FS001B · theorem body

four_square_descent_square_factor_cancel

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

A nonzero natural square factor cancels without subtraction or division axioms.

historical independently replay-verified empty-context experiment; the experiment itself persisted no certificate and granted no release authority; current checked use follows separately sealed proof bundles; no Stable promotion.

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 k s t. ~(k = 0) -> (k * k) * s = (k * k) * t -> s = t

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 k s t. ~(k = 0) -> (k * k) * s = (k * k) * t -> s = t

Proof neighborhood

Direct theorem prerequisites

FS0018 four_square_descent_nonzero_square mul_left_cancel_nonzero · Stable closed

Direct theorem dependents

FS001C four_square_descent_scaled_norm_quotient

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

15 script commands · 4 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 (1)
four square descent nonzero square
01Fix variables and assumptionsL1–5

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

  1. L1
    intro k
  2. L2
    intro s
  3. L3
    intro t
  4. L4
    intro hnonzero
  5. L5
    intro hequality
02Use earlier factsL6–10

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

  1. L6
    specialize mul_left_cancel_nonzero (k * k)
  2. L7
    specialize mul_left_cancel_nonzero s
  3. L8
    specialize mul_left_cancel_nonzero t
  4. L9
    apply mul_left_cancel_nonzero
  5. L10
    specialize four_square_descent_nonzero_square k
03Fix variables and assumptionsL11–11

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

  1. L11
    intro hsquare
04Use earlier factsL12–15

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

  1. L12
    apply four_square_descent_nonzero_square
  2. L13
    exact hnonzero
  3. L14
    exact hsquare
  4. L15
    exact hequality

Library-wide reading audit

Original defined command ledger · 15 lines
  1. 0001intro k
  2. 0002intro s
  3. 0003intro t
  4. 0004intro hnonzero
  5. 0005intro hequality
  6. 0006specialize mul_left_cancel_nonzero (k * k)
  7. 0007specialize mul_left_cancel_nonzero s
  8. 0008specialize mul_left_cancel_nonzero t
  9. 0009apply mul_left_cancel_nonzero
  10. 0010specialize four_square_descent_nonzero_square k
  11. 0011intro hsquare
  12. 0012apply four_square_descent_nonzero_square
  13. 0013exact hnonzero
  14. 0014exact hsquare
  15. 0015exact hequality