MV0012

mobius_positive_unit_negates_to_negative_unit

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

Canonical code two is positive one and code one is its genuine decoded additive inverse.

Exact expanded first-order arithmetic statement

exists mps_positive_unit_negation mps_negative_unit_negation. (((((2) = 2 * (mps_positive_unit_negation) /\ (mps_negative_unit_negation) = 0) \/ exists ge_signed_half_unit_negationsource. (((2) = 2 * ge_signed_half_unit_negationsource + 1 /\ (mps_positive_unit_negation) = 0) /\ (mps_negative_unit_negation) = S ge_signed_half_unit_negationsource))) /\ ((((1) = 2 * (mps_negative_unit_negation) /\ (mps_positive_unit_negation) = 0) \/ exists ge_signed_half_unit_negationtarget. (((1) = 2 * ge_signed_half_unit_negationtarget + 1 /\ (mps_negative_unit_negation) = 0) /\ (mps_positive_unit_negation) = S ge_signed_half_unit_negationtarget))))

Constructive proof overview

Generated structural guide

Canonical code two is positive one and code one is its genuine decoded additive inverse.

The unchanged tactic script uses 2 declared prerequisites and contains 17 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

mul_one Alpha theorem; checked-use authorized zero_add Alpha theorem; checked-use authorized

Direct dependents

MV0013 alternating_signed_unit_successor_negates

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

17 script commands · 11 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.

01Construct an explicit witnessL1–2

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

  1. L1
    exists 1
  2. L2
    exists 0
02Separate the logical casesL3–5

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

  1. L3
    split
  2. L4
    left
  3. L5
    split
03Calculate and transport equalitiesL6–6

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

  1. L6
    symm
04Use earlier factsL7–7

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

  1. L7
    apply mul_one
05Calculate and transport equalitiesL8–8

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

  1. L8
    refl
06Separate the logical casesL9–9

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

  1. L9
    right
07Construct an explicit witnessL10–10

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

  1. L10
    exists 0
08Separate the logical casesL11–12

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

  1. L11
    split
  2. L12
    split
09Calculate and transport equalitiesL13–14

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

  1. L13
    rewrite PA5
  2. L14
    symm
10Use earlier factsL15–15

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

  1. L15
    apply zero_add
11Calculate and transport equalitiesL16–17

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

  1. L16
    refl
  2. L17
    refl

Library-wide reading audit

Original exact command ledger · 17 lines
  1. 0001exists 1
  2. 0002exists 0
  3. 0003split
  4. 0004left
  5. 0005split
  6. 0006symm
  7. 0007apply mul_one
  8. 0008refl
  9. 0009right
  10. 0010exists 0
  11. 0011split
  12. 0012split
  13. 0013rewrite PA5
  14. 0014symm
  15. 0015apply zero_add
  16. 0016refl
  17. 0017refl