PC0023

double_successor_le_triple_above_one

Alpha v34 independently verified · alpha_closed; checked-use authorized; not Stable

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

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.

Exact expanded first-order arithmetic 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))

Constructive proof overview

Generated structural guide

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

The unchanged tactic script uses 5 declared prerequisites and contains 19 exact native proof lines.

Alpha v34 checked-use · first admitted v27 · independently kernel and Lean verified; not Stable

Proof neighborhood

Direct dependencies

euclidean_log_double_successor Alpha theorem; checked-use authorized add_le_add_left Stable theorem; checked-use authorized mul_comm Stable theorem; checked-use authorized zero_add Stable theorem; checked-use authorized add_assoc Stable theorem; checked-use authorized

Direct dependents

PC0026 binary_half_scale_bounds

Formal native tactic body

Dependencies are introduced as named hypotheses before line 1. Local theorem links identify exact declared prerequisites. This exact body belongs to a complete independently kernel-checked constructive proof bundle and has Alpha checked-use authority; it does not imply Stable membership.

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.

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 exact 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