SS0010

divisor_signed_balance_negate_intro

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

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. ∀ b. ∀ p. ∀ n. SignedBalance(a,p,n) → SignedBalance(b,n,p) → ∃ x. ∃ y. SignedDecode(a,x,y) ∧ SignedDecode(b,y,x)

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

Definition DAG

SignedDecode(z,p,n) · 4SignedBalance(z,p,n) · 2

Actual proof prerequisites

signed_negate_total · checked external prerequisitedivisor_signed_balance_negatesigned_balance_functional · checked external prerequisite
Original expanded first-order 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)))))

Complete tactic proof in conservative notation

All 27 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

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.

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

Named ingredients (1)
divisor signed balance negate
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(a,x,y)SignedDecode(c,y,x)Original native command in the exact edition
  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 defined 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 : ∃ c. ∃ x. ∃ y. SignedDecode(a,x,y) ∧ SignedDecode(c,y,x)
  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