FS005X · theorem body

four_square_signed_partition_balance

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

Positive and negative signed dot-product groups balance when their combined centered square norm vanishes.

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

∀ k. ∀ p. ∀ s. ∀ n. ∀ t. ModEq(k,p,s) → ModEq(k,n + t,0) → ModEq(k,s + t,0) → ModEq(k,p,n)

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

PD0008 ModEq

In local proof propositions

PD0008 ModEq
Exact expanded first-order statement
forall k p s n t. (exists ftcn_left_fssq_partition_positive ftcn_right_fssq_partition_positive. (p) + (k) * ftcn_left_fssq_partition_positive = (s) + (k) * ftcn_right_fssq_partition_positive) -> (exists ftcn_left_fssq_partition_negative ftcn_right_fssq_partition_negative. (n + t) + (k) * ftcn_left_fssq_partition_negative = (0) + (k) * ftcn_right_fssq_partition_negative) -> (exists ftcn_left_fssq_partition_norm ftcn_right_fssq_partition_norm. (s + t) + (k) * ftcn_left_fssq_partition_norm = (0) + (k) * ftcn_right_fssq_partition_norm) -> (exists ftcn_left_fssq_partition_result ftcn_right_fssq_partition_result. (p) + (k) * ftcn_left_fssq_partition_result = (n) + (k) * ftcn_right_fssq_partition_result)

Proof neighborhood

Direct theorem prerequisites

mod_eq_refl · Stable closed mod_eq_add · Stable closed mod_eq_trans · Stable closed FS005L four_square_signed_common_zero_cancel

Direct theorem dependents

FS004O four_square_signed_natural_positive_first_blocks FS005Y four_square_signed_conjugate_positive_blocks FS005Z four_square_signed_conjugate_mixed_blocks FS0060 four_square_signed_natural_negative_first_blocks

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

34 script commands · 5 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 (1)
four square signed common zero cancel
01Fix variables and assumptionsL1–8

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

  1. L1
    intro k
  2. L2
    intro p
  3. L3
    intro s
  4. L4
    intro n
  5. L5
    intro t
  6. L6
    intro hpositive
  7. L7
    intro hnegative
  8. L8
    intro hnorm
02Establish hpaddedL9–18

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

  1. L9
    have hpadded : ModEq(k,p + t,s + t)Definitions: ModEq(k,p + t,s + t)Original native command in the exact edition
  2. L10
    specialize mod_eq_add k
  3. L11
    specialize mod_eq_add p
  4. L12
    specialize mod_eq_add s
  5. L13
    specialize mod_eq_add t
  6. L14
    specialize mod_eq_add t
  7. L15
    apply mod_eq_add
  8. L16
    exact hpositive
  9. L17
    specialize mod_eq_refl k
  10. L18
    specialize mod_eq_refl t
03Use earlier factsL19–19

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

  1. L19
    exact mod_eq_refl
04Establish hzeroL20–29

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

  1. L20
    have hzero : ModEq(k,p + t,0)Definitions: ModEq(k,p + t,0)Original native command in the exact edition
  2. L21
    specialize mod_eq_trans k
  3. L22
    specialize mod_eq_trans (p + t)
  4. L23
    specialize mod_eq_trans (s + t)
  5. L24
    specialize mod_eq_trans 0
  6. L25
    apply mod_eq_trans
  7. L26
    exact hpadded
  8. L27
    exact hnorm
  9. L28
    specialize four_square_signed_common_zero_cancel k
  10. L29
    specialize four_square_signed_common_zero_cancel p
05Use earlier factsL30–34

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

  1. L30
    specialize four_square_signed_common_zero_cancel n
  2. L31
    specialize four_square_signed_common_zero_cancel t
  3. L32
    apply four_square_signed_common_zero_cancel
  4. L33
    exact hzero
  5. L34
    exact hnegative

Library-wide reading audit

Original defined command ledger · 34 lines
  1. 0001intro k
  2. 0002intro p
  3. 0003intro s
  4. 0004intro n
  5. 0005intro t
  6. 0006intro hpositive
  7. 0007intro hnegative
  8. 0008intro hnorm
  9. 0009have hpadded : ModEq(k,p + t,s + t)
    Exact native replay linehave hpadded : exists ftcn_left_fssq_partition_padded ftcn_right_fssq_partition_padded. (p + t) + (k) * ftcn_left_fssq_partition_padded = (s + t) + (k) * ftcn_right_fssq_partition_padded
  10. 0010specialize mod_eq_add k
  11. 0011specialize mod_eq_add p
  12. 0012specialize mod_eq_add s
  13. 0013specialize mod_eq_add t
  14. 0014specialize mod_eq_add t
  15. 0015apply mod_eq_add
  16. 0016exact hpositive
  17. 0017specialize mod_eq_refl k
  18. 0018specialize mod_eq_refl t
  19. 0019exact mod_eq_refl
  20. 0020have hzero : ModEq(k,p + t,0)
    Exact native replay linehave hzero : exists ftcn_left_fssq_partition_zero ftcn_right_fssq_partition_zero. (p + t) + (k) * ftcn_left_fssq_partition_zero = (0) + (k) * ftcn_right_fssq_partition_zero
  21. 0021specialize mod_eq_trans k
  22. 0022specialize mod_eq_trans (p + t)
  23. 0023specialize mod_eq_trans (s + t)
  24. 0024specialize mod_eq_trans 0
  25. 0025apply mod_eq_trans
  26. 0026exact hpadded
  27. 0027exact hnorm
  28. 0028specialize four_square_signed_common_zero_cancel k
  29. 0029specialize four_square_signed_common_zero_cancel p
  30. 0030specialize four_square_signed_common_zero_cancel n
  31. 0031specialize four_square_signed_common_zero_cancel t
  32. 0032apply four_square_signed_common_zero_cancel
  33. 0033exact hzero
  34. 0034exact hnegative