EL0001

euclidean_log_double_monotone

Constructive natural doubling preserves witnessed non-strict order.

Alpha v34 checked-use · first admitted v23 · 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.

The exact G101 milestone is fully proved, including the stronger checked bound k≤2·BitLen(b), a real beta-coded execution, and its actual terminal gcd. The independent T13 determinant/rank/integer-span substrate is now closed in the separate Alpha-v27 integer-linear-algebra branch. Full T13 proof · Alpha v27

Exact theorem in conservative defined notation

∀ a. ∀ b. Le(a,b) → Le(a + a,b + b)

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

Definition DAG

Le(a,b) · 2

Actual proof prerequisites

add_le_add_right · checked external prerequisiteadd_le_add_left · checked external prerequisitele_trans · checked external prerequisite
Original expanded first-order statement 
forall a b. (exists gap. gap + a = b) -> exists gap. gap + (a + a) = b + b

Complete unchanged native tactic proof

All 21 lines are the exact independently kernel-checked original script.

Read the argument

Proof checkpoints

21 script commands · 4 reading checkpoints · 2 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 assumptionsL1–3

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

  1. L1
    intro a
  2. L2
    intro b
  3. L3
    intro hle
02Establish hfirstL4–9

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

  1. L4
    have hfirst : exists gap. gap + (a + a) = b + a
  2. L5
    specialize add_le_add_right a
  3. L6
    specialize add_le_add_right b
  4. L7
    specialize add_le_add_right a
  5. L8
    apply add_le_add_right
  6. L9
    exact hle
03Establish hsecondL10–19

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

  1. L10
    have hsecond : exists gap. gap + (b + a) = b + b
  2. L11
    specialize add_le_add_left a
  3. L12
    specialize add_le_add_left b
  4. L13
    specialize add_le_add_left b
  5. L14
    apply add_le_add_left
  6. L15
    exact hle
  7. L16
    specialize le_trans (a + a)
  8. L17
    specialize le_trans (b + a)
  9. L18
    specialize le_trans (b + b)
  10. L19
    apply le_trans
04Use earlier factsL20–21

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

  1. L20
    exact hfirst
  2. L21
    exact hsecond

Library-wide reading audit

Original defined command ledger · 21 lines
  1. 0001intro a
  2. 0002intro b
  3. 0003intro hle
  4. 0004have hfirst : exists gap. gap + (a + a) = b + a
  5. 0005specialize add_le_add_right a
  6. 0006specialize add_le_add_right b
  7. 0007specialize add_le_add_right a
  8. 0008apply add_le_add_right
  9. 0009exact hle
  10. 0010have hsecond : exists gap. gap + (b + a) = b + b
  11. 0011specialize add_le_add_left a
  12. 0012specialize add_le_add_left b
  13. 0013specialize add_le_add_left b
  14. 0014apply add_le_add_left
  15. 0015exact hle
  16. 0016specialize le_trans (a + a)
  17. 0017specialize le_trans (b + a)
  18. 0018specialize le_trans (b + b)
  19. 0019apply le_trans
  20. 0020exact hfirst
  21. 0021exact hsecond