PC0027

pow_four_equals_binary_double

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

Actual 4^n equals actual 2^(n+n), using only constructed powers and their checked product laws.

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 P Q. (exists pa_b_pc_four_double_four pa_c_pc_four_double_four. ((forall pa_i_pc_four_double_four_repeat. (exists pa_lt_pc_four_double_four_repeat_bound. pa_lt_pc_four_double_four_repeat_bound + S pa_i_pc_four_double_four_repeat = n) -> (((exists pa_h_pc_four_double_four_repeat_decoded. pa_h_pc_four_double_four_repeat_decoded + S (4) = S ((S (pa_i_pc_four_double_four_repeat)) * pa_c_pc_four_double_four)) /\ exists pa_q_pc_four_double_four_repeat_decoded. pa_b_pc_four_double_four = pa_q_pc_four_double_four_repeat_decoded * S ((S (pa_i_pc_four_double_four_repeat)) * pa_c_pc_four_double_four) + (4)))) /\ (exists pa_u_pc_four_double_four_product pa_v_pc_four_double_four_product. ((((exists pa_h_pc_four_double_four_product_start. pa_h_pc_four_double_four_product_start + S (1) = S ((S (0)) * pa_v_pc_four_double_four_product)) /\ exists pa_q_pc_four_double_four_product_start. pa_u_pc_four_double_four_product = pa_q_pc_four_double_four_product_start * S ((S (0)) * pa_v_pc_four_double_four_product) + (1))) /\ ((((exists pa_h_pc_four_double_four_product_terminal. pa_h_pc_four_double_four_product_terminal + S (P) = S ((S (n)) * pa_v_pc_four_double_four_product)) /\ exists pa_q_pc_four_double_four_product_terminal. pa_u_pc_four_double_four_product = pa_q_pc_four_double_four_product_terminal * S ((S (n)) * pa_v_pc_four_double_four_product) + (P))) /\ forall pa_i_pc_four_double_four_product. (exists pa_lt_pc_four_double_four_product_bound. pa_lt_pc_four_double_four_product_bound + S pa_i_pc_four_double_four_product = n) -> exists pa_p_pc_four_double_four_product pa_r_pc_four_double_four_product pa_s_pc_four_double_four_product. ((((exists pa_h_pc_four_double_four_product_factor. pa_h_pc_four_double_four_product_factor + S (pa_p_pc_four_double_four_product) = S ((S (pa_i_pc_four_double_four_product)) * pa_c_pc_four_double_four)) /\ exists pa_q_pc_four_double_four_product_factor. pa_b_pc_four_double_four = pa_q_pc_four_double_four_product_factor * S ((S (pa_i_pc_four_double_four_product)) * pa_c_pc_four_double_four) + (pa_p_pc_four_double_four_product))) /\ ((((exists pa_h_pc_four_double_four_product_partial. pa_h_pc_four_double_four_product_partial + S (pa_r_pc_four_double_four_product) = S ((S (pa_i_pc_four_double_four_product)) * pa_v_pc_four_double_four_product)) /\ exists pa_q_pc_four_double_four_product_partial. pa_u_pc_four_double_four_product = pa_q_pc_four_double_four_product_partial * S ((S (pa_i_pc_four_double_four_product)) * pa_v_pc_four_double_four_product) + (pa_r_pc_four_double_four_product))) /\ ((((exists pa_h_pc_four_double_four_product_successor. pa_h_pc_four_double_four_product_successor + S (pa_s_pc_four_double_four_product) = S ((S (S pa_i_pc_four_double_four_product)) * pa_v_pc_four_double_four_product)) /\ exists pa_q_pc_four_double_four_product_successor. pa_u_pc_four_double_four_product = pa_q_pc_four_double_four_product_successor * S ((S (S pa_i_pc_four_double_four_product)) * pa_v_pc_four_double_four_product) + (pa_s_pc_four_double_four_product))) /\ pa_s_pc_four_double_four_product = pa_r_pc_four_double_four_product * pa_p_pc_four_double_four_product)))))))) -> (exists pa_b_pc_four_double_two pa_c_pc_four_double_two. ((forall pa_i_pc_four_double_two_repeat. (exists pa_lt_pc_four_double_two_repeat_bound. pa_lt_pc_four_double_two_repeat_bound + S pa_i_pc_four_double_two_repeat = n + n) -> (((exists pa_h_pc_four_double_two_repeat_decoded. pa_h_pc_four_double_two_repeat_decoded + S (2) = S ((S (pa_i_pc_four_double_two_repeat)) * pa_c_pc_four_double_two)) /\ exists pa_q_pc_four_double_two_repeat_decoded. pa_b_pc_four_double_two = pa_q_pc_four_double_two_repeat_decoded * S ((S (pa_i_pc_four_double_two_repeat)) * pa_c_pc_four_double_two) + (2)))) /\ (exists pa_u_pc_four_double_two_product pa_v_pc_four_double_two_product. ((((exists pa_h_pc_four_double_two_product_start. pa_h_pc_four_double_two_product_start + S (1) = S ((S (0)) * pa_v_pc_four_double_two_product)) /\ exists pa_q_pc_four_double_two_product_start. pa_u_pc_four_double_two_product = pa_q_pc_four_double_two_product_start * S ((S (0)) * pa_v_pc_four_double_two_product) + (1))) /\ ((((exists pa_h_pc_four_double_two_product_terminal. pa_h_pc_four_double_two_product_terminal + S (Q) = S ((S (n + n)) * pa_v_pc_four_double_two_product)) /\ exists pa_q_pc_four_double_two_product_terminal. pa_u_pc_four_double_two_product = pa_q_pc_four_double_two_product_terminal * S ((S (n + n)) * pa_v_pc_four_double_two_product) + (Q))) /\ forall pa_i_pc_four_double_two_product. (exists pa_lt_pc_four_double_two_product_bound. pa_lt_pc_four_double_two_product_bound + S pa_i_pc_four_double_two_product = n + n) -> exists pa_p_pc_four_double_two_product pa_r_pc_four_double_two_product pa_s_pc_four_double_two_product. ((((exists pa_h_pc_four_double_two_product_factor. pa_h_pc_four_double_two_product_factor + S (pa_p_pc_four_double_two_product) = S ((S (pa_i_pc_four_double_two_product)) * pa_c_pc_four_double_two)) /\ exists pa_q_pc_four_double_two_product_factor. pa_b_pc_four_double_two = pa_q_pc_four_double_two_product_factor * S ((S (pa_i_pc_four_double_two_product)) * pa_c_pc_four_double_two) + (pa_p_pc_four_double_two_product))) /\ ((((exists pa_h_pc_four_double_two_product_partial. pa_h_pc_four_double_two_product_partial + S (pa_r_pc_four_double_two_product) = S ((S (pa_i_pc_four_double_two_product)) * pa_v_pc_four_double_two_product)) /\ exists pa_q_pc_four_double_two_product_partial. pa_u_pc_four_double_two_product = pa_q_pc_four_double_two_product_partial * S ((S (pa_i_pc_four_double_two_product)) * pa_v_pc_four_double_two_product) + (pa_r_pc_four_double_two_product))) /\ ((((exists pa_h_pc_four_double_two_product_successor. pa_h_pc_four_double_two_product_successor + S (pa_s_pc_four_double_two_product) = S ((S (S pa_i_pc_four_double_two_product)) * pa_v_pc_four_double_two_product)) /\ exists pa_q_pc_four_double_two_product_successor. pa_u_pc_four_double_two_product = pa_q_pc_four_double_two_product_successor * S ((S (S pa_i_pc_four_double_two_product)) * pa_v_pc_four_double_two_product) + (pa_s_pc_four_double_two_product))) /\ pa_s_pc_four_double_two_product = pa_r_pc_four_double_two_product * pa_p_pc_four_double_two_product)))))))) -> P = Q

Constructive proof overview

Generated structural guide

Actual 4^n equals actual 2^(n+n), using only constructed powers and their checked product laws.

The unchanged tactic script uses 3 declared prerequisites and contains 30 exact native proof lines.

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

Proof neighborhood

Direct dependencies

pow_exists Stable theorem; checked-use authorized PC0018 pow_four_is_square_of_pow_two pow_add Stable theorem; checked-use authorized

Direct dependents

PC0028 prime_cutoff_exponent_bound

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

30 script commands · 9 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.

Named ingredients (1)
pow four is square of pow two

Long local formulas use this family’s existing definitions. Each new abbreviation was expanded back to the identical native formula, including its free-variable context. The original edition is preserved below.

01Fix variables and assumptionsL1–5

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

  1. L1
    intro n
  2. L2
    intro P
  3. L3
    intro Q
  4. L4
    intro hP
  5. L5
    intro hQ
02Establish hpL6–9

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

  1. L6
    have hp : ∃ v. PowTwo(n,v)Definitions: PowTwo
  2. L7
    specialize pow_exists 2
  3. L8
    specialize pow_exists n
  4. L9
    apply pow_exists
03Separate the logical casesL10–10

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

  1. L10
    cases hp
04Calculate and transport equalitiesL11–11

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

  1. L11
    trans x * x
05Use earlier factsL12–17

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

  1. L12
    specialize pow_four_is_square_of_pow_two n
  2. L13
    specialize pow_four_is_square_of_pow_two x
  3. L14
    specialize pow_four_is_square_of_pow_two P
  4. L15
    apply pow_four_is_square_of_pow_two
  5. L16
    exact hp_witness
  6. L17
    exact hP
06Calculate and transport equalitiesL18–18

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

  1. L18
    symm
07Use earlier factsL19–26

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

  1. L19
    specialize pow_add 2
  2. L20
    specialize pow_add n
  3. L21
    specialize pow_add n
  4. L22
    specialize pow_add (n + n)
  5. L23
    specialize pow_add x
  6. L24
    specialize pow_add x
  7. L25
    specialize pow_add Q
  8. L26
    apply pow_add
08Calculate and transport equalitiesL27–27

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

  1. L27
    refl
09Use earlier factsL28–30

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

  1. L28
    exact hp_witness
  2. L29
    exact hp_witness
  3. L30
    exact hQ

Library-wide reading audit

Original exact command ledger · 30 lines
  1. 0001intro n
  2. 0002intro P
  3. 0003intro Q
  4. 0004intro hP
  5. 0005intro hQ
  6. 0006have hp : exists v. exists pa_b_pc_four_double_middle pa_c_pc_four_double_middle. ((forall pa_i_pc_four_double_middle_repeat. (exists pa_lt_pc_four_double_middle_repeat_bound. pa_lt_pc_four_double_middle_repeat_bound + S pa_i_pc_four_double_middle_repeat = n) -> (((exists pa_h_pc_four_double_middle_repeat_decoded. pa_h_pc_four_double_middle_repeat_decoded + S (2) = S ((S (pa_i_pc_four_double_middle_repeat)) * pa_c_pc_four_double_middle)) /\ exists pa_q_pc_four_double_middle_repeat_decoded. pa_b_pc_four_double_middle = pa_q_pc_four_double_middle_repeat_decoded * S ((S (pa_i_pc_four_double_middle_repeat)) * pa_c_pc_four_double_middle) + (2)))) /\ (exists pa_u_pc_four_double_middle_product pa_v_pc_four_double_middle_product. ((((exists pa_h_pc_four_double_middle_product_start. pa_h_pc_four_double_middle_product_start + S (1) = S ((S (0)) * pa_v_pc_four_double_middle_product)) /\ exists pa_q_pc_four_double_middle_product_start. pa_u_pc_four_double_middle_product = pa_q_pc_four_double_middle_product_start * S ((S (0)) * pa_v_pc_four_double_middle_product) + (1))) /\ ((((exists pa_h_pc_four_double_middle_product_terminal. pa_h_pc_four_double_middle_product_terminal + S (v) = S ((S (n)) * pa_v_pc_four_double_middle_product)) /\ exists pa_q_pc_four_double_middle_product_terminal. pa_u_pc_four_double_middle_product = pa_q_pc_four_double_middle_product_terminal * S ((S (n)) * pa_v_pc_four_double_middle_product) + (v))) /\ forall pa_i_pc_four_double_middle_product. (exists pa_lt_pc_four_double_middle_product_bound. pa_lt_pc_four_double_middle_product_bound + S pa_i_pc_four_double_middle_product = n) -> exists pa_p_pc_four_double_middle_product pa_r_pc_four_double_middle_product pa_s_pc_four_double_middle_product. ((((exists pa_h_pc_four_double_middle_product_factor. pa_h_pc_four_double_middle_product_factor + S (pa_p_pc_four_double_middle_product) = S ((S (pa_i_pc_four_double_middle_product)) * pa_c_pc_four_double_middle)) /\ exists pa_q_pc_four_double_middle_product_factor. pa_b_pc_four_double_middle = pa_q_pc_four_double_middle_product_factor * S ((S (pa_i_pc_four_double_middle_product)) * pa_c_pc_four_double_middle) + (pa_p_pc_four_double_middle_product))) /\ ((((exists pa_h_pc_four_double_middle_product_partial. pa_h_pc_four_double_middle_product_partial + S (pa_r_pc_four_double_middle_product) = S ((S (pa_i_pc_four_double_middle_product)) * pa_v_pc_four_double_middle_product)) /\ exists pa_q_pc_four_double_middle_product_partial. pa_u_pc_four_double_middle_product = pa_q_pc_four_double_middle_product_partial * S ((S (pa_i_pc_four_double_middle_product)) * pa_v_pc_four_double_middle_product) + (pa_r_pc_four_double_middle_product))) /\ ((((exists pa_h_pc_four_double_middle_product_successor. pa_h_pc_four_double_middle_product_successor + S (pa_s_pc_four_double_middle_product) = S ((S (S pa_i_pc_four_double_middle_product)) * pa_v_pc_four_double_middle_product)) /\ exists pa_q_pc_four_double_middle_product_successor. pa_u_pc_four_double_middle_product = pa_q_pc_four_double_middle_product_successor * S ((S (S pa_i_pc_four_double_middle_product)) * pa_v_pc_four_double_middle_product) + (pa_s_pc_four_double_middle_product))) /\ pa_s_pc_four_double_middle_product = pa_r_pc_four_double_middle_product * pa_p_pc_four_double_middle_product)))))))
  7. 0007specialize pow_exists 2
  8. 0008specialize pow_exists n
  9. 0009apply pow_exists
  10. 0010cases hp
  11. 0011trans x * x
  12. 0012specialize pow_four_is_square_of_pow_two n
  13. 0013specialize pow_four_is_square_of_pow_two x
  14. 0014specialize pow_four_is_square_of_pow_two P
  15. 0015apply pow_four_is_square_of_pow_two
  16. 0016exact hp_witness
  17. 0017exact hP
  18. 0018symm
  19. 0019specialize pow_add 2
  20. 0020specialize pow_add n
  21. 0021specialize pow_add n
  22. 0022specialize pow_add (n + n)
  23. 0023specialize pow_add x
  24. 0024specialize pow_add x
  25. 0025specialize pow_add Q
  26. 0026apply pow_add
  27. 0027refl
  28. 0028exact hp_witness
  29. 0029exact hp_witness
  30. 0030exact hQ