PF000L · theorem body

pythagorean_opposite_parity_square_gap_odd

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

Statement with defined notation

∀ m. ∀ n. ∀ d. m · m = n · n + d → OppositeParity(m,n) → Odd(d)

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

CF0016 OppositeParity PD0010 Odd

In local proof propositions

none
Exact expanded first-order 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

Proof neighborhood

Direct theorem prerequisites

PF000J pythagorean_even_odd_square_gap_odd PF000K pythagorean_odd_even_square_gap_odd

Direct theorem dependents

PF000S pythagorean_primitive_euclidean_legs

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

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.

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)
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 defined 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