FS003U · theorem body

four_square_parity_even_norm_coordinate_sum

Alpha v34 checked-use · independently kernel and Lean verified; 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.

Statement with defined notation

∀ n. ∀ a. ∀ b. ∀ c. ∀ d. n · 2 = a · a + b · b + c · c + d · d → Even(a + b + c + d)

Every purple notation token opens its conservative definition. This reading surface never changes the unchanged intuitionistic kernel or confers checked-use authority.

Definitions used by this theorem

In the theorem statement

PD0009 Even

In local proof propositions

PD0009 Even PD0010 Odd
Exact expanded first-order 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)

Proof neighborhood

Direct theorem prerequisites

FS003T four_square_parity_norm_mod_two_sum mod_two_preserves_parity · Alpha closed mul_comm · Stable closed

Direct theorem dependents

FS003X four_square_parity_even_norm_pair_selection

Definition-aware tactic body

Only propositions whose conservative expansion has been checked for exact first-order equivalence are compacted. Every changed line retains its immutable exact replay command.

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.

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)
four square parity norm mod two sum
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: Even(a · a + b · b + c · c + d · d)Even(a + b + c + d)Odd(a · a + b · b + c · c + d · d)Odd(a + b + c + d)Original native command in the exact edition
  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 defined 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 : (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)))
    Exact native replay linehave 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