MV0003

alternating_signed_unit_zero

Public research checkpoint, not admitted to Alpha or Stable. Alpha v30 remains 3222 checked-use theorems; Stable remains 432. The on-demand Alpha Lean service does not yet expose these checkpoint theorems; their independently checked literal bundles and unchanged sources are available below.

Exponent zero has canonical positive-unit code two, not code one.

Public research checkpoint: original HA and independently compiled Lean verified; not Alpha-enrolled, no Alpha checked-use authority; not Stable

Mobius(n,z) is defined only for positive n. The canonical signed codes are 0 for zero, 2 for +1, and 1 for −1. The function is defined independently of any divisor-sum identity. These values and prime-step lemmas are prerequisites for G007; divisor-sum cancellation and full Möbius inversion remain open in this checkpoint. No signed-table proof is included here.

Exact theorem in conservative defined notation

AlternatingSignedUnit(0,2)

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

Definition DAG

AlternatingSignedUnit(n,z) · 1

Actual proof prerequisites

none
Original expanded first-order statement 
(((exists mv_even_half_zeroeven. (0) = 2 * mv_even_half_zeroeven) /\ ((2) = 2))) \/ (((exists mv_odd_half_zeroodd. (0) = 2 * mv_odd_half_zeroodd + 1) /\ ((2) = 1)))

Complete tactic proof in conservative notation

All 6 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

6 script commands · 5 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.

Definition notation is shown below. Open the paired exact edition for the original native formulas. Source pairing is not a new equivalence certificate.

01Separate the logical casesL1–2

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

  1. L1
    left
  2. L2
    split
02Construct an explicit witnessL3–3

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

  1. L3
    exists 0
03Calculate and transport equalitiesL4–4

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

  1. L4
    symm
04Use earlier factsL5–5

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

  1. L5
    apply PA5
05Calculate and transport equalitiesL6–6

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

  1. L6
    refl

Library-wide reading audit

Original defined command ledger · 6 lines
  1. 0001left
  2. 0002split
  3. 0003exists 0
  4. 0004symm
  5. 0005apply PA5
  6. 0006refl