BX0006

binary_exponent_doubled_power

The relational power at an even exponent is the square of its half power.

Alpha v34 checked-use · first admitted v21 · 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.

G102 was OPEN when this family was first admitted in Alpha v21. It is now CLOSED in Alpha v23: every arbitrary exponent has actual canonical beta-coded digits, a complete modular execution, and exact counted bound operations≤3*BitLen(e)+2.

Exact theorem in conservative defined notation

∀ a. ∀ h. ∀ e. ∀ x. ∀ y. BinaryDoubledPower(a,h,e,x,y) → y = x · x

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

Definition DAG

BinaryDoubledPower(a,h,e,u,z) · 1

Actual proof prerequisites

pow_add · checked external prerequisite
Original expanded first-order statement 
forall a h e x y. (((e = h + h) /\ ((exists ff_b_binary_double_half ff_c_binary_double_half. ((forall ff_i_binary_double_half_repeat. (exists ff_lt_binary_double_half_repeat_bound. ff_lt_binary_double_half_repeat_bound + S ff_i_binary_double_half_repeat = h) -> (((exists ff_h_binary_double_half_repeat_decoded. ff_h_binary_double_half_repeat_decoded + S (a) = S ((S (ff_i_binary_double_half_repeat)) * ff_c_binary_double_half)) /\ exists ff_q_binary_double_half_repeat_decoded. ff_b_binary_double_half = ff_q_binary_double_half_repeat_decoded * S ((S (ff_i_binary_double_half_repeat)) * ff_c_binary_double_half) + (a)))) /\ (exists ff_u_binary_double_half_product ff_v_binary_double_half_product. ((((exists ff_h_binary_double_half_product_start. ff_h_binary_double_half_product_start + S (1) = S ((S (0)) * ff_v_binary_double_half_product)) /\ exists ff_q_binary_double_half_product_start. ff_u_binary_double_half_product = ff_q_binary_double_half_product_start * S ((S (0)) * ff_v_binary_double_half_product) + (1))) /\ ((((exists ff_h_binary_double_half_product_terminal. ff_h_binary_double_half_product_terminal + S (x) = S ((S (h)) * ff_v_binary_double_half_product)) /\ exists ff_q_binary_double_half_product_terminal. ff_u_binary_double_half_product = ff_q_binary_double_half_product_terminal * S ((S (h)) * ff_v_binary_double_half_product) + (x))) /\ forall ff_i_binary_double_half_product. (exists ff_lt_binary_double_half_product_bound. ff_lt_binary_double_half_product_bound + S ff_i_binary_double_half_product = h) -> exists ff_p_binary_double_half_product ff_r_binary_double_half_product ff_s_binary_double_half_product. ((((exists ff_h_binary_double_half_product_factor. ff_h_binary_double_half_product_factor + S (ff_p_binary_double_half_product) = S ((S (ff_i_binary_double_half_product)) * ff_c_binary_double_half)) /\ exists ff_q_binary_double_half_product_factor. ff_b_binary_double_half = ff_q_binary_double_half_product_factor * S ((S (ff_i_binary_double_half_product)) * ff_c_binary_double_half) + (ff_p_binary_double_half_product))) /\ ((((exists ff_h_binary_double_half_product_partial. ff_h_binary_double_half_product_partial + S (ff_r_binary_double_half_product) = S ((S (ff_i_binary_double_half_product)) * ff_v_binary_double_half_product)) /\ exists ff_q_binary_double_half_product_partial. ff_u_binary_double_half_product = ff_q_binary_double_half_product_partial * S ((S (ff_i_binary_double_half_product)) * ff_v_binary_double_half_product) + (ff_r_binary_double_half_product))) /\ ((((exists ff_h_binary_double_half_product_successor. ff_h_binary_double_half_product_successor + S (ff_s_binary_double_half_product) = S ((S (S ff_i_binary_double_half_product)) * ff_v_binary_double_half_product)) /\ exists ff_q_binary_double_half_product_successor. ff_u_binary_double_half_product = ff_q_binary_double_half_product_successor * S ((S (S ff_i_binary_double_half_product)) * ff_v_binary_double_half_product) + (ff_s_binary_double_half_product))) /\ ff_s_binary_double_half_product = ff_r_binary_double_half_product * ff_p_binary_double_half_product)))))))) /\ (exists ff_b_binary_double_full ff_c_binary_double_full. ((forall ff_i_binary_double_full_repeat. (exists ff_lt_binary_double_full_repeat_bound. ff_lt_binary_double_full_repeat_bound + S ff_i_binary_double_full_repeat = e) -> (((exists ff_h_binary_double_full_repeat_decoded. ff_h_binary_double_full_repeat_decoded + S (a) = S ((S (ff_i_binary_double_full_repeat)) * ff_c_binary_double_full)) /\ exists ff_q_binary_double_full_repeat_decoded. ff_b_binary_double_full = ff_q_binary_double_full_repeat_decoded * S ((S (ff_i_binary_double_full_repeat)) * ff_c_binary_double_full) + (a)))) /\ (exists ff_u_binary_double_full_product ff_v_binary_double_full_product. ((((exists ff_h_binary_double_full_product_start. ff_h_binary_double_full_product_start + S (1) = S ((S (0)) * ff_v_binary_double_full_product)) /\ exists ff_q_binary_double_full_product_start. ff_u_binary_double_full_product = ff_q_binary_double_full_product_start * S ((S (0)) * ff_v_binary_double_full_product) + (1))) /\ ((((exists ff_h_binary_double_full_product_terminal. ff_h_binary_double_full_product_terminal + S (y) = S ((S (e)) * ff_v_binary_double_full_product)) /\ exists ff_q_binary_double_full_product_terminal. ff_u_binary_double_full_product = ff_q_binary_double_full_product_terminal * S ((S (e)) * ff_v_binary_double_full_product) + (y))) /\ forall ff_i_binary_double_full_product. (exists ff_lt_binary_double_full_product_bound. ff_lt_binary_double_full_product_bound + S ff_i_binary_double_full_product = e) -> exists ff_p_binary_double_full_product ff_r_binary_double_full_product ff_s_binary_double_full_product. ((((exists ff_h_binary_double_full_product_factor. ff_h_binary_double_full_product_factor + S (ff_p_binary_double_full_product) = S ((S (ff_i_binary_double_full_product)) * ff_c_binary_double_full)) /\ exists ff_q_binary_double_full_product_factor. ff_b_binary_double_full = ff_q_binary_double_full_product_factor * S ((S (ff_i_binary_double_full_product)) * ff_c_binary_double_full) + (ff_p_binary_double_full_product))) /\ ((((exists ff_h_binary_double_full_product_partial. ff_h_binary_double_full_product_partial + S (ff_r_binary_double_full_product) = S ((S (ff_i_binary_double_full_product)) * ff_v_binary_double_full_product)) /\ exists ff_q_binary_double_full_product_partial. ff_u_binary_double_full_product = ff_q_binary_double_full_product_partial * S ((S (ff_i_binary_double_full_product)) * ff_v_binary_double_full_product) + (ff_r_binary_double_full_product))) /\ ((((exists ff_h_binary_double_full_product_successor. ff_h_binary_double_full_product_successor + S (ff_s_binary_double_full_product) = S ((S (S ff_i_binary_double_full_product)) * ff_v_binary_double_full_product)) /\ exists ff_q_binary_double_full_product_successor. ff_u_binary_double_full_product = ff_q_binary_double_full_product_successor * S ((S (S ff_i_binary_double_full_product)) * ff_v_binary_double_full_product) + (ff_s_binary_double_full_product))) /\ ff_s_binary_double_full_product = ff_r_binary_double_full_product * ff_p_binary_double_full_product))))))))))) -> y = x * x

Complete unchanged native tactic proof

All 20 lines are the exact independently kernel-checked original script.

Read the argument

Proof checkpoints

20 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.

01Fix variables and assumptionsL1–6

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

  1. L1
    intro a
  2. L2
    intro h
  3. L3
    intro e
  4. L4
    intro x
  5. L5
    intro y
  6. L6
    intro hdouble
02Separate the logical casesL7–8

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

  1. L7
    cases hdouble
  2. L8
    cases hdouble_right
03Use earlier factsL9–18

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

  1. L9
    specialize pow_add a
  2. L10
    specialize pow_add h
  3. L11
    specialize pow_add h
  4. L12
    specialize pow_add e
  5. L13
    specialize pow_add x
  6. L14
    specialize pow_add x
  7. L15
    specialize pow_add y
  8. L16
    apply pow_add
  9. L17
    exact hdouble_left
  10. L18
    exact hdouble_right_left
04Use earlier factsL19–20

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

  1. L19
    exact hdouble_right_left
  2. L20
    exact hdouble_right_right

Library-wide reading audit

Original defined command ledger · 20 lines
  1. 0001intro a
  2. 0002intro h
  3. 0003intro e
  4. 0004intro x
  5. 0005intro y
  6. 0006intro hdouble
  7. 0007cases hdouble
  8. 0008cases hdouble_right
  9. 0009specialize pow_add a
  10. 0010specialize pow_add h
  11. 0011specialize pow_add h
  12. 0012specialize pow_add e
  13. 0013specialize pow_add x
  14. 0014specialize pow_add x
  15. 0015specialize pow_add y
  16. 0016apply pow_add
  17. 0017exact hdouble_left
  18. 0018exact hdouble_right_left
  19. 0019exact hdouble_right_left
  20. 0020exact hdouble_right_right