FS003U

four_square_parity_even_norm_coordinate_sum

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

An actually even represented norm forces the ordinary sum of its coordinates to be even.

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 n a b c d. n * 2 = a * a + b * b + c * c + d * d -> (exists fsps_even_coordinate_sum. (a + b + c + d) = 2 * fsps_even_coordinate_sum)

Constructive proof overview

Generated structural guide

An actually even represented norm forces the ordinary sum of its coordinates to be even.

The unchanged tactic script uses 3 declared prerequisites and contains 19 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

FS003T four_square_parity_norm_mod_two_sum mod_two_preserves_parity Alpha theorem; checked-use authorized mul_comm Stable theorem; checked-use authorized

Direct dependents

FS003X four_square_parity_even_norm_pair_selection

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

19 script commands · 7 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)
four square parity norm mod two sum

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 n
  2. L2
    intro a
  3. L3
    intro b
  4. L4
    intro c
  5. L5
    intro d
  6. L6
    intro hnorm
02Establish htransportL7–11

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply mod two preserves parity.

  1. L7
    have htransport : (Even(a · a + b · b + c · c + d · d) → Even(a + b + c + d)) ∧ (Even(a + b + c + d) → Even(a · a + b · b + c · c + d · d)) ∧ ((Odd(a · a + b · b + c · c + d · d) → Odd(a + b + c + d)) ∧ (Odd(a + b + c + d) → Odd(a · a + b · b + c · c + d · d)))Definitions: EvenOdd
  2. L8
    specialize mod_two_preserves_parity (a * a + b * b + c * c + d * d)
  3. L9
    specialize mod_two_preserves_parity (a + b + c + d)
  4. L10
    apply mod_two_preserves_parity
  5. L11
    apply four_square_parity_norm_mod_two_sum
03Separate the logical casesL12–13

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

  1. L12
    cases htransport
  2. L13
    cases htransport_left
04Use earlier factsL14–14

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

  1. L14
    apply htransport_left_left
05Construct an explicit witnessL15–15

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

  1. L15
    exists n
06Calculate and transport equalitiesL16–17

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

  1. L16
    trans n * 2
  2. L17
    symm
07Use earlier factsL18–19

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

  1. L18
    exact hnorm
  2. L19
    apply mul_comm

Library-wide reading audit

Original exact command ledger · 19 lines
  1. 0001intro n
  2. 0002intro a
  3. 0003intro b
  4. 0004intro c
  5. 0005intro d
  6. 0006intro hnorm
  7. 0007have htransport : ((((exists fsps_even_transport_norm. (a * a + b * b + c * c + d * d) = 2 * fsps_even_transport_norm) -> (exists fsps_even_transport_sum. (a + b + c + d) = 2 * fsps_even_transport_sum)) /\ ((exists fsps_even_transport_back. (a + b + c + d) = 2 * fsps_even_transport_back) -> (exists fsps_even_transport_back_norm. (a * a + b * b + c * c + d * d) = 2 * fsps_even_transport_back_norm))) /\ (((exists fsps_odd_transport_odd_norm. (a * a + b * b + c * c + d * d) = 2 * fsps_odd_transport_odd_norm + 1) -> (exists fsps_odd_transport_odd_sum. (a + b + c + d) = 2 * fsps_odd_transport_odd_sum + 1)) /\ ((exists fsps_odd_transport_odd_back. (a + b + c + d) = 2 * fsps_odd_transport_odd_back + 1) -> (exists fsps_odd_transport_odd_back_norm. (a * a + b * b + c * c + d * d) = 2 * fsps_odd_transport_odd_back_norm + 1))))
  8. 0008specialize mod_two_preserves_parity (a * a + b * b + c * c + d * d)
  9. 0009specialize mod_two_preserves_parity (a + b + c + d)
  10. 0010apply mod_two_preserves_parity
  11. 0011apply four_square_parity_norm_mod_two_sum
  12. 0012cases htransport
  13. 0013cases htransport_left
  14. 0014apply htransport_left_left
  15. 0015exists n
  16. 0016trans n * 2
  17. 0017symm
  18. 0018exact hnorm
  19. 0019apply mul_comm