FS001B

four_square_descent_square_factor_cancel

Alpha v34 independently verified · alpha_closed; checked-use authorized; not Stable

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

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.

Exact expanded first-order arithmetic statement

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

Constructive proof overview

Generated structural guide

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

The unchanged tactic script uses 2 declared prerequisites and contains 15 exact native proof lines.

dependency-curried kernel-checked theorem body; Alpha enrollment and checked-use authority follow separately sealed release evidence; Stable membership remains unchanged

Historical empty-context replay experiment only; that experiment persisted no certificate and granted no release authority. Current checked use follows separately sealed, independently verified proof bundles; there is no Stable promotion.

Proof neighborhood

Direct dependencies

FS0018 four_square_descent_nonzero_square mul_left_cancel_nonzero Stable theorem; checked-use authorized

Direct dependents

FS001C four_square_descent_scaled_norm_quotient

Formal native tactic body

Dependencies are introduced as named hypotheses before line 1. Local theorem links identify exact declared prerequisites. This exact body belongs to a complete independently kernel-checked constructive proof bundle and has Alpha checked-use authority; it does not imply Stable membership.

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.

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 exact 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