FS005A · theorem body

four_square_signed_opposite_remainder_square_congruent

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

Oppositely signed modular representatives nevertheless have congruent natural squares.

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. ∀ q. a + e = k · q → 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

PD0008 ModEq

In local proof propositions

PD0008 ModEq
Exact expanded first-order statement
forall k a e q. a + e = k * q -> (exists ftcn_left_fssq_opposite_square ftcn_right_fssq_opposite_square. (a * a) + (k) * ftcn_left_fssq_opposite_square = (e * e) + (k) * ftcn_right_fssq_opposite_square)

Proof neighborhood

Direct theorem prerequisites

multiple_implies_balanced_zero_congruence · Alpha closed balanced_zero_sum_implies_squared_congruence · Alpha closed

Direct theorem dependents

FS005B four_square_signed_centered_square_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

16 script commands · 4 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.

01Fix variables and assumptionsL1–5

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 q
  5. L5
    intro hrepresentation
02Establish hzeroL6–9

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply multiple implies balanced zero congruence.

  1. L6
    have hzero : ModEq(k,a + e,0)Definitions: ModEq(k,a + e,0)Original native command in the exact edition
  2. L7
    specialize multiple_implies_balanced_zero_congruence k
  3. L8
    specialize multiple_implies_balanced_zero_congruence (a + e)
  4. L9
    apply multiple_implies_balanced_zero_congruence
03Construct an explicit witnessL10–10

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

  1. L10
    exists q
04Use earlier factsL11–16

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

  1. L11
    exact hrepresentation
  2. L12
    specialize balanced_zero_sum_implies_squared_congruence k
  3. L13
    specialize balanced_zero_sum_implies_squared_congruence a
  4. L14
    specialize balanced_zero_sum_implies_squared_congruence e
  5. L15
    apply balanced_zero_sum_implies_squared_congruence
  6. L16
    exact hzero

Library-wide reading audit

Original defined command ledger · 16 lines
  1. 0001intro k
  2. 0002intro a
  3. 0003intro e
  4. 0004intro q
  5. 0005intro hrepresentation
  6. 0006have hzero : ModEq(k,a + e,0)
    Exact native replay linehave hzero : exists ftcn_left_fssq_opposite_zero ftcn_right_fssq_opposite_zero. (a + e) + (k) * ftcn_left_fssq_opposite_zero = (0) + (k) * ftcn_right_fssq_opposite_zero
  7. 0007specialize multiple_implies_balanced_zero_congruence k
  8. 0008specialize multiple_implies_balanced_zero_congruence (a + e)
  9. 0009apply multiple_implies_balanced_zero_congruence
  10. 0010exists q
  11. 0011exact hrepresentation
  12. 0012specialize balanced_zero_sum_implies_squared_congruence k
  13. 0013specialize balanced_zero_sum_implies_squared_congruence a
  14. 0014specialize balanced_zero_sum_implies_squared_congruence e
  15. 0015apply balanced_zero_sum_implies_squared_congruence
  16. 0016exact hzero