BT009W · Bertrand theorem

pow_two

Stable checked-use theorem · independently kernel verified

The relational second power is exactly the square.

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. ∀ n. e = 2 → Pow(a,e,n) → n = a · 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

PD0020 Pow

1 occurrences

In local proof propositions

none

0 occurrences

Exact expanded native-PA statement
forall a e n. e = 2 -> (exists ff_b_two ff_c_two. ((forall ff_i_two_repeat. (exists ff_lt_two_repeat_bound. ff_lt_two_repeat_bound + S ff_i_two_repeat = e) -> (((exists ff_h_two_repeat_decoded. ff_h_two_repeat_decoded + S (a) = S ((S (ff_i_two_repeat)) * ff_c_two)) /\ exists ff_q_two_repeat_decoded. ff_b_two = ff_q_two_repeat_decoded * S ((S (ff_i_two_repeat)) * ff_c_two) + (a)))) /\ (exists ff_u_two_product ff_v_two_product. ((((exists ff_h_two_product_start. ff_h_two_product_start + S (1) = S ((S (0)) * ff_v_two_product)) /\ exists ff_q_two_product_start. ff_u_two_product = ff_q_two_product_start * S ((S (0)) * ff_v_two_product) + (1))) /\ ((((exists ff_h_two_product_terminal. ff_h_two_product_terminal + S (n) = S ((S (e)) * ff_v_two_product)) /\ exists ff_q_two_product_terminal. ff_u_two_product = ff_q_two_product_terminal * S ((S (e)) * ff_v_two_product) + (n))) /\ forall ff_i_two_product. (exists ff_lt_two_product_bound. ff_lt_two_product_bound + S ff_i_two_product = e) -> exists ff_p_two_product ff_r_two_product ff_s_two_product. ((((exists ff_h_two_product_factor. ff_h_two_product_factor + S (ff_p_two_product) = S ((S (ff_i_two_product)) * ff_c_two)) /\ exists ff_q_two_product_factor. ff_b_two = ff_q_two_product_factor * S ((S (ff_i_two_product)) * ff_c_two) + (ff_p_two_product))) /\ ((((exists ff_h_two_product_partial. ff_h_two_product_partial + S (ff_r_two_product) = S ((S (ff_i_two_product)) * ff_v_two_product)) /\ exists ff_q_two_product_partial. ff_u_two_product = ff_q_two_product_partial * S ((S (ff_i_two_product)) * ff_v_two_product) + (ff_r_two_product))) /\ ((((exists ff_h_two_product_successor. ff_h_two_product_successor + S (ff_s_two_product) = S ((S (S ff_i_two_product)) * ff_v_two_product)) /\ exists ff_q_two_product_successor. ff_u_two_product = ff_q_two_product_successor * S ((S (S ff_i_two_product)) * ff_v_two_product) + (ff_s_two_product))) /\ ff_s_two_product = ff_r_two_product * ff_p_two_product)))))))) -> n = a * a

Proof neighborhood

Direct theorem prerequisites

BT009V pow_two_from_one_successor

Direct theorem dependents

BT00QW pow_two_base_two_value_four BT00U1 pow_four_four_exact BT00W5 pow_eleven_two_le_pow_two_seven_from_total BT00WO pow_thirty_six_double_block_eq_pow_six_four_block_from_total BT00X4 bertrand_floor_power_product_le_h_from_total BT00YA prime_square_tail_of_two_three_range BT00YH floor_sqrt_above_root_power_two_strict

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

13 script commands · 4 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.

Named ingredients (1)
pow two from one successor
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 n
  4. L4
    intro he
  5. L5
    intro hpow
02Use earlier factsL6–10

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

  1. L6
    specialize pow_two_from_one_successor a
  2. L7
    specialize pow_two_from_one_successor 1
  3. L8
    specialize pow_two_from_one_successor e
  4. L9
    specialize pow_two_from_one_successor n
  5. L10
    apply pow_two_from_one_successor
03Calculate and transport equalitiesL11–11

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

  1. L11
    refl
04Use earlier factsL12–13

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

  1. L12
    exact he
  2. L13
    exact hpow

Library-wide reading audit

Original defined command ledger · 13 lines
  1. 0001intro a
  2. 0002intro e
  3. 0003intro n
  4. 0004intro he
  5. 0005intro hpow
  6. 0006specialize pow_two_from_one_successor a
  7. 0007specialize pow_two_from_one_successor 1
  8. 0008specialize pow_two_from_one_successor e
  9. 0009specialize pow_two_from_one_successor n
  10. 0010apply pow_two_from_one_successor
  11. 0011refl
  12. 0012exact he
  13. 0013exact hpow