PF002I

fermat_four_primitive_square_triangle

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

A primitive fourth-power counterexample is an actual primitive triangle with square legs.

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) /\ (~((b) = 0) /\ (~((h) = 0) /\ ((a) * (a) * (a) * (a) + (b) * (b) * (b) * (b) = (h) * (h)))))) /\ (forall pff_divisor_triangle_source. (exists pff_left_triangle_source. (a) = pff_divisor_triangle_source * pff_left_triangle_source) -> (exists pff_right_triangle_source. (b) = pff_divisor_triangle_source * pff_right_triangle_source) -> pff_divisor_triangle_source = 1))) -> ((((a * a) * (a * a) + (b * b) * (b * b) = (h) * (h)) /\ (forall pff_divisor_ffd_square_triangle. (exists pff_left_ffd_square_triangle. (a * a) = pff_divisor_ffd_square_triangle * pff_left_ffd_square_triangle) -> (exists pff_right_ffd_square_triangle. (b * b) = pff_divisor_ffd_square_triangle * pff_right_ffd_square_triangle) -> pff_divisor_ffd_square_triangle = 1)))

Constructive proof overview

Generated structural guide

A primitive fourth-power counterexample is an actual primitive triangle with square legs.

The unchanged tactic script uses 2 declared prerequisites and contains 13 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

PF000D fermat_four_counterexample_is_pythagorean PF0026 fermat_four_coprime_squares

Direct dependents

PF002K fermat_four_primitive_odd_even_descent PF002L fermat_four_primitive_descent

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

13 script commands · 3 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 (2)
fermat four counterexample is pythagoreanfermat four coprime squares
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 hprimitive
02Separate the logical casesL5–9

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

  1. L5
    cases hprimitive
  2. L6
    cases hprimitive_left
  3. L7
    cases hprimitive_left_right
  4. L8
    cases hprimitive_left_right_right
  5. L9
    split
03Use earlier factsL10–13

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

  1. L10
    apply fermat_four_counterexample_is_pythagorean
  2. L11
    exact hprimitive_left_right_right_right
  3. L12
    apply fermat_four_coprime_squares
  4. L13
    exact hprimitive_right

Library-wide reading audit

Original exact command ledger · 13 lines
  1. 0001intro a
  2. 0002intro b
  3. 0003intro h
  4. 0004intro hprimitive
  5. 0005cases hprimitive
  6. 0006cases hprimitive_left
  7. 0007cases hprimitive_left_right
  8. 0008cases hprimitive_left_right_right
  9. 0009split
  10. 0010apply fermat_four_counterexample_is_pythagorean
  11. 0011exact hprimitive_left_right_right_right
  12. 0012apply fermat_four_coprime_squares
  13. 0013exact hprimitive_right