PF002T

fermat_four_positive_sum_not_square

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

Exact campaign G078: the sum of two positive fourth powers is never a square, for every natural z including zero.

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 x y z. ~(x = 0) -> ~(y = 0) -> ~(x * x * x * x + y * y * y * y = z * z)

Constructive proof overview

Generated structural guide

Exact campaign G078: the sum of two positive fourth powers is never a square, for every natural z including zero.

The unchanged tactic script uses 1 declared prerequisite and contains 17 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

Direct dependents

none

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

17 script commands · 4 reading checkpoints · 1 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 assumptionsL1–6

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

  1. L1
    intro x
  2. L2
    intro y
  3. L3
    intro z
  4. L4
    intro hx
  5. L5
    intro hy
  6. L6
    intro hequation
02Establish hzeroL7–12

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply fermat four square solutions have zero coordinate.

  1. L7
    have hzero : x = 0 \/ y = 0
  2. L8
    specialize fermat_four_square_solutions_have_zero_coordinate (x)
  3. L9
    specialize fermat_four_square_solutions_have_zero_coordinate (y)
  4. L10
    specialize fermat_four_square_solutions_have_zero_coordinate (z)
  5. L11
    apply fermat_four_square_solutions_have_zero_coordinate
  6. L12
    exact hequation
03Separate the logical casesL13–13

Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.

  1. L13
    cases hzero
04Use earlier factsL14–17

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

  1. L14
    apply hx
  2. L15
    exact hzero_left
  3. L16
    apply hy
  4. L17
    exact hzero_right

Library-wide reading audit

Original exact command ledger · 17 lines
  1. 0001intro x
  2. 0002intro y
  3. 0003intro z
  4. 0004intro hx
  5. 0005intro hy
  6. 0006intro hequation
  7. 0007have hzero : x = 0 \/ y = 0
  8. 0008specialize fermat_four_square_solutions_have_zero_coordinate (x)
  9. 0009specialize fermat_four_square_solutions_have_zero_coordinate (y)
  10. 0010specialize fermat_four_square_solutions_have_zero_coordinate (z)
  11. 0011apply fermat_four_square_solutions_have_zero_coordinate
  12. 0012exact hequation
  13. 0013cases hzero
  14. 0014apply hx
  15. 0015exact hzero_left
  16. 0016apply hy
  17. 0017exact hzero_right