SS0011

divisor_signed_negate_fixed_zero

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

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

Exact expanded first-order arithmetic 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

Constructive proof overview

Generated structural guide

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

The unchanged tactic script uses 3 declared prerequisites and contains 30 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

signed_decode_functional Alpha theorem; checked-use authorized signed_balance_zero_iff Alpha theorem; checked-use authorized add_comm Alpha theorem; checked-use authorized

Direct dependents

none

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

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.

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 : exists ge_balance_positive_fixed_balance ge_balance_negative_fixed_balance. (((((a) = 2 * (ge_balance_positive_fixed_balance) /\ (ge_balance_negative_fixed_balance) = 0) \/ exists ge_signed_half_fixed_balancedecode. (((a) = 2 * ge_signed_half_fixed_balancedecode + 1 /\ (ge_balance_positive_fixed_balance) = 0) /\ (ge_balance_negative_fixed_balance) = S ge_signed_half_fixed_balancedecode))) /\ ((x) + ge_balance_negative_fixed_balance = (x1) + ge_balance_positive_fixed_balance))
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 exact 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 : exists ge_balance_positive_fixed_balance ge_balance_negative_fixed_balance. (((((a) = 2 * (ge_balance_positive_fixed_balance) /\ (ge_balance_negative_fixed_balance) = 0) \/ exists ge_signed_half_fixed_balancedecode. (((a) = 2 * ge_signed_half_fixed_balancedecode + 1 /\ (ge_balance_positive_fixed_balance) = 0) /\ (ge_balance_negative_fixed_balance) = S ge_signed_half_fixed_balancedecode))) /\ ((x) + ge_balance_negative_fixed_balance = (x1) + ge_balance_positive_fixed_balance))
  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