FS003S

four_square_parity_triple_mod_two_sum

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

Three coordinate squares preserve their coordinate-sum parity.

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 a b c. (exists fsps_u_triple fsps_v_triple. (a * a + b * b + c * c) + 2 * fsps_u_triple = (a + b + c) + 2 * fsps_v_triple)

Constructive proof overview

Generated structural guide

Three coordinate squares preserve their coordinate-sum parity.

The unchanged tactic script uses 3 declared prerequisites and contains 11 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

Proof neighborhood

Direct dependencies

FS003R four_square_parity_pair_mod_two_sum FS003Q four_square_parity_square_mod_two_self mod_eq_add Stable theorem; checked-use authorized

Direct dependents

FS003T four_square_parity_norm_mod_two_sum

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

11 script commands · 2 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.

Named ingredients (2)
four square parity pair mod two sumfour square parity square mod two self
01Fix variables and assumptionsL1–3

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

  1. L1
    intro a
  2. L2
    intro b
  3. L3
    intro c
02Use earlier factsL4–11

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

  1. L4
    specialize mod_eq_add 2
  2. L5
    specialize mod_eq_add (a * a + b * b)
  3. L6
    specialize mod_eq_add (a + b)
  4. L7
    specialize mod_eq_add (c * c)
  5. L8
    specialize mod_eq_add c
  6. L9
    apply mod_eq_add
  7. L10
    apply four_square_parity_pair_mod_two_sum
  8. L11
    apply four_square_parity_square_mod_two_self

Library-wide reading audit

Original exact command ledger · 11 lines
  1. 0001intro a
  2. 0002intro b
  3. 0003intro c
  4. 0004specialize mod_eq_add 2
  5. 0005specialize mod_eq_add (a * a + b * b)
  6. 0006specialize mod_eq_add (a + b)
  7. 0007specialize mod_eq_add (c * c)
  8. 0008specialize mod_eq_add c
  9. 0009apply mod_eq_add
  10. 0010apply four_square_parity_pair_mod_two_sum
  11. 0011apply four_square_parity_square_mod_two_self