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.

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

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

Exact expanded first-order arithmetic 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))))

Constructive proof overview

Generated structural guide

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

The unchanged tactic script uses 1 declared prerequisite and contains 18 exact native proof lines.

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

Proof neighborhood

Direct dependencies

parity_cases Alpha theorem; checked-use authorized

Direct dependents

MV0009 mobius_value_exists

Formal native tactic body

Dependencies are introduced as named hypotheses before line 1. Local theorem links identify exact declared prerequisites. The literal dependency-closed bundle is checked by original HA and the independently compiled Lean verifier. Public delivery grants no Alpha checked-use authority or Stable membership.

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.

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 exact 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