PF002I · theorem body

fermat_four_primitive_square_triangle

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

Statement with defined notation

∀ a. ∀ b. ∀ h. PrimitiveFermatFourCounterexample(a,b,h) → PrimitivePythagorean(a · a,b · b,h)

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

CF0013 PrimitivePythagorean ND0071 PrimitiveFermatFourCounterexample

In local proof propositions

none
Exact expanded first-order 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)))

Proof neighborhood

Direct theorem prerequisites

PF000D fermat_four_counterexample_is_pythagorean PF0026 fermat_four_coprime_squares

Direct theorem dependents

PF002K fermat_four_primitive_odd_even_descent PF002L fermat_four_primitive_descent

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

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.

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 (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 defined 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