PF000D

fermat_four_counterexample_is_pythagorean

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

A positive fourth-power square counterexample necessarily supplies a Pythagorean triple whose two legs are natural squares.

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 b h. a * a * a * a + b * b * b * b = h * h -> ((a * a) * (a * a) + (b * b) * (b * b) = (h) * (h))

Constructive proof overview

Generated structural guide

A positive fourth-power square counterexample necessarily supplies a Pythagorean triple whose two legs are natural squares.

The unchanged tactic script uses 1 declared prerequisite and contains 11 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

fourth_power_regroup Stable theorem; checked-use authorized

Direct dependents

PF0028 fermat_four_scaled_equation PF002I fermat_four_primitive_square_triangle

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

11 script commands · 3 reading checkpoints · 2 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.

01Fix variables and assumptionsL1–4

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

  1. L1
    intro a
  2. L2
    intro b
  3. L3
    intro h
  4. L4
    intro hequation
02Establish hfirstL5–6

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply fourth power regroup.

  1. L5
    have hfirst : a * a * a * a = (a * a) * (a * a)
  2. L6
    apply fourth_power_regroup
03Establish hsecondL7–11

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply fourth power regroup.

  1. L7
    have hsecond : b * b * b * b = (b * b) * (b * b)
  2. L8
    apply fourth_power_regroup
  3. L9
    rewrite hfirst at hequation
  4. L10
    rewrite hsecond at hequation
  5. L11
    exact hequation

Library-wide reading audit

Original exact command ledger · 11 lines
  1. 0001intro a
  2. 0002intro b
  3. 0003intro h
  4. 0004intro hequation
  5. 0005have hfirst : a * a * a * a = (a * a) * (a * a)
  6. 0006apply fourth_power_regroup
  7. 0007have hsecond : b * b * b * b = (b * b) * (b * b)
  8. 0008apply fourth_power_regroup
  9. 0009rewrite hfirst at hequation
  10. 0010rewrite hsecond at hequation
  11. 0011exact hequation