FS005J · theorem body

four_square_signed_centered_orientation

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

Each centered remainder constructively supplies its positive congruence or its opposite signed congruence.

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,e) ∨ ModEq(k,a + e,0)

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

none
Exact expanded first-order statement
forall k a e. (((exists fsd_center_bound_fssq_orientation. fsd_center_bound_fssq_orientation + (e + e) = k) /\ ((exists fsd_center_lower_fssq_orientation. a = k * fsd_center_lower_fssq_orientation + e) \/ (exists fsd_center_upper_fssq_orientation. a + e = k * fsd_center_upper_fssq_orientation)))) -> ((exists ftcn_left_fssq_orientation_positive ftcn_right_fssq_orientation_positive. (a) + (k) * ftcn_left_fssq_orientation_positive = (e) + (k) * ftcn_right_fssq_orientation_positive) \/ (exists ftcn_left_fssq_orientation_negative ftcn_right_fssq_orientation_negative. (a + e) + (k) * ftcn_left_fssq_orientation_negative = (0) + (k) * ftcn_right_fssq_orientation_negative))

Proof neighborhood

Direct theorem prerequisites

FS0059 four_square_signed_lower_remainder_congruent multiple_implies_balanced_zero_congruence · Alpha closed

Direct theorem dependents

FS0058 four_square_signed_centered_representation

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

21 script commands · 7 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 signed lower remainder 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 casesL5–8

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
  4. L8
    left
03Use earlier factsL9–14

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

  1. L9
    specialize four_square_signed_lower_remainder_congruent k
  2. L10
    specialize four_square_signed_lower_remainder_congruent a
  3. L11
    specialize four_square_signed_lower_remainder_congruent e
  4. L12
    specialize four_square_signed_lower_remainder_congruent x
  5. L13
    apply four_square_signed_lower_remainder_congruent
  6. L14
    exact hcenter_right_left_witness
04Separate the logical casesL15–16

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

  1. L15
    cases hcenter_right_right
  2. L16
    right
05Use earlier factsL17–19

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

  1. L17
    specialize multiple_implies_balanced_zero_congruence k
  2. L18
    specialize multiple_implies_balanced_zero_congruence (a + e)
  3. L19
    apply multiple_implies_balanced_zero_congruence
06Construct an explicit witnessL20–20

Supply the displayed value, then prove that it has the required property.

  1. L20
    exists x
07Use earlier factsL21–21

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

  1. L21
    exact hcenter_right_right_witness

Library-wide reading audit

Original defined command ledger · 21 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. 0008left
  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. 0015cases hcenter_right_right
  16. 0016right
  17. 0017specialize multiple_implies_balanced_zero_congruence k
  18. 0018specialize multiple_implies_balanced_zero_congruence (a + e)
  19. 0019apply multiple_implies_balanced_zero_congruence
  20. 0020exists x
  21. 0021exact hcenter_right_right_witness