PF000Z

pythagorean_primitive_legs_not_both_even

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

The two legs of a primitive Pythagorean triple cannot both have even witnesses.

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) * (x) + (y) * (y) = (z) * (z)) /\ (forall pff_divisor_pairwise_source. (exists pff_left_pairwise_source. (x) = pff_divisor_pairwise_source * pff_left_pairwise_source) -> (exists pff_right_pairwise_source. (y) = pff_divisor_pairwise_source * pff_right_pairwise_source) -> pff_divisor_pairwise_source = 1))) -> (exists a. x = 2 * a) -> (exists b. y = 2 * b) -> false

Constructive proof overview

Generated structural guide

The two legs of a primitive Pythagorean triple cannot both have even witnesses.

The unchanged tactic script uses 1 declared prerequisite 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

prime_two Stable theorem; checked-use authorized

Direct dependents

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.

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 hprimitive
  5. L5
    intro hx
  6. L6
    intro hy
02Separate the logical casesL7–8

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

  1. L7
    cases hprimitive
  2. L8
    cases prime_two
03Use earlier factsL9–13

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

  1. L9
    apply prime_two_left
  2. L10
    specialize hprimitive_right 2
  3. L11
    apply hprimitive_right
  4. L12
    exact hx
  5. L13
    exact hy

Library-wide reading audit

Original exact command ledger · 13 lines
  1. 0001intro x
  2. 0002intro y
  3. 0003intro z
  4. 0004intro hprimitive
  5. 0005intro hx
  6. 0006intro hy
  7. 0007cases hprimitive
  8. 0008cases prime_two
  9. 0009apply prime_two_left
  10. 0010specialize hprimitive_right 2
  11. 0011apply hprimitive_right
  12. 0012exact hx
  13. 0013exact hy