PC002C

double_successor_le_triple_of_positive

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

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. Lt(0,A) → Lt(A + A,3 · A)

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

Definition DAG

Lt(a,b) · 2

Actual proof prerequisites

mul_comm · checked external prerequisitezero_add · checked external prerequisiteadd_assoc · checked external prerequisiteadd_le_add_left · checked external prerequisite
Original expanded first-order 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))

Complete tactic proof in conservative notation

All 15 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

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.

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 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 defined 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