_s_projection_power_product = ff_r_projection_power_product * ff_p_projection_power_product)))))))) /\ (exists bpv_factor_projection_divides. a = bpv_result_projection * bpv_factor_projection_divides)))

Proof neighborhood

Direct theorem prerequisites

none

Direct theorem dependents

BT00QI power_valuation_selected_and_successor_not_divides BT00QR power_valuation_mul_lower BT00QS power_valuation_mul_upper BT00RL prime_power_valuation_one_zero BT00SC power_divides_of_exponent_le_valuation BT00W0 power_valuation_nonzero_exponent_divides_base BT00YX prime_contribution_factor_divides

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

7 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.

01Fix variables and assumptionsL1–4

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

  1. L1
    intro p
  2. L2
    intro a
  3. L3
    intro e
  4. L4
    intro hvaluation
02Separate the logical casesL5–6

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

  1. L5
    cases hvaluation
  2. L6
    cases hvaluation_left
03Use earlier factsL7–7

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

  1. L7
    exact hvaluation_left_right

Library-wide reading audit

Original defined command ledger · 7 lines
  1. 0001intro p
  2. 0002intro a
  3. 0003intro e
  4. 0004intro hvaluation
  5. 0005cases hvaluation
  6. 0006cases hvaluation_left
  7. 0007exact hvaluation_left_right