PC0023

double_successor_le_triple_above_one

For h at least two, twice its successor is at most three times h.

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

∀ h. Lt(1,h) → Le(S h + S h,3 · h)

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

Definition DAG

Le(a,b) · 1Lt(a,b) · 1

Actual proof prerequisites

euclidean_log_double_successor · checked external prerequisiteadd_le_add_left · checked external prerequisitemul_comm · checked external prerequisitezero_add · checked external prerequisiteadd_assoc · checked external prerequisite
Original expanded first-order statement 
forall h. (exists pc_le_triple_input. pc_le_triple_input + (2) = (h)) -> (exists pc_le_triple_result. pc_le_triple_result + (S h + S h) = (3 * h))

Complete tactic proof in conservative notation

All 19 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

19 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 assumptionsL1–2

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

  1. L1
    intro h
  2. L2
    intro hh
02Establish hdoubleL3–8

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply euclidean log double successor.

  1. L3
    have hdouble : S h + S h = (h + h) + 2
  2. L4
    trans S (S (h + h))
  3. L5
    specialize euclidean_log_double_successor h
  4. L6
    apply euclidean_log_double_successor
  5. L7
    symm
  6. L8
    simp
03Establish htripleL9–18

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

  1. L9
    have htriple : 3 * h = (h + h) + h
  2. L10
    trans h * 3
  3. L11
    apply mul_comm
  4. L12
    simp [zero_add, add_assoc]
  5. L13
    rewrite hdouble
  6. L14
    rewrite htriple
  7. L15
    specialize add_le_add_left 2
  8. L16
    specialize add_le_add_left h
  9. L17
    specialize add_le_add_left (h + h)
  10. L18
    apply add_le_add_left
04Use earlier factsL19–19

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

  1. L19
    exact hh

Library-wide reading audit

Original defined command ledger · 19 lines
  1. 0001intro h
  2. 0002intro hh
  3. 0003have hdouble : S h + S h = (h + h) + 2
  4. 0004trans S (S (h + h))
  5. 0005specialize euclidean_log_double_successor h
  6. 0006apply euclidean_log_double_successor
  7. 0007symm
  8. 0008simp
  9. 0009have htriple : 3 * h = (h + h) + h
  10. 0010trans h * 3
  11. 0011apply mul_comm
  12. 0012simp [zero_add, add_assoc]
  13. 0013rewrite hdouble
  14. 0014rewrite htriple
  15. 0015specialize add_le_add_left 2
  16. 0016specialize add_le_add_left h
  17. 0017specialize add_le_add_left (h + h)
  18. 0018apply add_le_add_left
  19. 0019exact hh