TS0014 · theorem body

negative_one_scaled_square_congruent_zero

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

A witnessed root of minus one forces its scaled linear square plus the coordinate square to vanish modulo the modulus.

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

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

PD0008 ModEq
Exact expanded first-order statement
forall p r x. (exists ftcn_factor_root. (r * r + 1) = (p) * ftcn_factor_root) -> (exists ftcn_left_scaled_root ftcn_right_scaled_root. ((r * x) * (r * x) + x * x) + (p) * ftcn_left_scaled_root = (0) + (p) * ftcn_right_scaled_root)

Proof neighborhood

Direct theorem prerequisites

TS0012 multiple_implies_balanced_zero_congruence mod_eq_mul_right · Stable closed TS0013 negative_one_scaled_square_identity mul_zero_left · Stable closed

Direct theorem dependents

TS0017 negative_one_congruent_square_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

22 script commands · 6 reading checkpoints · 2 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)
multiple implies balanced zero congruencenegative one scaled square identity
01Fix variables and assumptionsL1–4

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 hroot
02Establish hbaseL5–9

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

  1. L5
    have hbase : ModEq(p,r · r + 1,0)Definitions: ModEq(p,r · r + 1,0)Original native command in the exact edition
  2. L6
    specialize multiple_implies_balanced_zero_congruence p
  3. L7
    specialize multiple_implies_balanced_zero_congruence (r * r + 1)
  4. L8
    apply multiple_implies_balanced_zero_congruence
  5. L9
    exact hroot
03Establish hscaledL10–19

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply mod eq mul right.

  1. L10
    have hscaled : ModEq(p,(r · r + 1) · (x · x),0 · (x · x))Definitions: ModEq(p,(r · r + 1) · (x · x),0 · (x · x))Original native command in the exact edition
  2. L11
    specialize mod_eq_mul_right p
  3. L12
    specialize mod_eq_mul_right (r * r + 1)
  4. L13
    specialize mod_eq_mul_right 0
  5. L14
    specialize mod_eq_mul_right (x * x)
  6. L15
    apply mod_eq_mul_right
  7. L16
    exact hbase
  8. L17
    specialize negative_one_scaled_square_identity r
  9. L18
    specialize negative_one_scaled_square_identity x
  10. L19
    rewrite negative_one_scaled_square_identity at hscaled
04Use earlier factsL20–20

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

  1. L20
    specialize mul_zero_left (x * x)
05Calculate and transport equalitiesL21–21

Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.

  1. L21
    rewrite mul_zero_left at hscaled
06Use earlier factsL22–22

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

  1. L22
    exact hscaled

Library-wide reading audit

Original defined command ledger · 22 lines
  1. 0001intro p
  2. 0002intro r
  3. 0003intro x
  4. 0004intro hroot
  5. 0005have hbase : ModEq(p,r · r + 1,0)
    Exact native replay linehave hbase : exists ftcn_left_scaled_base ftcn_right_scaled_base. (r * r + 1) + (p) * ftcn_left_scaled_base = (0) + (p) * ftcn_right_scaled_base
  6. 0006specialize multiple_implies_balanced_zero_congruence p
  7. 0007specialize multiple_implies_balanced_zero_congruence (r * r + 1)
  8. 0008apply multiple_implies_balanced_zero_congruence
  9. 0009exact hroot
  10. 0010have hscaled : ModEq(p,(r · r + 1) · (x · x),0 · (x · x))
    Exact native replay linehave hscaled : exists ftcn_left_scaled_step ftcn_right_scaled_step. ((r * r + 1) * (x * x)) + (p) * ftcn_left_scaled_step = (0 * (x * x)) + (p) * ftcn_right_scaled_step
  11. 0011specialize mod_eq_mul_right p
  12. 0012specialize mod_eq_mul_right (r * r + 1)
  13. 0013specialize mod_eq_mul_right 0
  14. 0014specialize mod_eq_mul_right (x * x)
  15. 0015apply mod_eq_mul_right
  16. 0016exact hbase
  17. 0017specialize negative_one_scaled_square_identity r
  18. 0018specialize negative_one_scaled_square_identity x
  19. 0019rewrite negative_one_scaled_square_identity at hscaled
  20. 0020specialize mul_zero_left (x * x)
  21. 0021rewrite mul_zero_left at hscaled
  22. 0022exact hscaled