TS0019 · theorem body

negative_one_opposite_linear_congruence_norm_multiple

Alpha v34 checked-use · independently kernel and Lean verified; 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.

Statement with defined notation

∀ p. ∀ r. ∀ x. ∀ y. Dvd(p,r · r + 1) → ModEq(p,r · x + y,0) → Dvd(p,x · x + y · y)

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

PD0003 Dvd PD0008 ModEq

In local proof propositions

none
Exact expanded first-order 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)

Proof neighborhood

Direct theorem prerequisites

TS0016 balanced_zero_sum_implies_squared_congruence TS0017 negative_one_congruent_square_norm_multiple

Direct theorem dependents

TS001D affine_collision_absolute_difference_norm_multiple

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

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.

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)
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 defined 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