PF000R · theorem body

pythagorean_square_gap_coprime_parameter_product

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

A square-difference leg is coprime to the complete Euclidean parameter product.

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) → Coprime(d,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

PD0005 Coprime

In local proof propositions

none
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) -> (forall pff_divisor_gap_product_result. (exists pff_left_gap_product_result. (d) = pff_divisor_gap_product_result * pff_left_gap_product_result) -> (exists pff_right_gap_product_result. (m * n) = pff_divisor_gap_product_result * pff_right_gap_product_result) -> pff_divisor_gap_product_result = 1)

Proof neighborhood

Direct theorem prerequisites

PF000P pythagorean_square_gap_coprime_first_parameter PF000Q pythagorean_square_gap_coprime_second_parameter coprime_mul_right · Stable closed

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

21 script commands · 3 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 square gap coprime first parameterpythagorean square gap coprime second parameter
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 hcoprime
02Use earlier factsL6–15

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

  1. L6
    specialize coprime_mul_right d
  2. L7
    specialize coprime_mul_right m
  3. L8
    specialize coprime_mul_right n
  4. L9
    apply coprime_mul_right
  5. L10
    specialize pythagorean_square_gap_coprime_first_parameter m
  6. L11
    specialize pythagorean_square_gap_coprime_first_parameter n
  7. L12
    specialize pythagorean_square_gap_coprime_first_parameter d
  8. L13
    apply pythagorean_square_gap_coprime_first_parameter
  9. L14
    exact hgap
  10. L15
    exact hcoprime
03Use earlier factsL16–21

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

  1. L16
    specialize pythagorean_square_gap_coprime_second_parameter m
  2. L17
    specialize pythagorean_square_gap_coprime_second_parameter n
  3. L18
    specialize pythagorean_square_gap_coprime_second_parameter d
  4. L19
    apply pythagorean_square_gap_coprime_second_parameter
  5. L20
    exact hgap
  6. L21
    exact hcoprime

Library-wide reading audit

Original defined command ledger · 21 lines
  1. 0001intro m
  2. 0002intro n
  3. 0003intro d
  4. 0004intro hgap
  5. 0005intro hcoprime
  6. 0006specialize coprime_mul_right d
  7. 0007specialize coprime_mul_right m
  8. 0008specialize coprime_mul_right n
  9. 0009apply coprime_mul_right
  10. 0010specialize pythagorean_square_gap_coprime_first_parameter m
  11. 0011specialize pythagorean_square_gap_coprime_first_parameter n
  12. 0012specialize pythagorean_square_gap_coprime_first_parameter d
  13. 0013apply pythagorean_square_gap_coprime_first_parameter
  14. 0014exact hgap
  15. 0015exact hcoprime
  16. 0016specialize pythagorean_square_gap_coprime_second_parameter m
  17. 0017specialize pythagorean_square_gap_coprime_second_parameter n
  18. 0018specialize pythagorean_square_gap_coprime_second_parameter d
  19. 0019apply pythagorean_square_gap_coprime_second_parameter
  20. 0020exact hgap
  21. 0021exact hcoprime