BT00Q1 · Bertrand theorem

pow_nonzero_of_one_le

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

A relational power of a base at least one cannot be zero.

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

∀ a. ∀ e. ∀ x. Lt(0,a) → Pow(a,e,x) → ¬x = 0

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

PD0002 Lt PD0020 Pow

2 occurrences

In local proof propositions

PD0002 Lt

1 occurrences

Exact expanded native-PA statement
forall a e x. (exists bpg_gap_base. bpg_gap_base + (1) = (a)) -> (exists ff_b_bpg_value ff_c_bpg_value. ((forall ff_i_bpg_value_repeat. (exists ff_lt_bpg_value_repeat_bound. ff_lt_bpg_value_repeat_bound + S ff_i_bpg_value_repeat = e) -> (((exists ff_h_bpg_value_repeat_decoded. ff_h_bpg_value_repeat_decoded + S (a) = S ((S (ff_i_bpg_value_repeat)) * ff_c_bpg_value)) /\ exists ff_q_bpg_value_repeat_decoded. ff_b_bpg_value = ff_q_bpg_value_repeat_decoded * S ((S (ff_i_bpg_value_repeat)) * ff_c_bpg_value) + (a)))) /\ (exists ff_u_bpg_value_product ff_v_bpg_value_product. ((((exists ff_h_bpg_value_product_start. ff_h_bpg_value_product_start + S (1) = S ((S (0)) * ff_v_bpg_value_product)) /\ exists ff_q_bpg_value_product_start. ff_u_bpg_value_product = ff_q_bpg_value_product_start * S ((S (0)) * ff_v_bpg_value_product) + (1))) /\ ((((exists ff_h_bpg_value_product_terminal. ff_h_bpg_value_product_terminal + S (x) = S ((S (e)) * ff_v_bpg_value_product)) /\ exists ff_q_bpg_value_product_terminal. ff_u_bpg_value_product = ff_q_bpg_value_product_terminal * S ((S (e)) * ff_v_bpg_value_product) + (x))) /\ forall ff_i_bpg_value_product. (exists ff_lt_bpg_value_product_bound. ff_lt_bpg_value_product_bound + S ff_i_bpg_value_product = e) -> exists ff_p_bpg_value_product ff_r_bpg_value_product ff_s_bpg_value_product. ((((exists ff_h_bpg_value_product_factor. ff_h_bpg_value_product_factor + S (ff_p_bpg_value_product) = S ((S (ff_i_bpg_value_product)) * ff_c_bpg_value)) /\ exists ff_q_bpg_value_product_factor. ff_b_bpg_value = ff_q_bpg_value_product_factor * S ((S (ff_i_bpg_value_product)) * ff_c_bpg_value) + (ff_p_bpg_value_product))) /\ ((((exists ff_h_bpg_value_product_partial. ff_h_bpg_value_product_partial + S (ff_r_bpg_value_product) = S ((S (ff_i_bpg_value_product)) * ff_v_bpg_value_product)) /\ exists ff_q_bpg_value_product_partial. ff_u_bpg_value_product = ff_q_bpg_value_product_partial * S ((S (ff_i_bpg_value_product)) * ff_v_bpg_value_product) + (ff_r_bpg_value_product))) /\ ((((exists ff_h_bpg_value_product_successor. ff_h_bpg_value_product_successor + S (ff_s_bpg_value_product) = S ((S (S ff_i_bpg_value_product)) * ff_v_bpg_value_product)) /\ exists ff_q_bpg_value_product_successor. ff_u_bpg_value_product = ff_q_bpg_value_product_successor * S ((S (S ff_i_bpg_value_product)) * ff_v_bpg_value_product) + (ff_s_bpg_value_product))) /\ ff_s_bpg_value_product = ff_r_bpg_value_product * ff_p_bpg_value_product)))))))) -> ~(x = 0)

Proof neighborhood

Direct theorem prerequisites

BT00Q0 one_le_pow BT0011 ne_zero_of_one_le

Direct theorem dependents

BT00QF prime_power_exponent_le BT00QN prime_power_successor_cancel_cofactor BT00S0 prime_power_quotient_prefix_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

17 script commands · 3 reading checkpoints · 1 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 (2)
one le powne zero of one le
01Fix variables and assumptionsL1–5

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

  1. L1
    intro a
  2. L2
    intro e
  3. L3
    intro x
  4. L4
    intro ha
  5. L5
    intro hx
02Establish hx1L6–15

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

  1. L6
  2. L7
    specialize one_le_pow a
  3. L8
    specialize one_le_pow e
  4. L9
    specialize one_le_pow x
  5. L10
    apply one_le_pow
  6. L11
    exact ha
  7. L12
    exact hx
  8. L13
    intro hx0
  9. L14
    specialize ne_zero_of_one_le x
  10. L15
    apply ne_zero_of_one_le
03Use earlier factsL16–17

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

  1. L16
    exact hx1
  2. L17
    exact hx0

Library-wide reading audit

Original defined command ledger · 17 lines
  1. 0001intro a
  2. 0002intro e
  3. 0003intro x
  4. 0004intro ha
  5. 0005intro hx
  6. 0006have hx1 : Lt(0,x)
    Exact native replay linehave hx1 : exists bpg_gap_value. bpg_gap_value + (1) = (x)
  7. 0007specialize one_le_pow a
  8. 0008specialize one_le_pow e
  9. 0009specialize one_le_pow x
  10. 0010apply one_le_pow
  11. 0011exact ha
  12. 0012exact hx
  13. 0013intro hx0
  14. 0014specialize ne_zero_of_one_le x
  15. 0015apply ne_zero_of_one_le
  16. 0016exact hx1
  17. 0017exact hx0