PF000R

pythagorean_square_gap_coprime_parameter_product

Alpha v34 independently verified · alpha_closed; checked-use authorized; 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.

Exact expanded first-order arithmetic 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)

Constructive proof overview

Generated structural guide

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

The unchanged tactic script uses 3 declared prerequisites and contains 21 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

PF000P pythagorean_square_gap_coprime_first_parameter PF000Q pythagorean_square_gap_coprime_second_parameter coprime_mul_right Stable theorem; checked-use authorized

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

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.

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 exact 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