PF0025

fermat_four_square_nonzero

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

A positive coordinate has a positive square.

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 a. ~(a = 0) -> ~(a * a = 0)

Constructive proof overview

Generated structural guide

A positive coordinate has a positive square.

The unchanged tactic script uses 1 declared prerequisite and contains 9 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

Proof neighborhood

Direct dependencies

PF0024 fermat_four_product_nonzero

Direct dependents

PF0029 fermat_four_cancel_scaled_equation PF002B fermat_four_root_lt_norm PF002F fermat_four_primitive_normalization PF002K fermat_four_primitive_odd_even_descent PF002P fermat_four_equation_height_nonzero

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

9 script commands · 2 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)
fermat four product nonzero
01Fix variables and assumptionsL1–3

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

  1. L1
    intro a
  2. L2
    intro ha
  3. L3
    intro hzero
02Use earlier factsL4–9

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

  1. L4
    specialize fermat_four_product_nonzero (a)
  2. L5
    specialize fermat_four_product_nonzero (a)
  3. L6
    apply fermat_four_product_nonzero
  4. L7
    exact ha
  5. L8
    exact ha
  6. L9
    exact hzero

Library-wide reading audit

Original exact command ledger · 9 lines
  1. 0001intro a
  2. 0002intro ha
  3. 0003intro hzero
  4. 0004specialize fermat_four_product_nonzero (a)
  5. 0005specialize fermat_four_product_nonzero (a)
  6. 0006apply fermat_four_product_nonzero
  7. 0007exact ha
  8. 0008exact ha
  9. 0009exact hzero