PF002P

fermat_four_equation_height_nonzero

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

A nonzero first fourth-power summand forces the square hypotenuse to be nonzero, so its positivity need not be assumed.

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 = 0) -> a * a * a * a + b * b * b * b = h * h -> ~(h = 0)

Constructive proof overview

Generated structural guide

A nonzero first fourth-power summand forces the square hypotenuse to be nonzero, so its positivity need not be assumed.

The unchanged tactic script uses 4 declared prerequisites and contains 28 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

PF0025 fermat_four_square_nonzero fourth_power_regroup Stable theorem; checked-use authorized add_eq_zero_left Stable theorem; checked-use authorized mul_zero_left Stable theorem; checked-use authorized

Direct dependents

PF002Q fermat_four_square_solutions_have_zero_coordinate

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

28 script commands · 7 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.

Named ingredients (1)
fermat four square nonzero
01Fix variables and assumptionsL1–6

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 ha
  5. L5
    intro hequation
  6. L6
    intro hzero
02Establish hsquareL7–12

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

  1. L7
    have hsquare : ~(a * a = 0)
  2. L8
    intro hz
  3. L9
    specialize fermat_four_square_nonzero (a)
  4. L10
    apply fermat_four_square_nonzero
  5. L11
    exact ha
  6. L12
    exact hz
03Establish hfourthL13–22

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

  1. L13
    have hfourth : ~((a * a) * (a * a) = 0)
  2. L14
    intro hz
  3. L15
    specialize fermat_four_square_nonzero (a * a)
  4. L16
    apply fermat_four_square_nonzero
  5. L17
    exact hsquare
  6. L18
    exact hz
  7. L19
    apply hfourth
  8. L20
    trans a * a * a * a
  9. L21
    symm
  10. L22
    apply fourth_power_regroup
04Use earlier factsL23–25

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

  1. L23
    specialize add_eq_zero_left (a * a * a * a)
  2. L24
    specialize add_eq_zero_left (b * b * b * b)
  3. L25
    apply add_eq_zero_left
05Calculate and transport equalitiesL26–26

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

  1. L26
    trans h * h
06Use earlier factsL27–27

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

  1. L27
    exact hequation
07Calculate and transport equalitiesL28–28

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

  1. L28
    simp [hzero, mul_zero_left]

Library-wide reading audit

Original exact command ledger · 28 lines
  1. 0001intro a
  2. 0002intro b
  3. 0003intro h
  4. 0004intro ha
  5. 0005intro hequation
  6. 0006intro hzero
  7. 0007have hsquare : ~(a * a = 0)
  8. 0008intro hz
  9. 0009specialize fermat_four_square_nonzero (a)
  10. 0010apply fermat_four_square_nonzero
  11. 0011exact ha
  12. 0012exact hz
  13. 0013have hfourth : ~((a * a) * (a * a) = 0)
  14. 0014intro hz
  15. 0015specialize fermat_four_square_nonzero (a * a)
  16. 0016apply fermat_four_square_nonzero
  17. 0017exact hsquare
  18. 0018exact hz
  19. 0019apply hfourth
  20. 0020trans a * a * a * a
  21. 0021symm
  22. 0022apply fourth_power_regroup
  23. 0023specialize add_eq_zero_left (a * a * a * a)
  24. 0024specialize add_eq_zero_left (b * b * b * b)
  25. 0025apply add_eq_zero_left
  26. 0026trans h * h
  27. 0027exact hequation
  28. 0028simp [hzero, mul_zero_left]