PF000L

pythagorean_opposite_parity_square_gap_odd

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

Opposite-parity Euclidean parameters force the subtraction-free square-difference leg to be odd in either parity orientation.

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 d. m * m = n * n + d -> ((((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. d = 2 * q + 1

Constructive proof overview

Generated structural guide

Opposite-parity Euclidean parameters force the subtraction-free square-difference leg to be odd in either parity orientation.

The unchanged tactic script uses 2 declared prerequisites and contains 16 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

PF000J pythagorean_even_odd_square_gap_odd PF000K pythagorean_odd_even_square_gap_odd

Direct dependents

PF000S pythagorean_primitive_euclidean_legs

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

16 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 even odd square gap oddpythagorean odd even square gap odd
01Fix variables and assumptionsL1–5

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

  1. L1
    intro m
  2. L2
    intro n
  3. L3
    intro d
  4. L4
    intro hgap
  5. L5
    intro hopposite
02Separate the logical casesL6–7

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

  1. L6
    cases hopposite
  2. L7
    cases hopposite_left
03Use earlier factsL8–11

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

  1. L8
    apply pythagorean_even_odd_square_gap_odd
  2. L9
    exact hgap
  3. L10
    exact hopposite_left_left
  4. L11
    exact hopposite_left_right
04Separate the logical casesL12–12

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

  1. L12
    cases hopposite_right
05Use earlier factsL13–16

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

  1. L13
    apply pythagorean_odd_even_square_gap_odd
  2. L14
    exact hgap
  3. L15
    exact hopposite_right_left
  4. L16
    exact hopposite_right_right

Library-wide reading audit

Original exact command ledger · 16 lines
  1. 0001intro m
  2. 0002intro n
  3. 0003intro d
  4. 0004intro hgap
  5. 0005intro hopposite
  6. 0006cases hopposite
  7. 0007cases hopposite_left
  8. 0008apply pythagorean_even_odd_square_gap_odd
  9. 0009exact hgap
  10. 0010exact hopposite_left_left
  11. 0011exact hopposite_left_right
  12. 0012cases hopposite_right
  13. 0013apply pythagorean_odd_even_square_gap_odd
  14. 0014exact hgap
  15. 0015exact hopposite_right_left
  16. 0016exact hopposite_right_right