PF000T

pythagorean_primitive_euclidean_constructor

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

The complete forward primitive Euclid constructor proves both the exact Pythagorean identity and coprimality of its two displayed 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.

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) -> ((((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)))) -> ((((d) * (d) + (2 * (m * n)) * (2 * (m * n)) = (m * m + n * n) * (m * m + n * n)) /\ (forall pff_divisor_euclidean_result. (exists pff_left_euclidean_result. (d) = pff_divisor_euclidean_result * pff_left_euclidean_result) -> (exists pff_right_euclidean_result. (2 * (m * n)) = pff_divisor_euclidean_result * pff_right_euclidean_result) -> pff_divisor_euclidean_result = 1)))

Constructive proof overview

Generated structural guide

The complete forward primitive Euclid constructor proves both the exact Pythagorean identity and coprimality of its two displayed legs.

The unchanged tactic script uses 2 declared prerequisites and contains 19 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

PF0001 pythagorean_euclidean_identity PF000S pythagorean_primitive_euclidean_legs

Direct dependents

PF000U pythagorean_primitive_euclidean_from_order PF000V pythagorean_primitive_euclidean_swapped_constructor PF0021 pythagorean_euclidean_parameters_positive_constructor

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

19 script commands · 4 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 euclidean identitypythagorean primitive euclidean legs
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
02Separate the logical casesL7–7

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

  1. L7
    split
03Use earlier factsL8–17

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

  1. L8
    specialize pythagorean_euclidean_identity m
  2. L9
    specialize pythagorean_euclidean_identity n
  3. L10
    specialize pythagorean_euclidean_identity d
  4. L11
    apply pythagorean_euclidean_identity
  5. L12
    exact hgap
  6. L13
    specialize pythagorean_primitive_euclidean_legs m
  7. L14
    specialize pythagorean_primitive_euclidean_legs n
  8. L15
    specialize pythagorean_primitive_euclidean_legs d
  9. L16
    apply pythagorean_primitive_euclidean_legs
  10. L17
    exact hgap
04Use earlier factsL18–19

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

  1. L18
    exact hcoprime
  2. L19
    exact hopposite

Library-wide reading audit

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