PF000S · theorem body

pythagorean_primitive_euclidean_legs

Alpha v34 checked-use · independently kernel and Lean verified; not Stable

Coprime opposite-parity Euclidean parameters yield genuinely coprime square-difference and doubled-product legs.

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 → Coprime(m,n) → OppositeParity(m,n) → Coprime(d,2 · (m · n))

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 PD0005 Coprime

In local proof propositions

PD0010 Odd
Exact expanded first-order statement
forall m n d. m * m = n * n + d -> (forall pff_divisor_primitive_parameters. (exists pff_left_primitive_parameters. (m) = pff_divisor_primitive_parameters * pff_left_primitive_parameters) -> (exists pff_right_primitive_parameters. (n) = pff_divisor_primitive_parameters * pff_right_primitive_parameters) -> pff_divisor_primitive_parameters = 1) -> ((((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)))) -> (forall pff_divisor_euclidean_legs. (exists pff_left_euclidean_legs. (d) = pff_divisor_euclidean_legs * pff_left_euclidean_legs) -> (exists pff_right_euclidean_legs. (2 * (m * n)) = pff_divisor_euclidean_legs * pff_right_euclidean_legs) -> pff_divisor_euclidean_legs = 1)

Proof neighborhood

Direct theorem prerequisites

PF000L pythagorean_opposite_parity_square_gap_odd PF000N pythagorean_odd_coordinate_coprime_two PF000R pythagorean_square_gap_coprime_parameter_product coprime_mul_right · Stable closed

Direct theorem dependents

PF000T pythagorean_primitive_euclidean_constructor

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

26 script commands · 3 reading checkpoints · 1 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 (3)
pythagorean opposite parity square gap oddpythagorean odd coordinate coprime twopythagorean square gap coprime parameter product
01Fix variables and assumptionsL1–6

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 hcoprime
  6. L6
    intro hopposite
02Establish hoddL7–16

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply pythagorean opposite parity square gap odd.

  1. L7
  2. L8
    specialize pythagorean_opposite_parity_square_gap_odd m
  3. L9
    specialize pythagorean_opposite_parity_square_gap_odd n
  4. L10
    specialize pythagorean_opposite_parity_square_gap_odd d
  5. L11
    apply pythagorean_opposite_parity_square_gap_odd
  6. L12
    exact hgap
  7. L13
    exact hopposite
  8. L14
    specialize coprime_mul_right d
  9. L15
    specialize coprime_mul_right 2
  10. L16
    specialize coprime_mul_right (m * n)
03Use earlier factsL17–26

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

  1. L17
    apply coprime_mul_right
  2. L18
    specialize pythagorean_odd_coordinate_coprime_two d
  3. L19
    apply pythagorean_odd_coordinate_coprime_two
  4. L20
    exact hodd
  5. L21
    specialize pythagorean_square_gap_coprime_parameter_product m
  6. L22
    specialize pythagorean_square_gap_coprime_parameter_product n
  7. L23
    specialize pythagorean_square_gap_coprime_parameter_product d
  8. L24
    apply pythagorean_square_gap_coprime_parameter_product
  9. L25
    exact hgap
  10. L26
    exact hcoprime

Library-wide reading audit

Original defined command ledger · 26 lines
  1. 0001intro m
  2. 0002intro n
  3. 0003intro d
  4. 0004intro hgap
  5. 0005intro hcoprime
  6. 0006intro hopposite
  7. 0007have hodd : Odd(d)
    Exact native replay linehave hodd : exists q. d = 2 * q + 1
  8. 0008specialize pythagorean_opposite_parity_square_gap_odd m
  9. 0009specialize pythagorean_opposite_parity_square_gap_odd n
  10. 0010specialize pythagorean_opposite_parity_square_gap_odd d
  11. 0011apply pythagorean_opposite_parity_square_gap_odd
  12. 0012exact hgap
  13. 0013exact hopposite
  14. 0014specialize coprime_mul_right d
  15. 0015specialize coprime_mul_right 2
  16. 0016specialize coprime_mul_right (m * n)
  17. 0017apply coprime_mul_right
  18. 0018specialize pythagorean_odd_coordinate_coprime_two d
  19. 0019apply pythagorean_odd_coordinate_coprime_two
  20. 0020exact hodd
  21. 0021specialize pythagorean_square_gap_coprime_parameter_product m
  22. 0022specialize pythagorean_square_gap_coprime_parameter_product n
  23. 0023specialize pythagorean_square_gap_coprime_parameter_product d
  24. 0024apply pythagorean_square_gap_coprime_parameter_product
  25. 0025exact hgap
  26. 0026exact hcoprime