SS000F

divisor_signed_balance_negate

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.

Canonical signed negation swaps arbitrary natural-component balances, not merely normalized representatives.

Public research checkpoint: original HA and independently compiled Lean verified; not Alpha-enrolled, no Alpha checked-use authority; not Stable

These are genuine signed-table and finite-sum foundations, not full divisor-sum cancellation or Möbius inversion. G007 remains open. The historical MatrixMinorFourCode definition is reused solely as generic nested pairing of four beta parameters; no matrix-specific hypothesis is imported. Equality is equality of represented signed values, not equality of arbitrary component codes.

Exact theorem in conservative defined notation

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

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

Definition DAG

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

Actual proof prerequisites

signed_negate_to_swapped_decode · checked external prerequisite
Original expanded first-order statement 
forall a b p n. (exists ge_balance_positive_negate_balance ge_balance_negative_negate_balance. (((((a) = 2 * (ge_balance_positive_negate_balance) /\ (ge_balance_negative_negate_balance) = 0) \/ exists ge_signed_half_negate_balancedecode. (((a) = 2 * ge_signed_half_negate_balancedecode + 1 /\ (ge_balance_positive_negate_balance) = 0) /\ (ge_balance_negative_negate_balance) = S ge_signed_half_negate_balancedecode))) /\ ((p) + ge_balance_negative_negate_balance = (n) + ge_balance_positive_negate_balance))) -> (exists mps_positive_negate_graph mps_negative_negate_graph. (((((a) = 2 * (mps_positive_negate_graph) /\ (mps_negative_negate_graph) = 0) \/ exists ge_signed_half_negate_graphsource. (((a) = 2 * ge_signed_half_negate_graphsource + 1 /\ (mps_positive_negate_graph) = 0) /\ (mps_negative_negate_graph) = S ge_signed_half_negate_graphsource))) /\ ((((b) = 2 * (mps_negative_negate_graph) /\ (mps_positive_negate_graph) = 0) \/ exists ge_signed_half_negate_graphtarget. (((b) = 2 * ge_signed_half_negate_graphtarget + 1 /\ (mps_negative_negate_graph) = 0) /\ (mps_positive_negate_graph) = S ge_signed_half_negate_graphtarget))))) -> (exists ge_balance_positive_negate_result ge_balance_negative_negate_result. (((((b) = 2 * (ge_balance_positive_negate_result) /\ (ge_balance_negative_negate_result) = 0) \/ exists ge_signed_half_negate_resultdecode. (((b) = 2 * ge_signed_half_negate_resultdecode + 1 /\ (ge_balance_positive_negate_result) = 0) /\ (ge_balance_negative_negate_result) = S ge_signed_half_negate_resultdecode))) /\ ((n) + ge_balance_negative_negate_result = (p) + ge_balance_positive_negate_result)))

Complete tactic proof in conservative notation

All 21 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

21 script commands · 7 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.

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 hbalance
  6. L6
    intro hneg
02Separate the logical casesL7–9

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

  1. L7
    cases hbalance
  2. L8
    cases hbalance_witness
  3. L9
    cases hbalance_witness_witness
03Construct an explicit witnessL10–11

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

  1. L10
    exists x1
  2. L11
    exists x
04Separate the logical casesL12–12

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

  1. L12
    split
05Use earlier factsL13–19

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

  1. L13
    specialize signed_negate_to_swapped_decode (a)
  2. L14
    specialize signed_negate_to_swapped_decode (b)
  3. L15
    specialize signed_negate_to_swapped_decode (x)
  4. L16
    specialize signed_negate_to_swapped_decode (x1)
  5. L17
    apply signed_negate_to_swapped_decode
  6. L18
    exact hbalance_witness_witness_left
  7. L19
    exact hneg
06Calculate and transport equalitiesL20–20

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

  1. L20
    symm
07Use earlier factsL21–21

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

  1. L21
    exact hbalance_witness_witness_right

Library-wide reading audit

Original defined command ledger · 21 lines
  1. 0001intro a
  2. 0002intro b
  3. 0003intro p
  4. 0004intro n
  5. 0005intro hbalance
  6. 0006intro hneg
  7. 0007cases hbalance
  8. 0008cases hbalance_witness
  9. 0009cases hbalance_witness_witness
  10. 0010exists x1
  11. 0011exists x
  12. 0012split
  13. 0013specialize signed_negate_to_swapped_decode (a)
  14. 0014specialize signed_negate_to_swapped_decode (b)
  15. 0015specialize signed_negate_to_swapped_decode (x)
  16. 0016specialize signed_negate_to_swapped_decode (x1)
  17. 0017apply signed_negate_to_swapped_decode
  18. 0018exact hbalance_witness_witness_left
  19. 0019exact hneg
  20. 0020symm
  21. 0021exact hbalance_witness_witness_right