TS0019

negative_one_opposite_linear_congruence_norm_multiple

Alpha v34 independently verified · alpha_closed; checked-use authorized; not Stable

The opposite-sign coordinate-difference branch also produces a witnessed divisible norm.

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 p r x y. (exists ftcn_factor_root. (r * r + 1) = (p) * ftcn_factor_root) -> (exists ftcn_left_opposite_linear ftcn_right_opposite_linear. (r * x + y) + (p) * ftcn_left_opposite_linear = (0) + (p) * ftcn_right_opposite_linear) -> (exists ftcn_factor_norm. (x * x + y * y) = (p) * ftcn_factor_norm)

Constructive proof overview

Generated structural guide

The opposite-sign coordinate-difference branch also produces a witnessed divisible norm.

The unchanged tactic script uses 2 declared prerequisites and contains 17 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

TS0016 balanced_zero_sum_implies_squared_congruence TS0017 negative_one_congruent_square_norm_multiple

Direct dependents

TS001D affine_collision_absolute_difference_norm_multiple

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

17 script commands · 3 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.

Named ingredients (2)
balanced zero sum implies squared congruencenegative one congruent square norm multiple
01Fix variables and assumptionsL1–6

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

  1. L1
    intro p
  2. L2
    intro r
  3. L3
    intro x
  4. L4
    intro y
  5. L5
    intro hroot
  6. L6
    intro hopposite
02Use earlier factsL7–16

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

  1. L7
    specialize negative_one_congruent_square_norm_multiple p
  2. L8
    specialize negative_one_congruent_square_norm_multiple r
  3. L9
    specialize negative_one_congruent_square_norm_multiple x
  4. L10
    specialize negative_one_congruent_square_norm_multiple y
  5. L11
    apply negative_one_congruent_square_norm_multiple
  6. L12
    exact hroot
  7. L13
    specialize balanced_zero_sum_implies_squared_congruence p
  8. L14
    specialize balanced_zero_sum_implies_squared_congruence (r * x)
  9. L15
    specialize balanced_zero_sum_implies_squared_congruence y
  10. L16
    apply balanced_zero_sum_implies_squared_congruence
03Use earlier factsL17–17

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

  1. L17
    exact hopposite

Library-wide reading audit

Original exact command ledger · 17 lines
  1. 0001intro p
  2. 0002intro r
  3. 0003intro x
  4. 0004intro y
  5. 0005intro hroot
  6. 0006intro hopposite
  7. 0007specialize negative_one_congruent_square_norm_multiple p
  8. 0008specialize negative_one_congruent_square_norm_multiple r
  9. 0009specialize negative_one_congruent_square_norm_multiple x
  10. 0010specialize negative_one_congruent_square_norm_multiple y
  11. 0011apply negative_one_congruent_square_norm_multiple
  12. 0012exact hroot
  13. 0013specialize balanced_zero_sum_implies_squared_congruence p
  14. 0014specialize balanced_zero_sum_implies_squared_congruence (r * x)
  15. 0015specialize balanced_zero_sum_implies_squared_congruence y
  16. 0016apply balanced_zero_sum_implies_squared_congruence
  17. 0017exact hopposite