PF002T · theorem body

fermat_four_positive_sum_not_square

Alpha v34 checked-use · independently kernel and Lean verified; 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.

Statement with defined notation

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

Every purple notation token opens its conservative definition. This reading surface never changes the unchanged intuitionistic kernel or confers checked-use authority.

Definitions used by this theorem

In the theorem statement

none

In local proof propositions

none
Exact expanded first-order statement
forall x y z. ~(x = 0) -> ~(y = 0) -> ~(x * x * x * x + y * y * y * y = z * z)

Proof neighborhood

Direct theorem prerequisites

PF002Q fermat_four_square_solutions_have_zero_coordinate

Direct theorem dependents

none

Definition-aware tactic body

Only propositions whose conservative expansion has been checked for exact first-order equivalence are compacted. Every changed line retains its immutable exact replay command.

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.

Definition notation is shown below. Open the paired exact edition for the original native formulas. Source pairing is not a new equivalence certificate.

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 defined 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