PC002C

double_successor_le_triple_of_positive

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

For positive A, twice A plus one is at most three times A.

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 A. (exists pc_le_positive_triple_input. pc_le_positive_triple_input + (1) = (A)) -> (exists pc_le_positive_triple_result. pc_le_positive_triple_result + (S (A + A)) = (3 * A))

Constructive proof overview

Generated structural guide

For positive A, twice A plus one is at most three times A.

The unchanged tactic script uses 4 declared prerequisites and contains 15 exact native proof lines.

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

Proof neighborhood

Direct dependencies

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

Direct dependents

PC002D binary_split_eight_bound

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

15 script commands · 3 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 A
  2. L2
    intro hA
02Establish htripleL3–6

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

  1. L3
    have htriple : 3 * A = (A + A) + A
  2. L4
    trans A * 3
  3. L5
    apply mul_comm
  4. L6
    simp [zero_add, add_assoc]
03Establish honeL7–15

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

  1. L7
    have hone : (A + A) + 1 = S (A + A)
  2. L8
    simp
  3. L9
    rewrite <- hone
  4. L10
    rewrite htriple
  5. L11
    specialize add_le_add_left 1
  6. L12
    specialize add_le_add_left A
  7. L13
    specialize add_le_add_left (A + A)
  8. L14
    apply add_le_add_left
  9. L15
    exact hA

Library-wide reading audit

Original exact command ledger · 15 lines
  1. 0001intro A
  2. 0002intro hA
  3. 0003have htriple : 3 * A = (A + A) + A
  4. 0004trans A * 3
  5. 0005apply mul_comm
  6. 0006simp [zero_add, add_assoc]
  7. 0007have hone : (A + A) + 1 = S (A + A)
  8. 0008simp
  9. 0009rewrite <- hone
  10. 0010rewrite htriple
  11. 0011specialize add_le_add_left 1
  12. 0012specialize add_le_add_left A
  13. 0013specialize add_le_add_left (A + A)
  14. 0014apply add_le_add_left
  15. 0015exact hA