TS0033

pairing_double_equals_two_mul

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

The additive double of a natural equals its standard multiplicative parity witness.

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. a + a = 2 * a

Constructive proof overview

Generated structural guide

The additive double of a natural equals its standard multiplicative parity witness.

The unchanged tactic script uses 2 declared prerequisites and contains 7 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

mul_succ_left Stable theorem; checked-use authorized one_mul Stable theorem; checked-use authorized

Direct dependents

TS0034 even_double_sum_reflects_even_tail

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

7 script commands · 6 reading checkpoints · 0 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 assumptionsL1–1

Work with arbitrary variables or the premises of the current implication.

  1. L1
    intro a
02Calculate and transport equalitiesL2–3

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

  1. L2
    symm
  2. L3
    trans 1 * a + a
03Use earlier factsL4–4

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

  1. L4
    apply mul_succ_left
04Calculate and transport equalitiesL5–5

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

  1. L5
    congr
05Use earlier factsL6–6

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

  1. L6
    apply one_mul
06Calculate and transport equalitiesL7–7

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

  1. L7
    refl

Library-wide reading audit

Original exact command ledger · 7 lines
  1. 0001intro a
  2. 0002symm
  3. 0003trans 1 * a + a
  4. 0004apply mul_succ_left
  5. 0005congr
  6. 0006apply one_mul
  7. 0007refl