BT00Q4 · Bertrand theorem

power_divides_zero

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

The zeroth relational power divides every natural.

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. ∀ z. z = 0 → PowerDivides(p,z,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

1 occurrences

In local proof propositions

PD0020 Pow

1 occurrences

Exact expanded native-PA statement
forall p a z. z = 0 -> (exists bpv_result_zero. ((exists ff_b_zero_power ff_c_zero_power. ((forall ff_i_zero_power_repeat. (exists ff_lt_zero_power_repeat_bound. ff_lt_zero_power_repeat_bound + S ff_i_zero_power_repeat = z) -> (((exists ff_h_zero_power_repeat_decoded. ff_h_zero_power_repeat_decoded + S (p) = S ((S (ff_i_zero_power_repeat)) * ff_c_zero_power)) /\ exists ff_q_zero_power_repeat_decoded. ff_b_zero_power = ff_q_zero_power_repeat_decoded * S ((S (ff_i_zero_power_repeat)) * ff_c_zero_power) + (p)))) /\ (exists ff_u_zero_power_product ff_v_zero_power_product. ((((exists ff_h_zero_power_product_start. ff_h_zero_power_product_start + S (1) = S ((S (0)) * ff_v_zero_power_product)) /\ exists ff_q_zero_power_product_start. ff_u_zero_power_product = ff_q_zero_power_product_start * S ((S (0)) * ff_v_zero_power_product) + (1))) /\ ((((exists ff_h_zero_power_product_terminal. ff_h_zero_power_product_terminal + S (bpv_result_zero) = S ((S (z)) * ff_v_zero_power_product)) /\ exists ff_q_zero_power_product_terminal. ff_u_zero_power_product = ff_q_zero_power_product_terminal * S ((S (z)) * ff_v_zero_power_product) + (bpv_result_zero))) /\ forall ff_i_zero_power_product. (exists ff_lt_zero_power_product_bound. ff_lt_zero_power_product_bound + S ff_i_zero_power_product = z) -> exists ff_p_zero_power_product ff_r_zero_power_product ff_s_zero_power_product. ((((exists ff_h_zero_power_product_factor. ff_h_zero_power_product_factor + S (ff_p_zero_power_product) = S ((S (ff_i_zero_power_product)) * ff_c_zero_power)) /\ exists ff_q_zero_power_product_factor. ff_b_zero_power = ff_q_zero_power_product_factor * S ((S (ff_i_zero_power_product)) * ff_c_zero_power) + (ff_p_zero_power_product))) /\ ((((exists ff_h_zero_power_product_partial. ff_h_zero_power_product_partial + S (ff_r_zero_power_product) = S ((S (ff_i_zero_power_product)) * ff_v_zero_power_product)) /\ exists ff_q_zero_power_product_partial. ff_u_zero_power_product = ff_q_zero_power_product_partial * S ((S (ff_i_zero_power_product)) * ff_v_zero_power_product) + (ff_r_zero_power_product))) /\ ((((exists ff_h_zero_power_product_successor. ff_h_zero_power_product_successor + S (ff_s_zero_power_product) = S ((S (S ff_i_zero_power_product)) * ff_v_zero_power_product)) /\ exists ff_q_zero_power_product_successor. ff_u_zero_power_product = ff_q_zero_power_product_successor * S ((S (S ff_i_zero_power_product)) * ff_v_zero_power_product) + (ff_s_zero_power_product))) /\ ff_s_zero_power_product = ff_r_zero_power_product * ff_p_zero_power_product)))))))) /\ (exists bpv_factor_zero_divides. a = bpv_result_zero * bpv_factor_zero_divides)))

Proof neighborhood

Direct theorem prerequisites

BT0080 pow_exists BT0081 pow_zero BT0027 one_multiple

Direct theorem dependents

BT00Q6 bounded_power_valuation_exists

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

22 script commands · 9 reading checkpoints · 2 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 (3)
pow existspow zeroone multiple
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 z
  4. L4
    intro hz
02Establish hpowerL5–8

Establish this local claim before using it. It is not an additional assumption.

  1. L5
    have hpower : ∃ r. Pow(p,z,r)Definitions: Pow(p,z,r)Original native command in the exact edition
  2. L6
    specialize pow_exists p
  3. L7
    specialize pow_exists z
  4. L8
    exact pow_exists
03Separate the logical casesL9–9

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

  1. L9
    cases hpower
04Establish hrL10–16

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply pow zero.

  1. L10
    have hr : x = 1
  2. L11
    specialize pow_zero p
  3. L12
    specialize pow_zero z
  4. L13
    specialize pow_zero x
  5. L14
    apply pow_zero
  6. L15
    exact hz
  7. L16
    exact hpower_witness
05Construct an explicit witnessL17–17

Supply the displayed value, then prove that it has the required property.

  1. L17
    exists x
06Separate the logical casesL18–18

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

  1. L18
    split
07Use earlier factsL19–19

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

  1. L19
    exact hpower_witness
08Calculate and transport equalitiesL20–20

Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.

  1. L20
    rewrite hr
09Use earlier factsL21–22

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

  1. L21
    specialize one_multiple a
  2. L22
    exact one_multiple

Library-wide reading audit

Original defined command ledger · 22 lines
  1. 0001intro p
  2. 0002intro a
  3. 0003intro z
  4. 0004intro hz
  5. 0005have hpower : ∃ r. Pow(p,z,r)
    Exact native replay linehave hpower : exists r. (exists ff_b_zero_witness ff_c_zero_witness. ((forall ff_i_zero_witness_repeat. (exists ff_lt_zero_witness_repeat_bound. ff_lt_zero_witness_repeat_bound + S ff_i_zero_witness_repeat = z) -> (((exists ff_h_zero_witness_repeat_decoded. ff_h_zero_witness_repeat_decoded + S (p) = S ((S (ff_i_zero_witness_repeat)) * ff_c_zero_witness)) /\ exists ff_q_zero_witness_repeat_decoded. ff_b_zero_witness = ff_q_zero_witness_repeat_decoded * S ((S (ff_i_zero_witness_repeat)) * ff_c_zero_witness) + (p)))) /\ (exists ff_u_zero_witness_product ff_v_zero_witness_product. ((((exists ff_h_zero_witness_product_start. ff_h_zero_witness_product_start + S (1) = S ((S (0)) * ff_v_zero_witness_product)) /\ exists ff_q_zero_witness_product_start. ff_u_zero_witness_product = ff_q_zero_witness_product_start * S ((S (0)) * ff_v_zero_witness_product) + (1))) /\ ((((exists ff_h_zero_witness_product_terminal. ff_h_zero_witness_product_terminal + S (r) = S ((S (z)) * ff_v_zero_witness_product)) /\ exists ff_q_zero_witness_product_terminal. ff_u_zero_witness_product = ff_q_zero_witness_product_terminal * S ((S (z)) * ff_v_zero_witness_product) + (r))) /\ forall ff_i_zero_witness_product. (exists ff_lt_zero_witness_product_bound. ff_lt_zero_witness_product_bound + S ff_i_zero_witness_product = z) -> exists ff_p_zero_witness_product ff_r_zero_witness_product ff_s_zero_witness_product. ((((exists ff_h_zero_witness_product_factor. ff_h_zero_witness_product_factor + S (ff_p_zero_witness_product) = S ((S (ff_i_zero_witness_product)) * ff_c_zero_witness)) /\ exists ff_q_zero_witness_product_factor. ff_b_zero_witness = ff_q_zero_witness_product_factor * S ((S (ff_i_zero_witness_product)) * ff_c_zero_witness) + (ff_p_zero_witness_product))) /\ ((((exists ff_h_zero_witness_product_partial. ff_h_zero_witness_product_partial + S (ff_r_zero_witness_product) = S ((S (ff_i_zero_witness_product)) * ff_v_zero_witness_product)) /\ exists ff_q_zero_witness_product_partial. ff_u_zero_witness_product = ff_q_zero_witness_product_partial * S ((S (ff_i_zero_witness_product)) * ff_v_zero_witness_product) + (ff_r_zero_witness_product))) /\ ((((exists ff_h_zero_witness_product_successor. ff_h_zero_witness_product_successor + S (ff_s_zero_witness_product) = S ((S (S ff_i_zero_witness_product)) * ff_v_zero_witness_product)) /\ exists ff_q_zero_witness_product_successor. ff_u_zero_witness_product = ff_q_zero_witness_product_successor * S ((S (S ff_i_zero_witness_product)) * ff_v_zero_witness_product) + (ff_s_zero_witness_product))) /\ ff_s_zero_witness_product = ff_r_zero_witness_product * ff_p_zero_witness_product))))))))
  6. 0006specialize pow_exists p
  7. 0007specialize pow_exists z
  8. 0008exact pow_exists
  9. 0009cases hpower
  10. 0010have hr : x = 1
  11. 0011specialize pow_zero p
  12. 0012specialize pow_zero z
  13. 0013specialize pow_zero x
  14. 0014apply pow_zero
  15. 0015exact hz
  16. 0016exact hpower_witness
  17. 0017exists x
  18. 0018split
  19. 0019exact hpower_witness
  20. 0020rewrite hr
  21. 0021specialize one_multiple a
  22. 0022exact one_multiple