BL000B

binary_power_two_exponent_monotone

The relational powers of two preserve non-strict exponent ordering.

Alpha v34 checked-use · first admitted v22 · 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.

G101 and G102 were OPEN when these BitLen foundations were first admitted in Alpha v22. Both are now CLOSED in Alpha v23: complete canonical exponent digits and both exact logarithmic execution bounds are proved.

Exact theorem in conservative defined notation

∀ e. ∀ f. ∀ p. ∀ q. Le(e,f) → PowTwo(e,p) → PowTwo(f,q) → Le(p,q)

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

Definition DAG

PowTwo(e,p) · 2Le(a,b) · 2

Actual proof prerequisites

pow_exponent_monotone · checked external prerequisite
Original expanded first-order statement 
forall e f p q. (exists gap. gap + e = f) -> (exists pa_b_bl_power pa_c_bl_power. ((forall pa_i_bl_power_repeat. (exists pa_lt_bl_power_repeat_bound. pa_lt_bl_power_repeat_bound + S pa_i_bl_power_repeat = e) -> (((exists pa_h_bl_power_repeat_decoded. pa_h_bl_power_repeat_decoded + S (2) = S ((S (pa_i_bl_power_repeat)) * pa_c_bl_power)) /\ exists pa_q_bl_power_repeat_decoded. pa_b_bl_power = pa_q_bl_power_repeat_decoded * S ((S (pa_i_bl_power_repeat)) * pa_c_bl_power) + (2)))) /\ (exists pa_u_bl_power_product pa_v_bl_power_product. ((((exists pa_h_bl_power_product_start. pa_h_bl_power_product_start + S (1) = S ((S (0)) * pa_v_bl_power_product)) /\ exists pa_q_bl_power_product_start. pa_u_bl_power_product = pa_q_bl_power_product_start * S ((S (0)) * pa_v_bl_power_product) + (1))) /\ ((((exists pa_h_bl_power_product_terminal. pa_h_bl_power_product_terminal + S (p) = S ((S (e)) * pa_v_bl_power_product)) /\ exists pa_q_bl_power_product_terminal. pa_u_bl_power_product = pa_q_bl_power_product_terminal * S ((S (e)) * pa_v_bl_power_product) + (p))) /\ forall pa_i_bl_power_product. (exists pa_lt_bl_power_product_bound. pa_lt_bl_power_product_bound + S pa_i_bl_power_product = e) -> exists pa_p_bl_power_product pa_r_bl_power_product pa_s_bl_power_product. ((((exists pa_h_bl_power_product_factor. pa_h_bl_power_product_factor + S (pa_p_bl_power_product) = S ((S (pa_i_bl_power_product)) * pa_c_bl_power)) /\ exists pa_q_bl_power_product_factor. pa_b_bl_power = pa_q_bl_power_product_factor * S ((S (pa_i_bl_power_product)) * pa_c_bl_power) + (pa_p_bl_power_product))) /\ ((((exists pa_h_bl_power_product_partial. pa_h_bl_power_product_partial + S (pa_r_bl_power_product) = S ((S (pa_i_bl_power_product)) * pa_v_bl_power_product)) /\ exists pa_q_bl_power_product_partial. pa_u_bl_power_product = pa_q_bl_power_product_partial * S ((S (pa_i_bl_power_product)) * pa_v_bl_power_product) + (pa_r_bl_power_product))) /\ ((((exists pa_h_bl_power_product_successor. pa_h_bl_power_product_successor + S (pa_s_bl_power_product) = S ((S (S pa_i_bl_power_product)) * pa_v_bl_power_product)) /\ exists pa_q_bl_power_product_successor. pa_u_bl_power_product = pa_q_bl_power_product_successor * S ((S (S pa_i_bl_power_product)) * pa_v_bl_power_product) + (pa_s_bl_power_product))) /\ pa_s_bl_power_product = pa_r_bl_power_product * pa_p_bl_power_product)))))))) -> (exists pa_b_bl_later_power pa_c_bl_later_power. ((forall pa_i_bl_later_power_repeat. (exists pa_lt_bl_later_power_repeat_bound. pa_lt_bl_later_power_repeat_bound + S pa_i_bl_later_power_repeat = f) -> (((exists pa_h_bl_later_power_repeat_decoded. pa_h_bl_later_power_repeat_decoded + S (2) = S ((S (pa_i_bl_later_power_repeat)) * pa_c_bl_later_power)) /\ exists pa_q_bl_later_power_repeat_decoded. pa_b_bl_later_power = pa_q_bl_later_power_repeat_decoded * S ((S (pa_i_bl_later_power_repeat)) * pa_c_bl_later_power) + (2)))) /\ (exists pa_u_bl_later_power_product pa_v_bl_later_power_product. ((((exists pa_h_bl_later_power_product_start. pa_h_bl_later_power_product_start + S (1) = S ((S (0)) * pa_v_bl_later_power_product)) /\ exists pa_q_bl_later_power_product_start. pa_u_bl_later_power_product = pa_q_bl_later_power_product_start * S ((S (0)) * pa_v_bl_later_power_product) + (1))) /\ ((((exists pa_h_bl_later_power_product_terminal. pa_h_bl_later_power_product_terminal + S (q) = S ((S (f)) * pa_v_bl_later_power_product)) /\ exists pa_q_bl_later_power_product_terminal. pa_u_bl_later_power_product = pa_q_bl_later_power_product_terminal * S ((S (f)) * pa_v_bl_later_power_product) + (q))) /\ forall pa_i_bl_later_power_product. (exists pa_lt_bl_later_power_product_bound. pa_lt_bl_later_power_product_bound + S pa_i_bl_later_power_product = f) -> exists pa_p_bl_later_power_product pa_r_bl_later_power_product pa_s_bl_later_power_product. ((((exists pa_h_bl_later_power_product_factor. pa_h_bl_later_power_product_factor + S (pa_p_bl_later_power_product) = S ((S (pa_i_bl_later_power_product)) * pa_c_bl_later_power)) /\ exists pa_q_bl_later_power_product_factor. pa_b_bl_later_power = pa_q_bl_later_power_product_factor * S ((S (pa_i_bl_later_power_product)) * pa_c_bl_later_power) + (pa_p_bl_later_power_product))) /\ ((((exists pa_h_bl_later_power_product_partial. pa_h_bl_later_power_product_partial + S (pa_r_bl_later_power_product) = S ((S (pa_i_bl_later_power_product)) * pa_v_bl_later_power_product)) /\ exists pa_q_bl_later_power_product_partial. pa_u_bl_later_power_product = pa_q_bl_later_power_product_partial * S ((S (pa_i_bl_later_power_product)) * pa_v_bl_later_power_product) + (pa_r_bl_later_power_product))) /\ ((((exists pa_h_bl_later_power_product_successor. pa_h_bl_later_power_product_successor + S (pa_s_bl_later_power_product) = S ((S (S pa_i_bl_later_power_product)) * pa_v_bl_later_power_product)) /\ exists pa_q_bl_later_power_product_successor. pa_u_bl_later_power_product = pa_q_bl_later_power_product_successor * S ((S (S pa_i_bl_later_power_product)) * pa_v_bl_later_power_product) + (pa_s_bl_later_power_product))) /\ pa_s_bl_later_power_product = pa_r_bl_later_power_product * pa_p_bl_later_power_product)))))))) -> exists gap. gap + p = q

Complete unchanged native tactic proof

All 18 lines are the exact independently kernel-checked original script.

Read the argument

Proof checkpoints

18 script commands · 5 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 assumptionsL1–7

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

  1. L1
    intro e
  2. L2
    intro f
  3. L3
    intro p
  4. L4
    intro q
  5. L5
    intro horder
  6. L6
    intro hp
  7. L7
    intro hq
02Use earlier factsL8–13

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

  1. L8
    specialize pow_exponent_monotone 2
  2. L9
    specialize pow_exponent_monotone e
  3. L10
    specialize pow_exponent_monotone f
  4. L11
    specialize pow_exponent_monotone p
  5. L12
    specialize pow_exponent_monotone q
  6. L13
    apply pow_exponent_monotone
03Construct an explicit witnessL14–14

Supply the displayed value, then prove that it has the required property.

  1. L14
    exists 1
04Calculate and transport equalitiesL15–15

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

  1. L15
    simp
05Use earlier factsL16–18

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

  1. L16
    exact horder
  2. L17
    exact hp
  3. L18
    exact hq

Library-wide reading audit

Original defined command ledger · 18 lines
  1. 0001intro e
  2. 0002intro f
  3. 0003intro p
  4. 0004intro q
  5. 0005intro horder
  6. 0006intro hp
  7. 0007intro hq
  8. 0008specialize pow_exponent_monotone 2
  9. 0009specialize pow_exponent_monotone e
  10. 0010specialize pow_exponent_monotone f
  11. 0011specialize pow_exponent_monotone p
  12. 0012specialize pow_exponent_monotone q
  13. 0013apply pow_exponent_monotone
  14. 0014exists 1
  15. 0015simp
  16. 0016exact horder
  17. 0017exact hp
  18. 0018exact hq