PF002R

fermat_four_solutions_have_zero_coordinate

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

Fermat exponent four has no nontrivial natural solution, without any separate positivity assumptions.

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 * h * h -> (a = 0 \/ b = 0)

Constructive proof overview

Generated structural guide

Fermat exponent four has no nontrivial natural solution, without any separate positivity assumptions.

The unchanged tactic script uses 2 declared prerequisites 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

PF002Q fermat_four_square_solutions_have_zero_coordinate fourth_power_regroup Stable theorem; checked-use authorized

Direct dependents

PF002S fermat_four_complete_classification

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 · 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)
fermat four square solutions have zero coordinate
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
02Use earlier factsL5–8

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

  1. L5
    specialize fermat_four_square_solutions_have_zero_coordinate (a)
  2. L6
    specialize fermat_four_square_solutions_have_zero_coordinate (b)
  3. L7
    specialize fermat_four_square_solutions_have_zero_coordinate (h * h)
  4. L8
    apply fermat_four_square_solutions_have_zero_coordinate
03Calculate and transport equalitiesL9–9

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

  1. L9
    trans h * h * h * h
04Use earlier factsL10–11

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

  1. L10
    exact hequation
  2. L11
    apply fourth_power_regroup

Library-wide reading audit

Original exact command ledger · 11 lines
  1. 0001intro a
  2. 0002intro b
  3. 0003intro h
  4. 0004intro hequation
  5. 0005specialize fermat_four_square_solutions_have_zero_coordinate (a)
  6. 0006specialize fermat_four_square_solutions_have_zero_coordinate (b)
  7. 0007specialize fermat_four_square_solutions_have_zero_coordinate (h * h)
  8. 0008apply fermat_four_square_solutions_have_zero_coordinate
  9. 0009trans h * h * h * h
  10. 0010exact hequation
  11. 0011apply fourth_power_regroup