SS0010

divisor_signed_balance_negate_intro

Alpha v34 independently verified · alpha_closed; checked-use authorized; not Stable

Opposite arbitrary component balances imply actual canonical SignedNegate, using its constructed inverse and literal functionality.

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

Exact expanded first-order arithmetic statement

forall a b p n. (exists ge_balance_positive_negate_intro_first ge_balance_negative_negate_intro_first. (((((a) = 2 * (ge_balance_positive_negate_intro_first) /\ (ge_balance_negative_negate_intro_first) = 0) \/ exists ge_signed_half_negate_intro_firstdecode. (((a) = 2 * ge_signed_half_negate_intro_firstdecode + 1 /\ (ge_balance_positive_negate_intro_first) = 0) /\ (ge_balance_negative_negate_intro_first) = S ge_signed_half_negate_intro_firstdecode))) /\ ((p) + ge_balance_negative_negate_intro_first = (n) + ge_balance_positive_negate_intro_first))) -> (exists ge_balance_positive_negate_intro_second ge_balance_negative_negate_intro_second. (((((b) = 2 * (ge_balance_positive_negate_intro_second) /\ (ge_balance_negative_negate_intro_second) = 0) \/ exists ge_signed_half_negate_intro_seconddecode. (((b) = 2 * ge_signed_half_negate_intro_seconddecode + 1 /\ (ge_balance_positive_negate_intro_second) = 0) /\ (ge_balance_negative_negate_intro_second) = S ge_signed_half_negate_intro_seconddecode))) /\ ((n) + ge_balance_negative_negate_intro_second = (p) + ge_balance_positive_negate_intro_second))) -> (exists mps_positive_negate_intro_result mps_negative_negate_intro_result. (((((a) = 2 * (mps_positive_negate_intro_result) /\ (mps_negative_negate_intro_result) = 0) \/ exists ge_signed_half_negate_intro_resultsource. (((a) = 2 * ge_signed_half_negate_intro_resultsource + 1 /\ (mps_positive_negate_intro_result) = 0) /\ (mps_negative_negate_intro_result) = S ge_signed_half_negate_intro_resultsource))) /\ ((((b) = 2 * (mps_negative_negate_intro_result) /\ (mps_positive_negate_intro_result) = 0) \/ exists ge_signed_half_negate_intro_resulttarget. (((b) = 2 * ge_signed_half_negate_intro_resulttarget + 1 /\ (mps_negative_negate_intro_result) = 0) /\ (mps_positive_negate_intro_result) = S ge_signed_half_negate_intro_resulttarget)))))

Constructive proof overview

Generated structural guide

Opposite arbitrary component balances imply actual canonical SignedNegate, using its constructed inverse and literal functionality.

The unchanged tactic script uses 3 declared prerequisites and contains 27 exact native proof lines.

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

Proof neighborhood

Direct dependencies

signed_negate_total Alpha theorem; checked-use authorized SS000F divisor_signed_balance_negate signed_balance_functional 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. This exact body belongs to a complete independently kernel-checked constructive proof bundle and has Alpha checked-use authority; it does not imply Stable membership.

Read the argument

Proof checkpoints

27 script commands · 7 reading checkpoints · 2 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.

Named ingredients (1)
divisor signed balance negate

Long local formulas use this family’s existing definitions. Each new abbreviation was expanded back to the identical native formula, including its free-variable context. The original edition is preserved below.

01Fix variables and assumptionsL1–6

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

  1. L1
    intro a
  2. L2
    intro b
  3. L3
    intro p
  4. L4
    intro n
  5. L5
    intro ha
  6. L6
    intro hb
02Establish hnL7–9

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

  1. L7
    have hn : ∃ c. ∃ x. ∃ y. SignedDecode(a,x,y) ∧ SignedDecode(c,y,x)Definitions: SignedDecode
  2. L8
    specialize signed_negate_total (a)
  3. L9
    apply signed_negate_total
03Separate the logical casesL10–10

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

  1. L10
    cases hn
04Establish heqL11–20

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

  1. L11
    have heq : x = b
  2. L12
    specialize signed_balance_functional (n)
  3. L13
    specialize signed_balance_functional (p)
  4. L14
    specialize signed_balance_functional (x)
  5. L15
    specialize signed_balance_functional (b)
  6. L16
    apply signed_balance_functional
  7. L17
    specialize divisor_signed_balance_negate (a)
  8. L18
    specialize divisor_signed_balance_negate (x)
  9. L19
    specialize divisor_signed_balance_negate (p)
  10. L20
    specialize divisor_signed_balance_negate (n)
05Use earlier factsL21–24

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

  1. L21
    apply divisor_signed_balance_negate
  2. L22
    exact ha
  3. L23
    exact hn_witness
  4. L24
    exact hb
06Calculate and transport equalitiesL25–26

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

  1. L25
    rewrite heq at hn_witness
  2. L26
    rewrite heq at hn_witness
07Use earlier factsL27–27

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

  1. L27
    exact hn_witness

Library-wide reading audit

Original exact command ledger · 27 lines
  1. 0001intro a
  2. 0002intro b
  3. 0003intro p
  4. 0004intro n
  5. 0005intro ha
  6. 0006intro hb
  7. 0007have hn : exists c. (exists mps_positive_negate_intro_exists mps_negative_negate_intro_exists. (((((a) = 2 * (mps_positive_negate_intro_exists) /\ (mps_negative_negate_intro_exists) = 0) \/ exists ge_signed_half_negate_intro_existssource. (((a) = 2 * ge_signed_half_negate_intro_existssource + 1 /\ (mps_positive_negate_intro_exists) = 0) /\ (mps_negative_negate_intro_exists) = S ge_signed_half_negate_intro_existssource))) /\ ((((c) = 2 * (mps_negative_negate_intro_exists) /\ (mps_positive_negate_intro_exists) = 0) \/ exists ge_signed_half_negate_intro_existstarget. (((c) = 2 * ge_signed_half_negate_intro_existstarget + 1 /\ (mps_negative_negate_intro_exists) = 0) /\ (mps_positive_negate_intro_exists) = S ge_signed_half_negate_intro_existstarget)))))
  8. 0008specialize signed_negate_total (a)
  9. 0009apply signed_negate_total
  10. 0010cases hn
  11. 0011have heq : x = b
  12. 0012specialize signed_balance_functional (n)
  13. 0013specialize signed_balance_functional (p)
  14. 0014specialize signed_balance_functional (x)
  15. 0015specialize signed_balance_functional (b)
  16. 0016apply signed_balance_functional
  17. 0017specialize divisor_signed_balance_negate (a)
  18. 0018specialize divisor_signed_balance_negate (x)
  19. 0019specialize divisor_signed_balance_negate (p)
  20. 0020specialize divisor_signed_balance_negate (n)
  21. 0021apply divisor_signed_balance_negate
  22. 0022exact ha
  23. 0023exact hn_witness
  24. 0024exact hb
  25. 0025rewrite heq at hn_witness
  26. 0026rewrite heq at hn_witness
  27. 0027exact hn_witness