PF000M

pythagorean_opposite_parity_hypotenuse_odd

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

The Euclidean hypotenuse is explicitly odd whenever its two parameters have opposite parity.

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 m n. ((((exists pp_even_primitive_parameters_first_even. (m) = 2 * pp_even_primitive_parameters_first_even) /\ (exists pp_odd_primitive_parameters_second_odd. (n) = 2 * pp_odd_primitive_parameters_second_odd + 1)) \/ ((exists pp_odd_primitive_parameters_first_odd. (m) = 2 * pp_odd_primitive_parameters_first_odd + 1) /\ (exists pp_even_primitive_parameters_second_even. (n) = 2 * pp_even_primitive_parameters_second_even)))) -> exists q. m * m + n * n = 2 * q + 1

Constructive proof overview

Generated structural guide

The Euclidean hypotenuse is explicitly odd whenever its two parameters have opposite parity.

The unchanged tactic script uses 4 declared prerequisites and contains 20 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

PF000H pythagorean_parameter_even_square PF000I pythagorean_parameter_odd_square even_add_odd Stable theorem; checked-use authorized odd_add_even Stable theorem; checked-use authorized

Direct dependents

PF0016 pythagorean_primitive_hypotenuse_odd

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

20 script commands · 5 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 parameter even squarepythagorean parameter odd square
01Fix variables and assumptionsL1–3

Work with arbitrary variables or the premises of the current implication.

  1. L1
    intro m
  2. L2
    intro n
  3. L3
    intro hopposite
02Separate the logical casesL4–5

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

  1. L4
    cases hopposite
  2. L5
    cases hopposite_left
03Use earlier factsL6–12

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

  1. L6
    specialize even_add_odd (m * m)
  2. L7
    specialize even_add_odd (n * n)
  3. L8
    apply even_add_odd
  4. L9
    apply pythagorean_parameter_even_square
  5. L10
    exact hopposite_left_left
  6. L11
    apply pythagorean_parameter_odd_square
  7. L12
    exact hopposite_left_right
04Separate the logical casesL13–13

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

  1. L13
    cases hopposite_right
05Use earlier factsL14–20

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

  1. L14
    specialize odd_add_even (m * m)
  2. L15
    specialize odd_add_even (n * n)
  3. L16
    apply odd_add_even
  4. L17
    apply pythagorean_parameter_odd_square
  5. L18
    exact hopposite_right_left
  6. L19
    apply pythagorean_parameter_even_square
  7. L20
    exact hopposite_right_right

Library-wide reading audit

Original exact command ledger · 20 lines
  1. 0001intro m
  2. 0002intro n
  3. 0003intro hopposite
  4. 0004cases hopposite
  5. 0005cases hopposite_left
  6. 0006specialize even_add_odd (m * m)
  7. 0007specialize even_add_odd (n * n)
  8. 0008apply even_add_odd
  9. 0009apply pythagorean_parameter_even_square
  10. 0010exact hopposite_left_left
  11. 0011apply pythagorean_parameter_odd_square
  12. 0012exact hopposite_left_right
  13. 0013cases hopposite_right
  14. 0014specialize odd_add_even (m * m)
  15. 0015specialize odd_add_even (n * n)
  16. 0016apply odd_add_even
  17. 0017apply pythagorean_parameter_odd_square
  18. 0018exact hopposite_right_left
  19. 0019apply pythagorean_parameter_even_square
  20. 0020exact hopposite_right_right