FS000H

four_square_absolute_difference_total

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

Canonical signed totality yields a natural absolute difference for any ordered pair, without importing another candidate module.

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 original first-admission records.

Exact expanded first-order arithmetic statement

forall left right. exists magnitude. (left = right + magnitude \/ right = left + magnitude)

Constructive proof overview

Generated structural guide

Canonical signed totality yields a natural absolute difference for any ordered pair, without importing another candidate module.

The unchanged tactic script uses 2 declared prerequisites and contains 10 exact native proof lines.

dependency-curried kernel-checked theorem body; Alpha enrollment and checked-use authority follow separately sealed release evidence; Stable membership remains unchanged

Historical empty-context replay experiment only; that experiment persisted no certificate and granted no release authority. Current checked use follows separately sealed, independently verified proof bundles; there is no Stable promotion.

Proof neighborhood

Direct dependencies

signed_balance_total Alpha theorem; checked-use authorized FS0002 signed_balance_absolute_exists

Direct dependents

FS000J four_square_two_square_factor_total FS0013 four_square_conjugate_absolute_coordinates_total FS001Q four_square_descent_matching_parity_absolute_even

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

10 script commands · 4 reading checkpoints · 1 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)
signed balance absolute exists
01Fix variables and assumptionsL1–2

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

  1. L1
    intro left
  2. L2
    intro right
02Establish hbalanceL3–4

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

  1. L3
    have hbalance : exists code. (exists sb_pos_fs_difference sb_neg_fs_difference. (((code = 2 * sb_pos_fs_difference /\ sb_neg_fs_difference = 0) \/ exists sd_half_fs_difference. ((code = 2 * sd_half_fs_difference + 1 /\ sb_pos_fs_difference = 0) /\ sb_neg_fs_difference = S sd_half_fs_difference)) /\ (left) + sb_neg_fs_difference = (right) + sb_pos_fs_difference))
  2. L4
    apply signed_balance_total
03Separate the logical casesL5–5

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

  1. L5
    cases hbalance
04Use earlier factsL6–10

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

  1. L6
    specialize signed_balance_absolute_exists x
  2. L7
    specialize signed_balance_absolute_exists left
  3. L8
    specialize signed_balance_absolute_exists right
  4. L9
    apply signed_balance_absolute_exists
  5. L10
    exact hbalance_witness

Library-wide reading audit

Original exact command ledger · 10 lines
  1. 0001intro left
  2. 0002intro right
  3. 0003have hbalance : exists code. (exists sb_pos_fs_difference sb_neg_fs_difference. (((code = 2 * sb_pos_fs_difference /\ sb_neg_fs_difference = 0) \/ exists sd_half_fs_difference. ((code = 2 * sd_half_fs_difference + 1 /\ sb_pos_fs_difference = 0) /\ sb_neg_fs_difference = S sd_half_fs_difference)) /\ (left) + sb_neg_fs_difference = (right) + sb_pos_fs_difference))
  4. 0004apply signed_balance_total
  5. 0005cases hbalance
  6. 0006specialize signed_balance_absolute_exists x
  7. 0007specialize signed_balance_absolute_exists left
  8. 0008specialize signed_balance_absolute_exists right
  9. 0009apply signed_balance_absolute_exists
  10. 0010exact hbalance_witness