TS0018 · theorem body

negative_one_linear_congruence_norm_multiple

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

The same-sign coordinate-difference branch 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) → 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_linear ftcn_right_linear. (r * x) + (p) * ftcn_left_linear = (y) + (p) * ftcn_right_linear) -> (exists ftcn_factor_norm. (x * x + y * y) = (p) * ftcn_factor_norm)

Proof neighborhood

Direct theorem prerequisites

TS0015 balanced_linear_congruence_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

18 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 linear congruence 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 hlinear
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_linear_congruence_implies_squared_congruence p
  8. L14
    specialize balanced_linear_congruence_implies_squared_congruence r
  9. L15
    specialize balanced_linear_congruence_implies_squared_congruence x
  10. L16
    specialize balanced_linear_congruence_implies_squared_congruence y
03Use earlier factsL17–18

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

  1. L17
    apply balanced_linear_congruence_implies_squared_congruence
  2. L18
    exact hlinear

Library-wide reading audit

Original defined command ledger · 18 lines
  1. 0001intro p
  2. 0002intro r
  3. 0003intro x
  4. 0004intro y
  5. 0005intro hroot
  6. 0006intro hlinear
  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_linear_congruence_implies_squared_congruence p
  14. 0014specialize balanced_linear_congruence_implies_squared_congruence r
  15. 0015specialize balanced_linear_congruence_implies_squared_congruence x
  16. 0016specialize balanced_linear_congruence_implies_squared_congruence y
  17. 0017apply balanced_linear_congruence_implies_squared_congruence
  18. 0018exact hlinear