TS002Y · theorem body

positive_double_at_least_two

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

The double of a positive natural has an explicit witness for being at least two.

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.

Statement with defined notation

∀ h. ¬h = 0 → Lt(1,h + h)

Every purple notation token opens its conservative definition. This reading surface never changes the unchanged intuitionistic kernel or confers checked-use authority.

Definitions used by this theorem

In the theorem statement

PD0002 Lt

In local proof propositions

none
Exact expanded first-order statement
forall h. ~(h = 0) -> exists k. k + 2 = h + h

Proof neighborhood

Direct theorem prerequisites

nonzero_is_succ · Stable closed add_succ_left · Stable closed add_assoc · Stable closed add_comm · Stable closed

Direct theorem dependents

TS002Z even_positive_prime_valuation_has_square_divisor

Definition-aware tactic body

Only propositions whose conservative expansion has been checked for exact first-order equivalence are compacted. Every changed line retains its immutable exact replay command.

Read the argument

Proof checkpoints

11 script commands · 6 reading checkpoints · 1 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 hnonzero
02Use earlier factsL3–3

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

  1. L3
    specialize nonzero_is_succ h
03Establish hsuccessorL4–6

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

  1. L4
    have hsuccessor : exists k. h = S k
  2. L5
    apply nonzero_is_succ
  3. L6
    exact hnonzero
04Separate the logical casesL7–7

Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.

  1. L7
    cases hsuccessor
05Construct an explicit witnessL8–8

Supply the displayed value, then prove that it has the required property.

  1. L8
    exists x + x
06Calculate and transport equalitiesL9–11

Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.

  1. L9
    rewrite hsuccessor_witness
  2. L10
    rewrite hsuccessor_witness
  3. L11
    simp [add_succ_left, add_assoc, add_comm]

Library-wide reading audit

Original defined command ledger · 11 lines
  1. 0001intro h
  2. 0002intro hnonzero
  3. 0003specialize nonzero_is_succ h
  4. 0004have hsuccessor : exists k. h = S k
  5. 0005apply nonzero_is_succ
  6. 0006exact hnonzero
  7. 0007cases hsuccessor
  8. 0008exists x + x
  9. 0009rewrite hsuccessor_witness
  10. 0010rewrite hsuccessor_witness
  11. 0011simp [add_succ_left, add_assoc, add_comm]