FS001C · theorem body

four_square_descent_scaled_norm_quotient

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

An exact represented product of a prime multiple and a k-multiple descends through their common square factor.

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

forall p k r u v w x. ~(k = 0) -> (p * k) * (k * r) = ((k * u) * (k * u) + (k * v) * (k * v) + (k * w) * (k * w) + (k * x) * (k * x)) -> p * r = (u * u + v * v + w * w + x * 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

none

In local proof propositions

none
Exact expanded first-order statement
forall p k r u v w x. ~(k = 0) -> (p * k) * (k * r) = ((k * u) * (k * u) + (k * v) * (k * v) + (k * w) * (k * w) + (k * x) * (k * x)) -> p * r = (u * u + v * v + w * w + x * x)

Proof neighborhood

Direct theorem prerequisites

FS0019 four_square_descent_product_reassociate FS001A four_square_descent_square_factor_norm FS001B four_square_descent_square_factor_cancel

Direct theorem dependents

FS001D four_square_descent_quaternion_quotient FS001S four_square_descent_even_multiplier_paired_halving FS0056 four_square_signed_divisible_norm_product_representation

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

20 script commands · 6 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 (3)
four square descent product reassociatefour square descent square factor normfour square descent square factor cancel
01Fix variables and assumptionsL1–9

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

  1. L1
    intro p
  2. L2
    intro k
  3. L3
    intro r
  4. L4
    intro u
  5. L5
    intro v
  6. L6
    intro w
  7. L7
    intro x
  8. L8
    intro hnonzero
  9. L9
    intro hnorm
02Use earlier factsL10–14

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

  1. L10
    specialize four_square_descent_square_factor_cancel k
  2. L11
    specialize four_square_descent_square_factor_cancel (p * r)
  3. L12
    specialize four_square_descent_square_factor_cancel (u * u + v * v + w * w + x * x)
  4. L13
    apply four_square_descent_square_factor_cancel
  5. L14
    exact hnonzero
03Calculate and transport equalitiesL15–16

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

  1. L15
    trans (p * k) * (k * r)
  2. L16
    symm
04Use earlier factsL17–17

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

  1. L17
    apply four_square_descent_product_reassociate
05Calculate and transport equalitiesL18–18

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

  1. L18
    trans (k * u) * (k * u) + (k * v) * (k * v) + (k * w) * (k * w) + (k * x) * (k * x)
06Use earlier factsL19–20

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

  1. L19
    exact hnorm
  2. L20
    apply four_square_descent_square_factor_norm

Library-wide reading audit

Original defined command ledger · 20 lines
  1. 0001intro p
  2. 0002intro k
  3. 0003intro r
  4. 0004intro u
  5. 0005intro v
  6. 0006intro w
  7. 0007intro x
  8. 0008intro hnonzero
  9. 0009intro hnorm
  10. 0010specialize four_square_descent_square_factor_cancel k
  11. 0011specialize four_square_descent_square_factor_cancel (p * r)
  12. 0012specialize four_square_descent_square_factor_cancel (u * u + v * v + w * w + x * x)
  13. 0013apply four_square_descent_square_factor_cancel
  14. 0014exact hnonzero
  15. 0015trans (p * k) * (k * r)
  16. 0016symm
  17. 0017apply four_square_descent_product_reassociate
  18. 0018trans (k * u) * (k * u) + (k * v) * (k * v) + (k * w) * (k * w) + (k * x) * (k * x)
  19. 0019exact hnorm
  20. 0020apply four_square_descent_square_factor_norm