SS0011

divisor_signed_negate_fixed_zero

A canonical signed integer equal to its own additive inverse is zero; no characteristic-zero claim is assumed without proof.

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.

These are actual signed-table and finite-sum foundations. Equality compares represented signed values, not arbitrary encodings. MatrixMinorFourCode is reused solely as generic nested pairing, without a matrix hypothesis. Full finite signed G007 is established separately in the Möbius-inversion family.

Exact theorem in conservative defined notation

∀ a. (∃ x. ∃ y. SignedDecode(a,x,y) ∧ SignedDecode(a,y,x)) → a = 0

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

Definition DAG

SignedDecode(z,p,n) · 2SignedBalance(z,p,n) · 1

Actual proof prerequisites

signed_decode_functional · checked external prerequisitesigned_balance_zero_iff · checked external prerequisiteadd_comm · checked external prerequisite
Original expanded first-order statement 
forall a. (exists mps_positive_fixed_negation mps_negative_fixed_negation. (((((a) = 2 * (mps_positive_fixed_negation) /\ (mps_negative_fixed_negation) = 0) \/ exists ge_signed_half_fixed_negationsource. (((a) = 2 * ge_signed_half_fixed_negationsource + 1 /\ (mps_positive_fixed_negation) = 0) /\ (mps_negative_fixed_negation) = S ge_signed_half_fixed_negationsource))) /\ ((((a) = 2 * (mps_negative_fixed_negation) /\ (mps_positive_fixed_negation) = 0) \/ exists ge_signed_half_fixed_negationtarget. (((a) = 2 * ge_signed_half_fixed_negationtarget + 1 /\ (mps_negative_fixed_negation) = 0) /\ (mps_positive_fixed_negation) = S ge_signed_half_fixed_negationtarget))))) -> a = 0

Complete tactic proof in conservative notation

All 30 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

30 script commands · 11 reading checkpoints · 3 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 assumptionsL1–2

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

  1. L1
    intro a
  2. L2
    intro hneg
02Separate the logical casesL3–5

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

  1. L3
    cases hneg
  2. L4
    cases hneg_witness
  3. L5
    cases hneg_witness_witness
03Establish heqL6–14

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply signed decode functional.

  1. L6
    have heq : x = x1 /\ x1 = x
  2. L7
    specialize signed_decode_functional (a)
  3. L8
    specialize signed_decode_functional (x)
  4. L9
    specialize signed_decode_functional (x1)
  5. L10
    specialize signed_decode_functional (x1)
  6. L11
    specialize signed_decode_functional (x)
  7. L12
    apply signed_decode_functional
  8. L13
    exact hneg_witness_witness_left
  9. L14
    exact hneg_witness_witness_right
04Separate the logical casesL15–15

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

  1. L15
    cases heq
05Establish hbL16–16

Establish this local claim before using it. It is not an additional assumption.

  1. L16
    have hb : SignedBalance(a,x,x1)Definitions: SignedBalance(a,x,x1)Original native command in the exact edition
06Construct an explicit witnessL17–18

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

  1. L17
    exists x
  2. L18
    exists x1
07Separate the logical casesL19–19

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

  1. L19
    split
08Use earlier factsL20–21

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

  1. L20
    exact hneg_witness_witness_left
  2. L21
    apply add_comm
09Establish hzL22–27

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply signed balance zero iff.

  1. L22
    have hz : (a = 0 -> x = x1) /\ (x = x1 -> a = 0)
  2. L23
    specialize signed_balance_zero_iff (a)
  3. L24
    specialize signed_balance_zero_iff (x)
  4. L25
    specialize signed_balance_zero_iff (x1)
  5. L26
    apply signed_balance_zero_iff
  6. L27
    exact hb
10Separate the logical casesL28–28

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

  1. L28
    cases hz
11Use earlier factsL29–30

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

  1. L29
    apply hz_right
  2. L30
    exact heq_left

Library-wide reading audit

Original defined command ledger · 30 lines
  1. 0001intro a
  2. 0002intro hneg
  3. 0003cases hneg
  4. 0004cases hneg_witness
  5. 0005cases hneg_witness_witness
  6. 0006have heq : x = x1 /\ x1 = x
  7. 0007specialize signed_decode_functional (a)
  8. 0008specialize signed_decode_functional (x)
  9. 0009specialize signed_decode_functional (x1)
  10. 0010specialize signed_decode_functional (x1)
  11. 0011specialize signed_decode_functional (x)
  12. 0012apply signed_decode_functional
  13. 0013exact hneg_witness_witness_left
  14. 0014exact hneg_witness_witness_right
  15. 0015cases heq
  16. 0016have hb : SignedBalance(a,x,x1)
  17. 0017exists x
  18. 0018exists x1
  19. 0019split
  20. 0020exact hneg_witness_witness_left
  21. 0021apply add_comm
  22. 0022have hz : (a = 0 -> x = x1) /\ (x = x1 -> a = 0)
  23. 0023specialize signed_balance_zero_iff (a)
  24. 0024specialize signed_balance_zero_iff (x)
  25. 0025specialize signed_balance_zero_iff (x1)
  26. 0026apply signed_balance_zero_iff
  27. 0027exact hb
  28. 0028cases hz
  29. 0029apply hz_right
  30. 0030exact heq_left