PC0018

pow_four_is_square_of_pow_two

Alpha v34 independently verified · alpha_closed; checked-use authorized; not Stable

The actual fourth power-base value is the square of the actual binary power 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.

Exact expanded first-order arithmetic statement

forall n v w. (exists pa_b_pc_power_four_binary pa_c_pc_power_four_binary. ((forall pa_i_pc_power_four_binary_repeat. (exists pa_lt_pc_power_four_binary_repeat_bound. pa_lt_pc_power_four_binary_repeat_bound + S pa_i_pc_power_four_binary_repeat = n) -> (((exists pa_h_pc_power_four_binary_repeat_decoded. pa_h_pc_power_four_binary_repeat_decoded + S (2) = S ((S (pa_i_pc_power_four_binary_repeat)) * pa_c_pc_power_four_binary)) /\ exists pa_q_pc_power_four_binary_repeat_decoded. pa_b_pc_power_four_binary = pa_q_pc_power_four_binary_repeat_decoded * S ((S (pa_i_pc_power_four_binary_repeat)) * pa_c_pc_power_four_binary) + (2)))) /\ (exists pa_u_pc_power_four_binary_product pa_v_pc_power_four_binary_product. ((((exists pa_h_pc_power_four_binary_product_start. pa_h_pc_power_four_binary_product_start + S (1) = S ((S (0)) * pa_v_pc_power_four_binary_product)) /\ exists pa_q_pc_power_four_binary_product_start. pa_u_pc_power_four_binary_product = pa_q_pc_power_four_binary_product_start * S ((S (0)) * pa_v_pc_power_four_binary_product) + (1))) /\ ((((exists pa_h_pc_power_four_binary_product_terminal. pa_h_pc_power_four_binary_product_terminal + S (v) = S ((S (n)) * pa_v_pc_power_four_binary_product)) /\ exists pa_q_pc_power_four_binary_product_terminal. pa_u_pc_power_four_binary_product = pa_q_pc_power_four_binary_product_terminal * S ((S (n)) * pa_v_pc_power_four_binary_product) + (v))) /\ forall pa_i_pc_power_four_binary_product. (exists pa_lt_pc_power_four_binary_product_bound. pa_lt_pc_power_four_binary_product_bound + S pa_i_pc_power_four_binary_product = n) -> exists pa_p_pc_power_four_binary_product pa_r_pc_power_four_binary_product pa_s_pc_power_four_binary_product. ((((exists pa_h_pc_power_four_binary_product_factor. pa_h_pc_power_four_binary_product_factor + S (pa_p_pc_power_four_binary_product) = S ((S (pa_i_pc_power_four_binary_product)) * pa_c_pc_power_four_binary)) /\ exists pa_q_pc_power_four_binary_product_factor. pa_b_pc_power_four_binary = pa_q_pc_power_four_binary_product_factor * S ((S (pa_i_pc_power_four_binary_product)) * pa_c_pc_power_four_binary) + (pa_p_pc_power_four_binary_product))) /\ ((((exists pa_h_pc_power_four_binary_product_partial. pa_h_pc_power_four_binary_product_partial + S (pa_r_pc_power_four_binary_product) = S ((S (pa_i_pc_power_four_binary_product)) * pa_v_pc_power_four_binary_product)) /\ exists pa_q_pc_power_four_binary_product_partial. pa_u_pc_power_four_binary_product = pa_q_pc_power_four_binary_product_partial * S ((S (pa_i_pc_power_four_binary_product)) * pa_v_pc_power_four_binary_product) + (pa_r_pc_power_four_binary_product))) /\ ((((exists pa_h_pc_power_four_binary_product_successor. pa_h_pc_power_four_binary_product_successor + S (pa_s_pc_power_four_binary_product) = S ((S (S pa_i_pc_power_four_binary_product)) * pa_v_pc_power_four_binary_product)) /\ exists pa_q_pc_power_four_binary_product_successor. pa_u_pc_power_four_binary_product = pa_q_pc_power_four_binary_product_successor * S ((S (S pa_i_pc_power_four_binary_product)) * pa_v_pc_power_four_binary_product) + (pa_s_pc_power_four_binary_product))) /\ pa_s_pc_power_four_binary_product = pa_r_pc_power_four_binary_product * pa_p_pc_power_four_binary_product)))))))) -> (exists pa_b_pc_power_four_quaternary pa_c_pc_power_four_quaternary. ((forall pa_i_pc_power_four_quaternary_repeat. (exists pa_lt_pc_power_four_quaternary_repeat_bound. pa_lt_pc_power_four_quaternary_repeat_bound + S pa_i_pc_power_four_quaternary_repeat = n) -> (((exists pa_h_pc_power_four_quaternary_repeat_decoded. pa_h_pc_power_four_quaternary_repeat_decoded + S (4) = S ((S (pa_i_pc_power_four_quaternary_repeat)) * pa_c_pc_power_four_quaternary)) /\ exists pa_q_pc_power_four_quaternary_repeat_decoded. pa_b_pc_power_four_quaternary = pa_q_pc_power_four_quaternary_repeat_decoded * S ((S (pa_i_pc_power_four_quaternary_repeat)) * pa_c_pc_power_four_quaternary) + (4)))) /\ (exists pa_u_pc_power_four_quaternary_product pa_v_pc_power_four_quaternary_product. ((((exists pa_h_pc_power_four_quaternary_product_start. pa_h_pc_power_four_quaternary_product_start + S (1) = S ((S (0)) * pa_v_pc_power_four_quaternary_product)) /\ exists pa_q_pc_power_four_quaternary_product_start. pa_u_pc_power_four_quaternary_product = pa_q_pc_power_four_quaternary_product_start * S ((S (0)) * pa_v_pc_power_four_quaternary_product) + (1))) /\ ((((exists pa_h_pc_power_four_quaternary_product_terminal. pa_h_pc_power_four_quaternary_product_terminal + S (w) = S ((S (n)) * pa_v_pc_power_four_quaternary_product)) /\ exists pa_q_pc_power_four_quaternary_product_terminal. pa_u_pc_power_four_quaternary_product = pa_q_pc_power_four_quaternary_product_terminal * S ((S (n)) * pa_v_pc_power_four_quaternary_product) + (w))) /\ forall pa_i_pc_power_four_quaternary_product. (exists pa_lt_pc_power_four_quaternary_product_bound. pa_lt_pc_power_four_quaternary_product_bound + S pa_i_pc_power_four_quaternary_product = n) -> exists pa_p_pc_power_four_quaternary_product pa_r_pc_power_four_quaternary_product pa_s_pc_power_four_quaternary_product. ((((exists pa_h_pc_power_four_quaternary_product_factor. pa_h_pc_power_four_quaternary_product_factor + S (pa_p_pc_power_four_quaternary_product) = S ((S (pa_i_pc_power_four_quaternary_product)) * pa_c_pc_power_four_quaternary)) /\ exists pa_q_pc_power_four_quaternary_product_factor. pa_b_pc_power_four_quaternary = pa_q_pc_power_four_quaternary_product_factor * S ((S (pa_i_pc_power_four_quaternary_product)) * pa_c_pc_power_four_quaternary) + (pa_p_pc_power_four_quaternary_product))) /\ ((((exists pa_h_pc_power_four_quaternary_product_partial. pa_h_pc_power_four_quaternary_product_partial + S (pa_r_pc_power_four_quaternary_product) = S ((S (pa_i_pc_power_four_quaternary_product)) * pa_v_pc_power_four_quaternary_product)) /\ exists pa_q_pc_power_four_quaternary_product_partial. pa_u_pc_power_four_quaternary_product = pa_q_pc_power_four_quaternary_product_partial * S ((S (pa_i_pc_power_four_quaternary_product)) * pa_v_pc_power_four_quaternary_product) + (pa_r_pc_power_four_quaternary_product))) /\ ((((exists pa_h_pc_power_four_quaternary_product_successor. pa_h_pc_power_four_quaternary_product_successor + S (pa_s_pc_power_four_quaternary_product) = S ((S (S pa_i_pc_power_four_quaternary_product)) * pa_v_pc_power_four_quaternary_product)) /\ exists pa_q_pc_power_four_quaternary_product_successor. pa_u_pc_power_four_quaternary_product = pa_q_pc_power_four_quaternary_product_successor * S ((S (S pa_i_pc_power_four_quaternary_product)) * pa_v_pc_power_four_quaternary_product) + (pa_s_pc_power_four_quaternary_product))) /\ pa_s_pc_power_four_quaternary_product = pa_r_pc_power_four_quaternary_product * pa_p_pc_power_four_quaternary_product)))))))) -> w = v * v

Constructive proof overview

Generated structural guide

The actual fourth power-base value is the square of the actual binary power value.

The unchanged tactic script uses 1 declared prerequisite and contains 19 exact native proof lines.

Alpha v34 checked-use · first admitted v27 · independently kernel and Lean verified; not Stable

Proof neighborhood

Direct dependencies

pow_mul_base Alpha theorem; checked-use authorized

Direct dependents

PC0019 central_binom_dominates_pow_two PC0027 pow_four_equals_binary_double

Formal native tactic body

Dependencies are introduced as named hypotheses before line 1. Local theorem links identify exact declared prerequisites. This exact body belongs to a complete independently kernel-checked constructive proof bundle and has Alpha checked-use authority; it does not imply Stable membership.

Read the argument

Proof checkpoints

19 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.

01Fix variables and assumptionsL1–5

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

  1. L1
    intro n
  2. L2
    intro v
  3. L3
    intro w
  4. L4
    intro hv
  5. L5
    intro hw
02Use earlier factsL6–14

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

  1. L6
    specialize pow_mul_base 2
  2. L7
    specialize pow_mul_base 2
  3. L8
    specialize pow_mul_base n
  4. L9
    specialize pow_mul_base v
  5. L10
    specialize pow_mul_base v
  6. L11
    specialize pow_mul_base w
  7. L12
    apply pow_mul_base
  8. L13
    exact hv
  9. L14
    exact hv
03Establish hfourL15–19

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

  1. L15
    have hfour : 2 * 2 = 4
  2. L16
    norm_num
  3. L17
    rewrite hfour
  4. L18
    rewrite hfour
  5. L19
    exact hw

Library-wide reading audit

Original exact command ledger · 19 lines
  1. 0001intro n
  2. 0002intro v
  3. 0003intro w
  4. 0004intro hv
  5. 0005intro hw
  6. 0006specialize pow_mul_base 2
  7. 0007specialize pow_mul_base 2
  8. 0008specialize pow_mul_base n
  9. 0009specialize pow_mul_base v
  10. 0010specialize pow_mul_base v
  11. 0011specialize pow_mul_base w
  12. 0012apply pow_mul_base
  13. 0013exact hv
  14. 0014exact hv
  15. 0015have hfour : 2 * 2 = 4
  16. 0016norm_num
  17. 0017rewrite hfour
  18. 0018rewrite hfour
  19. 0019exact hw