FS0041 · theorem body

four_square_parity_represented_double_halving

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

Statement with defined notation

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)

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

none

In local proof propositions

none
Exact expanded first-order 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)

Proof neighborhood

Direct theorem prerequisites

FS0040 four_square_parity_even_multiplier_halving

Direct theorem dependents

FS000S four_square_branch_even_represented_strict_step

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

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.

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 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 defined 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