FS005A

four_square_signed_opposite_remainder_square_congruent

Alpha v34 independently verified · alpha_closed; checked-use authorized; 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.

Exact expanded first-order arithmetic 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)

Constructive proof overview

Generated structural guide

Oppositely signed modular representatives nevertheless have congruent natural squares.

The unchanged tactic script uses 2 declared prerequisites and contains 16 exact native proof lines.

dependency-curried kernel-checked theorem body; Alpha enrollment and checked-use authority follow separately sealed release evidence; Stable membership remains unchanged

Proof neighborhood

Direct dependencies

multiple_implies_balanced_zero_congruence Alpha theorem; checked-use authorized balanced_zero_sum_implies_squared_congruence Alpha theorem; checked-use authorized

Direct dependents

FS005B four_square_signed_centered_square_congruent

Formal native tactic body

Dependencies are introduced as named hypotheses before line 1. Local theorem links identify exact declared prerequisites. This exact body belongs to a complete independently kernel-checked constructive proof bundle and has Alpha checked-use authority; it does not imply Stable membership.

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.

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 : 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
  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 exact command ledger · 16 lines
  1. 0001intro k
  2. 0002intro a
  3. 0003intro e
  4. 0004intro q
  5. 0005intro hrepresentation
  6. 0006have 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