MV0012

mobius_positive_unit_negates_to_negative_unit

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

Alpha v34 checked-use · first admitted v31 · independently kernel and Lean verified; not Stable

Current library: Alpha v34, 4,223 checked-use theorems; Stable remains 432. Historical first admissions, original proof editions, and non-admitted aliases are preserved.

Mobius(n,z) is positive-domain only, independently defined from squarefreeness and actual prime-factor parity. Signed codes 0, 2 and 1 represent zero, +1 and -1. This family proves values and prime-adjunction laws; the separate Möbius-inversion family supplies the complete G007 endpoint.

Exact theorem in conservative defined notation

SignedNegate(2,1)

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

Definition DAG

SignedNegate(a,b) · 1

Actual proof prerequisites

mul_one · checked external prerequisitezero_add · checked external prerequisite
Original expanded first-order 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))))

Complete tactic proof in conservative notation

All 17 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

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.

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

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 defined 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