PF000X

pythagorean_primitive_hypotenuse_coprime_second_leg

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

In every primitive Pythagorean triple, the second leg and hypotenuse have no nonunit common divisor.

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))) -> (forall pff_divisor_second_hypotenuse_result. (exists pff_left_second_hypotenuse_result. (y) = pff_divisor_second_hypotenuse_result * pff_left_second_hypotenuse_result) -> (exists pff_right_second_hypotenuse_result. (z) = pff_divisor_second_hypotenuse_result * pff_right_second_hypotenuse_result) -> pff_divisor_second_hypotenuse_result = 1)

Constructive proof overview

Generated structural guide

In every primitive Pythagorean triple, the second leg and hypotenuse have no nonunit common divisor.

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

PF000C pythagorean_primitive_leg_swap PF000W pythagorean_primitive_hypotenuse_coprime_first_leg

Direct dependents

PF000Y pythagorean_primitive_pairwise_coprime

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 · 2 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)
pythagorean primitive leg swappythagorean primitive hypotenuse coprime first leg
01Fix variables and assumptionsL1–4

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
02Use earlier factsL5–13

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

  1. L5
    specialize pythagorean_primitive_hypotenuse_coprime_first_leg y
  2. L6
    specialize pythagorean_primitive_hypotenuse_coprime_first_leg x
  3. L7
    specialize pythagorean_primitive_hypotenuse_coprime_first_leg z
  4. L8
    apply pythagorean_primitive_hypotenuse_coprime_first_leg
  5. L9
    specialize pythagorean_primitive_leg_swap x
  6. L10
    specialize pythagorean_primitive_leg_swap y
  7. L11
    specialize pythagorean_primitive_leg_swap z
  8. L12
    apply pythagorean_primitive_leg_swap
  9. L13
    exact hprimitive

Library-wide reading audit

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