FS0048 · theorem body

four_square_ordered_square_congruence_factors

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

Congruent ordered squares force the prime modulus to divide the product of their gap and coordinate sum.

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. ∀ x. ∀ y. ∀ d. y = x + d → ModEq(p,x · x,y · y) → Dvd(p,d · (y + x))

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 x y d. y = x + d -> (exists fsri_left_ordered_squares fsri_right_ordered_squares. (x * x) + (p) * fsri_left_ordered_squares = (y * y) + (p) * fsri_right_ordered_squares) -> (exists fsri_factor_ordered_factor. (d * (y + x)) = (p) * fsri_factor_ordered_factor)

Proof neighborhood

Direct theorem prerequisites

FS0047 four_square_ordered_square_difference_factor add_comm · Stable closed mod_eq_ordered_gap_multiple · Stable closed

Direct theorem dependents

FS0049 four_square_ordered_half_square_injective

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

21 script commands · 5 reading checkpoints · 1 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 ordered square difference factor
01Fix variables and assumptionsL1–6

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

  1. L1
    intro p
  2. L2
    intro x
  3. L3
    intro y
  4. L4
    intro d
  5. L5
    intro horder
  6. L6
    intro hmod
02Establish hidentityL7–16

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply four square ordered square difference factor.

  1. L7
    have hidentity : x * x + d * (y + x) = y * y
  2. L8
    specialize four_square_ordered_square_difference_factor x
  3. L9
    specialize four_square_ordered_square_difference_factor y
  4. L10
    specialize four_square_ordered_square_difference_factor d
  5. L11
    apply four_square_ordered_square_difference_factor
  6. L12
    exact horder
  7. L13
    specialize mod_eq_ordered_gap_multiple p
  8. L14
    specialize mod_eq_ordered_gap_multiple (d * (y + x))
  9. L15
    specialize mod_eq_ordered_gap_multiple (x * x)
  10. L16
    specialize mod_eq_ordered_gap_multiple (y * y)
03Use earlier factsL17–17

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

  1. L17
    apply mod_eq_ordered_gap_multiple
04Calculate and transport equalitiesL18–18

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

  1. L18
    trans x * x + d * (y + x)
05Use earlier factsL19–21

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

  1. L19
    apply add_comm
  2. L20
    exact hidentity
  3. L21
    exact hmod

Library-wide reading audit

Original defined command ledger · 21 lines
  1. 0001intro p
  2. 0002intro x
  3. 0003intro y
  4. 0004intro d
  5. 0005intro horder
  6. 0006intro hmod
  7. 0007have hidentity : x * x + d * (y + x) = y * y
  8. 0008specialize four_square_ordered_square_difference_factor x
  9. 0009specialize four_square_ordered_square_difference_factor y
  10. 0010specialize four_square_ordered_square_difference_factor d
  11. 0011apply four_square_ordered_square_difference_factor
  12. 0012exact horder
  13. 0013specialize mod_eq_ordered_gap_multiple p
  14. 0014specialize mod_eq_ordered_gap_multiple (d * (y + x))
  15. 0015specialize mod_eq_ordered_gap_multiple (x * x)
  16. 0016specialize mod_eq_ordered_gap_multiple (y * y)
  17. 0017apply mod_eq_ordered_gap_multiple
  18. 0018trans x * x + d * (y + x)
  19. 0019apply add_comm
  20. 0020exact hidentity
  21. 0021exact hmod