MV0001

alternating_signed_unit_exists

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.

Parity constructs an actual canonical code for the alternating unit at every natural exponent.

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

∀ n. ∃ z. AlternatingSignedUnit(n,z)

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

Definition DAG

AlternatingSignedUnit(n,z) · 1

Actual proof prerequisites

parity_cases · checked external prerequisite
Original expanded first-order statement 
forall n. exists z. ((((exists mv_even_half_totaleven. (n) = 2 * mv_even_half_totaleven) /\ ((z) = 2))) \/ (((exists mv_odd_half_totalodd. (n) = 2 * mv_odd_half_totalodd + 1) /\ ((z) = 1))))

Complete tactic proof in conservative notation

All 18 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

18 script commands · 13 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 assumptionsL1–1

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

  1. L1
    intro n
02Establish hL2–4

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

  1. L2
    have h : exists k. n = 2 * k \/ n = 2 * k + 1
  2. L3
    specialize parity_cases (n)
  3. L4
    apply parity_cases
03Separate the logical casesL5–6

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

  1. L5
    cases h
  2. L6
    cases h_witness
04Construct an explicit witnessL7–7

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

  1. L7
    exists 2
05Separate the logical casesL8–9

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

  1. L8
    left
  2. L9
    split
06Construct an explicit witnessL10–10

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

  1. L10
    exists x
07Use earlier factsL11–11

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

  1. L11
    exact h_witness_left
08Calculate and transport equalitiesL12–12

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

  1. L12
    refl
09Construct an explicit witnessL13–13

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

  1. L13
    exists 1
10Separate the logical casesL14–15

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

  1. L14
    right
  2. L15
    split
11Construct an explicit witnessL16–16

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

  1. L16
    exists x
12Use earlier factsL17–17

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

  1. L17
    exact h_witness_right
13Calculate and transport equalitiesL18–18

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

  1. L18
    refl

Library-wide reading audit

Original defined command ledger · 18 lines
  1. 0001intro n
  2. 0002have h : exists k. n = 2 * k \/ n = 2 * k + 1
  3. 0003specialize parity_cases (n)
  4. 0004apply parity_cases
  5. 0005cases h
  6. 0006cases h_witness
  7. 0007exists 2
  8. 0008left
  9. 0009split
  10. 0010exists x
  11. 0011exact h_witness_left
  12. 0012refl
  13. 0013exists 1
  14. 0014right
  15. 0015split
  16. 0016exists x
  17. 0017exact h_witness_right
  18. 0018refl