PC0018

pow_four_is_square_of_pow_two

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

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

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.

These are the exact finite integer inequalities with constant 8, for every N≥2. The proof uses constructive binomial and primorial infrastructure; it does not assume logarithms, asymptotic estimates, the prime number theorem, or a factorization oracle.

Exact theorem in conservative defined notation

∀ n. ∀ v. ∀ w. PowTwo(n,v) → Pow(4,n,w) → w = v · v

Every linked abbreviation expands hygienically to the identical original native formula.

Definition DAG

PowTwo(e,p) · 1Pow(a,e,z) · 1

Actual proof prerequisites

pow_mul_base · checked external prerequisite
Original expanded first-order 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

Complete tactic proof in conservative notation

All 19 original proof lines are preserved. Only local proposition formulas are abbreviated; every abbreviation has an exact binder-safe expansion check. The linked exact edition contains the unchanged replay script.

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.

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 assumptionsL1–5

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 defined 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