PC0016

binary_power_two_order_reflects_exponent

Weak order between actual powers of two reflects weak order of the exponents.

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

∀ a. ∀ b. ∀ x. ∀ y. PowTwo(a,x) → PowTwo(b,y) → Le(x,y) → Le(a,b)

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

Definition DAG

PowTwo(e,p) · 2Le(a,b) · 3Lt(a,b) · 1

Actual proof prerequisites

le_or_lt · checked external prerequisitebinary_power_two_exponent_strict · checked external prerequisitelt_not_le · checked external prerequisite
Original expanded first-order statement 
forall a b x y. (exists pa_b_pc_power_reflect_left pa_c_pc_power_reflect_left. ((forall pa_i_pc_power_reflect_left_repeat. (exists pa_lt_pc_power_reflect_left_repeat_bound. pa_lt_pc_power_reflect_left_repeat_bound + S pa_i_pc_power_reflect_left_repeat = a) -> (((exists pa_h_pc_power_reflect_left_repeat_decoded. pa_h_pc_power_reflect_left_repeat_decoded + S (2) = S ((S (pa_i_pc_power_reflect_left_repeat)) * pa_c_pc_power_reflect_left)) /\ exists pa_q_pc_power_reflect_left_repeat_decoded. pa_b_pc_power_reflect_left = pa_q_pc_power_reflect_left_repeat_decoded * S ((S (pa_i_pc_power_reflect_left_repeat)) * pa_c_pc_power_reflect_left) + (2)))) /\ (exists pa_u_pc_power_reflect_left_product pa_v_pc_power_reflect_left_product. ((((exists pa_h_pc_power_reflect_left_product_start. pa_h_pc_power_reflect_left_product_start + S (1) = S ((S (0)) * pa_v_pc_power_reflect_left_product)) /\ exists pa_q_pc_power_reflect_left_product_start. pa_u_pc_power_reflect_left_product = pa_q_pc_power_reflect_left_product_start * S ((S (0)) * pa_v_pc_power_reflect_left_product) + (1))) /\ ((((exists pa_h_pc_power_reflect_left_product_terminal. pa_h_pc_power_reflect_left_product_terminal + S (x) = S ((S (a)) * pa_v_pc_power_reflect_left_product)) /\ exists pa_q_pc_power_reflect_left_product_terminal. pa_u_pc_power_reflect_left_product = pa_q_pc_power_reflect_left_product_terminal * S ((S (a)) * pa_v_pc_power_reflect_left_product) + (x))) /\ forall pa_i_pc_power_reflect_left_product. (exists pa_lt_pc_power_reflect_left_product_bound. pa_lt_pc_power_reflect_left_product_bound + S pa_i_pc_power_reflect_left_product = a) -> exists pa_p_pc_power_reflect_left_product pa_r_pc_power_reflect_left_product pa_s_pc_power_reflect_left_product. ((((exists pa_h_pc_power_reflect_left_product_factor. pa_h_pc_power_reflect_left_product_factor + S (pa_p_pc_power_reflect_left_product) = S ((S (pa_i_pc_power_reflect_left_product)) * pa_c_pc_power_reflect_left)) /\ exists pa_q_pc_power_reflect_left_product_factor. pa_b_pc_power_reflect_left = pa_q_pc_power_reflect_left_product_factor * S ((S (pa_i_pc_power_reflect_left_product)) * pa_c_pc_power_reflect_left) + (pa_p_pc_power_reflect_left_product))) /\ ((((exists pa_h_pc_power_reflect_left_product_partial. pa_h_pc_power_reflect_left_product_partial + S (pa_r_pc_power_reflect_left_product) = S ((S (pa_i_pc_power_reflect_left_product)) * pa_v_pc_power_reflect_left_product)) /\ exists pa_q_pc_power_reflect_left_product_partial. pa_u_pc_power_reflect_left_product = pa_q_pc_power_reflect_left_product_partial * S ((S (pa_i_pc_power_reflect_left_product)) * pa_v_pc_power_reflect_left_product) + (pa_r_pc_power_reflect_left_product))) /\ ((((exists pa_h_pc_power_reflect_left_product_successor. pa_h_pc_power_reflect_left_product_successor + S (pa_s_pc_power_reflect_left_product) = S ((S (S pa_i_pc_power_reflect_left_product)) * pa_v_pc_power_reflect_left_product)) /\ exists pa_q_pc_power_reflect_left_product_successor. pa_u_pc_power_reflect_left_product = pa_q_pc_power_reflect_left_product_successor * S ((S (S pa_i_pc_power_reflect_left_product)) * pa_v_pc_power_reflect_left_product) + (pa_s_pc_power_reflect_left_product))) /\ pa_s_pc_power_reflect_left_product = pa_r_pc_power_reflect_left_product * pa_p_pc_power_reflect_left_product)))))))) -> (exists pa_b_pc_power_reflect_right pa_c_pc_power_reflect_right. ((forall pa_i_pc_power_reflect_right_repeat. (exists pa_lt_pc_power_reflect_right_repeat_bound. pa_lt_pc_power_reflect_right_repeat_bound + S pa_i_pc_power_reflect_right_repeat = b) -> (((exists pa_h_pc_power_reflect_right_repeat_decoded. pa_h_pc_power_reflect_right_repeat_decoded + S (2) = S ((S (pa_i_pc_power_reflect_right_repeat)) * pa_c_pc_power_reflect_right)) /\ exists pa_q_pc_power_reflect_right_repeat_decoded. pa_b_pc_power_reflect_right = pa_q_pc_power_reflect_right_repeat_decoded * S ((S (pa_i_pc_power_reflect_right_repeat)) * pa_c_pc_power_reflect_right) + (2)))) /\ (exists pa_u_pc_power_reflect_right_product pa_v_pc_power_reflect_right_product. ((((exists pa_h_pc_power_reflect_right_product_start. pa_h_pc_power_reflect_right_product_start + S (1) = S ((S (0)) * pa_v_pc_power_reflect_right_product)) /\ exists pa_q_pc_power_reflect_right_product_start. pa_u_pc_power_reflect_right_product = pa_q_pc_power_reflect_right_product_start * S ((S (0)) * pa_v_pc_power_reflect_right_product) + (1))) /\ ((((exists pa_h_pc_power_reflect_right_product_terminal. pa_h_pc_power_reflect_right_product_terminal + S (y) = S ((S (b)) * pa_v_pc_power_reflect_right_product)) /\ exists pa_q_pc_power_reflect_right_product_terminal. pa_u_pc_power_reflect_right_product = pa_q_pc_power_reflect_right_product_terminal * S ((S (b)) * pa_v_pc_power_reflect_right_product) + (y))) /\ forall pa_i_pc_power_reflect_right_product. (exists pa_lt_pc_power_reflect_right_product_bound. pa_lt_pc_power_reflect_right_product_bound + S pa_i_pc_power_reflect_right_product = b) -> exists pa_p_pc_power_reflect_right_product pa_r_pc_power_reflect_right_product pa_s_pc_power_reflect_right_product. ((((exists pa_h_pc_power_reflect_right_product_factor. pa_h_pc_power_reflect_right_product_factor + S (pa_p_pc_power_reflect_right_product) = S ((S (pa_i_pc_power_reflect_right_product)) * pa_c_pc_power_reflect_right)) /\ exists pa_q_pc_power_reflect_right_product_factor. pa_b_pc_power_reflect_right = pa_q_pc_power_reflect_right_product_factor * S ((S (pa_i_pc_power_reflect_right_product)) * pa_c_pc_power_reflect_right) + (pa_p_pc_power_reflect_right_product))) /\ ((((exists pa_h_pc_power_reflect_right_product_partial. pa_h_pc_power_reflect_right_product_partial + S (pa_r_pc_power_reflect_right_product) = S ((S (pa_i_pc_power_reflect_right_product)) * pa_v_pc_power_reflect_right_product)) /\ exists pa_q_pc_power_reflect_right_product_partial. pa_u_pc_power_reflect_right_product = pa_q_pc_power_reflect_right_product_partial * S ((S (pa_i_pc_power_reflect_right_product)) * pa_v_pc_power_reflect_right_product) + (pa_r_pc_power_reflect_right_product))) /\ ((((exists pa_h_pc_power_reflect_right_product_successor. pa_h_pc_power_reflect_right_product_successor + S (pa_s_pc_power_reflect_right_product) = S ((S (S pa_i_pc_power_reflect_right_product)) * pa_v_pc_power_reflect_right_product)) /\ exists pa_q_pc_power_reflect_right_product_successor. pa_u_pc_power_reflect_right_product = pa_q_pc_power_reflect_right_product_successor * S ((S (S pa_i_pc_power_reflect_right_product)) * pa_v_pc_power_reflect_right_product) + (pa_s_pc_power_reflect_right_product))) /\ pa_s_pc_power_reflect_right_product = pa_r_pc_power_reflect_right_product * pa_p_pc_power_reflect_right_product)))))))) -> (exists pc_le_power_reflect_values. pc_le_power_reflect_values + (x) = (y)) -> (exists pc_le_power_reflect_result. pc_le_power_reflect_result + (a) = (b))

Complete tactic proof in conservative notation

All 26 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

26 script commands · 7 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 assumptionsL1–7

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

  1. L1
    intro a
  2. L2
    intro b
  3. L3
    intro x
  4. L4
    intro y
  5. L5
    intro hx
  6. L6
    intro hy
  7. L7
    intro hle
02Establish hcL8–11

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply le or lt.

  1. L8
    have hc : Le(a,b) ∨ Lt(b,a)Definitions: Le(a,b)Lt(b,a)Original native command in the exact edition
  2. L9
    specialize le_or_lt a
  3. L10
    specialize le_or_lt b
  4. L11
    apply le_or_lt
03Separate the logical casesL12–12

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

  1. L12
    cases hc
04Use earlier factsL13–13

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

  1. L13
    exact hc_left
05Separate the logical casesL14–14

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

  1. L14
    exfalso
06Use earlier factsL15–24

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

  1. L15
    specialize lt_not_le y
  2. L16
    specialize lt_not_le x
  3. L17
    apply lt_not_le
  4. L18
    specialize binary_power_two_exponent_strict b
  5. L19
    specialize binary_power_two_exponent_strict a
  6. L20
    specialize binary_power_two_exponent_strict y
  7. L21
    specialize binary_power_two_exponent_strict x
  8. L22
    apply binary_power_two_exponent_strict
  9. L23
    exact hc_right
  10. L24
    exact hy
07Use earlier factsL25–26

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

  1. L25
    exact hx
  2. L26
    exact hle

Library-wide reading audit

Original defined command ledger · 26 lines
  1. 0001intro a
  2. 0002intro b
  3. 0003intro x
  4. 0004intro y
  5. 0005intro hx
  6. 0006intro hy
  7. 0007intro hle
  8. 0008have hc : Le(a,b) ∨ Lt(b,a)
  9. 0009specialize le_or_lt a
  10. 0010specialize le_or_lt b
  11. 0011apply le_or_lt
  12. 0012cases hc
  13. 0013exact hc_left
  14. 0014exfalso
  15. 0015specialize lt_not_le y
  16. 0016specialize lt_not_le x
  17. 0017apply lt_not_le
  18. 0018specialize binary_power_two_exponent_strict b
  19. 0019specialize binary_power_two_exponent_strict a
  20. 0020specialize binary_power_two_exponent_strict y
  21. 0021specialize binary_power_two_exponent_strict x
  22. 0022apply binary_power_two_exponent_strict
  23. 0023exact hc_right
  24. 0024exact hy
  25. 0025exact hx
  26. 0026exact hle