FS005B · theorem body

four_square_signed_centered_square_congruent

Alpha v34 checked-use · independently kernel and Lean verified; not Stable

Every centered signed remainder has square congruent to the original coordinate, independently of either sign branch.

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

∀ k. ∀ a. ∀ e. Le(e + e,k) ∧ ((∃ x. a = k · x + e) ∨ Dvd(k,a + e)) → ModEq(k,a · a,e · e)

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

PD0001 Le PD0003 Dvd PD0008 ModEq

In local proof propositions

PD0008 ModEq
Exact expanded first-order statement
forall k a e. (((exists fsd_center_bound_fssq_center. fsd_center_bound_fssq_center + (e + e) = k) /\ ((exists fsd_center_lower_fssq_center. a = k * fsd_center_lower_fssq_center + e) \/ (exists fsd_center_upper_fssq_center. a + e = k * fsd_center_upper_fssq_center)))) -> (exists ftcn_left_fssq_center_square ftcn_right_fssq_center_square. (a * a) + (k) * ftcn_left_fssq_center_square = (e * e) + (k) * ftcn_right_fssq_center_square)

Proof neighborhood

Direct theorem prerequisites

FS0059 four_square_signed_lower_remainder_congruent mod_eq_mul · Stable closed FS005A four_square_signed_opposite_remainder_square_congruent

Direct theorem dependents

FS005C four_square_signed_centered_norm_congruent

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

29 script commands · 6 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 (2)
four square signed lower remainder congruentfour square signed opposite remainder square congruent
01Fix variables and assumptionsL1–4

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

  1. L1
    intro k
  2. L2
    intro a
  3. L3
    intro e
  4. L4
    intro hcenter
02Separate the logical casesL5–7

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

  1. L5
    cases hcenter
  2. L6
    cases hcenter_right
  3. L7
    cases hcenter_right_left
03Establish hmodL8–17

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply four square signed lower remainder congruent.

  1. L8
  2. L9
    specialize four_square_signed_lower_remainder_congruent k
  3. L10
    specialize four_square_signed_lower_remainder_congruent a
  4. L11
    specialize four_square_signed_lower_remainder_congruent e
  5. L12
    specialize four_square_signed_lower_remainder_congruent x
  6. L13
    apply four_square_signed_lower_remainder_congruent
  7. L14
    exact hcenter_right_left_witness
  8. L15
    specialize mod_eq_mul k
  9. L16
    specialize mod_eq_mul a
  10. L17
    specialize mod_eq_mul e
04Use earlier factsL18–22

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

  1. L18
    specialize mod_eq_mul a
  2. L19
    specialize mod_eq_mul e
  3. L20
    apply mod_eq_mul
  4. L21
    exact hmod
  5. L22
    exact hmod
05Separate the logical casesL23–23

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

  1. L23
    cases hcenter_right_right
06Use earlier factsL24–29

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

  1. L24
    specialize four_square_signed_opposite_remainder_square_congruent k
  2. L25
    specialize four_square_signed_opposite_remainder_square_congruent a
  3. L26
    specialize four_square_signed_opposite_remainder_square_congruent e
  4. L27
    specialize four_square_signed_opposite_remainder_square_congruent x
  5. L28
    apply four_square_signed_opposite_remainder_square_congruent
  6. L29
    exact hcenter_right_right_witness

Library-wide reading audit

Original defined command ledger · 29 lines
  1. 0001intro k
  2. 0002intro a
  3. 0003intro e
  4. 0004intro hcenter
  5. 0005cases hcenter
  6. 0006cases hcenter_right
  7. 0007cases hcenter_right_left
  8. 0008have hmod : ModEq(k,a,e)
    Exact native replay linehave hmod : exists ftcn_left_fssq_center_lower ftcn_right_fssq_center_lower. (a) + (k) * ftcn_left_fssq_center_lower = (e) + (k) * ftcn_right_fssq_center_lower
  9. 0009specialize four_square_signed_lower_remainder_congruent k
  10. 0010specialize four_square_signed_lower_remainder_congruent a
  11. 0011specialize four_square_signed_lower_remainder_congruent e
  12. 0012specialize four_square_signed_lower_remainder_congruent x
  13. 0013apply four_square_signed_lower_remainder_congruent
  14. 0014exact hcenter_right_left_witness
  15. 0015specialize mod_eq_mul k
  16. 0016specialize mod_eq_mul a
  17. 0017specialize mod_eq_mul e
  18. 0018specialize mod_eq_mul a
  19. 0019specialize mod_eq_mul e
  20. 0020apply mod_eq_mul
  21. 0021exact hmod
  22. 0022exact hmod
  23. 0023cases hcenter_right_right
  24. 0024specialize four_square_signed_opposite_remainder_square_congruent k
  25. 0025specialize four_square_signed_opposite_remainder_square_congruent a
  26. 0026specialize four_square_signed_opposite_remainder_square_congruent e
  27. 0027specialize four_square_signed_opposite_remainder_square_congruent x
  28. 0028apply four_square_signed_opposite_remainder_square_congruent
  29. 0029exact hcenter_right_right_witness