FS0042 · theorem body

four_square_parity_represented_additive_double_halving

dependency-curried kernel-checked candidate body; not enrolled in Alpha or Stable

For every natural n, an actual four-square representation of n+n constructively produces one of n.

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_add_double fsl_b_fsps_add_double fsl_c_fsps_add_double fsl_d_fsps_add_double. (n + n) = fsl_a_fsps_add_double * fsl_a_fsps_add_double + fsl_b_fsps_add_double * fsl_b_fsps_add_double + fsl_c_fsps_add_double * fsl_c_fsps_add_double + fsl_d_fsps_add_double * fsl_d_fsps_add_double) -> (exists fsl_a_fsps_add_half fsl_b_fsps_add_half fsl_c_fsps_add_half fsl_d_fsps_add_half. (n) = fsl_a_fsps_add_half * fsl_a_fsps_add_half + fsl_b_fsps_add_half * fsl_b_fsps_add_half + fsl_c_fsps_add_half * fsl_c_fsps_add_half + fsl_d_fsps_add_half * fsl_d_fsps_add_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_add_double fsl_b_fsps_add_double fsl_c_fsps_add_double fsl_d_fsps_add_double. (n + n) = fsl_a_fsps_add_double * fsl_a_fsps_add_double + fsl_b_fsps_add_double * fsl_b_fsps_add_double + fsl_c_fsps_add_double * fsl_c_fsps_add_double + fsl_d_fsps_add_double * fsl_d_fsps_add_double) -> (exists fsl_a_fsps_add_half fsl_b_fsps_add_half fsl_c_fsps_add_half fsl_d_fsps_add_half. (n) = fsl_a_fsps_add_half * fsl_a_fsps_add_half + fsl_b_fsps_add_half * fsl_b_fsps_add_half + fsl_c_fsps_add_half * fsl_c_fsps_add_half + fsl_d_fsps_add_half * fsl_d_fsps_add_half)

Proof neighborhood

Direct theorem prerequisites

mul_comm · Stable closed two_mul_eq_add_self · Alpha closed FS0040 four_square_parity_even_multiplier_halving

Direct theorem dependents

none

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

17 script commands · 5 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 factsL7–12

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
04Calculate and transport equalitiesL13–14

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

  1. L13
    trans n + n
  2. L14
    trans 2 * n
05Use earlier factsL15–17

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

  1. L15
    apply mul_comm
  2. L16
    apply two_mul_eq_add_self
  3. L17
    exact hdouble_witness_witness_witness_witness

Library-wide reading audit

Original defined command ledger · 17 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. 0013trans n + n
  14. 0014trans 2 * n
  15. 0015apply mul_comm
  16. 0016apply two_mul_eq_add_self
  17. 0017exact hdouble_witness_witness_witness_witness