FS0041

four_square_parity_represented_double_halving

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

Four-square representability is unconditionally closed under division of represented doubles by two.

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. (exists fsl_a_fsps_double fsl_b_fsps_double fsl_c_fsps_double fsl_d_fsps_double. (n * 2) = fsl_a_fsps_double * fsl_a_fsps_double + fsl_b_fsps_double * fsl_b_fsps_double + fsl_c_fsps_double * fsl_c_fsps_double + fsl_d_fsps_double * fsl_d_fsps_double) -> (exists fsl_a_fsps_double_half fsl_b_fsps_double_half fsl_c_fsps_double_half fsl_d_fsps_double_half. (n) = fsl_a_fsps_double_half * fsl_a_fsps_double_half + fsl_b_fsps_double_half * fsl_b_fsps_double_half + fsl_c_fsps_double_half * fsl_c_fsps_double_half + fsl_d_fsps_double_half * fsl_d_fsps_double_half)

Constructive proof overview

Generated structural guide

Four-square representability is unconditionally closed under division of represented doubles by two.

The unchanged tactic script uses 1 declared prerequisite and contains 13 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

FS0040 four_square_parity_even_multiplier_halving

Direct dependents

FS000S four_square_branch_even_represented_strict_step

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

13 script commands · 3 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 (1)
four square parity even multiplier halving
01Fix variables and assumptionsL1–2

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

  1. L1
    intro n
  2. L2
    intro hdouble
02Separate the logical casesL3–6

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

  1. L3
    cases hdouble
  2. L4
    cases hdouble_witness
  3. L5
    cases hdouble_witness_witness
  4. L6
    cases hdouble_witness_witness_witness
03Use earlier factsL7–13

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

  1. L7
    specialize four_square_parity_even_multiplier_halving n
  2. L8
    specialize four_square_parity_even_multiplier_halving x
  3. L9
    specialize four_square_parity_even_multiplier_halving x1
  4. L10
    specialize four_square_parity_even_multiplier_halving x2
  5. L11
    specialize four_square_parity_even_multiplier_halving x3
  6. L12
    apply four_square_parity_even_multiplier_halving
  7. L13
    exact hdouble_witness_witness_witness_witness

Library-wide reading audit

Original exact command ledger · 13 lines
  1. 0001intro n
  2. 0002intro hdouble
  3. 0003cases hdouble
  4. 0004cases hdouble_witness
  5. 0005cases hdouble_witness_witness
  6. 0006cases hdouble_witness_witness_witness
  7. 0007specialize four_square_parity_even_multiplier_halving n
  8. 0008specialize four_square_parity_even_multiplier_halving x
  9. 0009specialize four_square_parity_even_multiplier_halving x1
  10. 0010specialize four_square_parity_even_multiplier_halving x2
  11. 0011specialize four_square_parity_even_multiplier_halving x3
  12. 0012apply four_square_parity_even_multiplier_halving
  13. 0013exact hdouble_witness_witness_witness_witness