TS002Y

positive_double_at_least_two

Alpha v34 independently verified · alpha_closed; checked-use authorized; 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.

Exact expanded first-order arithmetic statement

forall h. ~(h = 0) -> exists k. k + 2 = h + h

Constructive proof overview

Generated structural guide

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

The unchanged tactic script uses 4 declared prerequisites and contains 11 exact native proof lines.

dependency-curried kernel-checked theorem body; Alpha enrollment and checked-use authority follow separately sealed release evidence; Stable membership remains unchanged

Proof neighborhood

Direct dependencies

nonzero_is_succ Stable theorem; checked-use authorized add_succ_left Stable theorem; checked-use authorized add_assoc Stable theorem; checked-use authorized add_comm Stable theorem; checked-use authorized

Direct dependents

TS002Z even_positive_prime_valuation_has_square_divisor

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

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.

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 exact 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]