PF000V

pythagorean_primitive_euclidean_swapped_constructor

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

The complete forward primitive Euclid constructor is valid in either leg 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.

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)))) -> ((((2 * (m * n)) * (2 * (m * n)) + (d) * (d) = (m * m + n * n) * (m * m + n * n)) /\ (forall pff_divisor_euclidean_swapped. (exists pff_left_euclidean_swapped. (2 * (m * n)) = pff_divisor_euclidean_swapped * pff_left_euclidean_swapped) -> (exists pff_right_euclidean_swapped. (d) = pff_divisor_euclidean_swapped * pff_right_euclidean_swapped) -> pff_divisor_euclidean_swapped = 1)))

Constructive proof overview

Generated structural guide

The complete forward primitive Euclid constructor is valid in either leg orientation.

The unchanged tactic script uses 2 declared prerequisites and contains 17 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

PF000T pythagorean_primitive_euclidean_constructor PF000C pythagorean_primitive_leg_swap

Direct dependents

none

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

17 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 primitive euclidean constructorpythagorean primitive leg swap
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
02Use earlier factsL7–16

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

  1. L7
    specialize pythagorean_primitive_leg_swap d
  2. L8
    specialize pythagorean_primitive_leg_swap (2 * (m * n))
  3. L9
    specialize pythagorean_primitive_leg_swap (m * m + n * n)
  4. L10
    apply pythagorean_primitive_leg_swap
  5. L11
    specialize pythagorean_primitive_euclidean_constructor m
  6. L12
    specialize pythagorean_primitive_euclidean_constructor n
  7. L13
    specialize pythagorean_primitive_euclidean_constructor d
  8. L14
    apply pythagorean_primitive_euclidean_constructor
  9. L15
    exact hgap
  10. L16
    exact hcoprime
03Use earlier factsL17–17

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

  1. L17
    exact hopposite

Library-wide reading audit

Original exact command ledger · 17 lines
  1. 0001intro m
  2. 0002intro n
  3. 0003intro d
  4. 0004intro hgap
  5. 0005intro hcoprime
  6. 0006intro hopposite
  7. 0007specialize pythagorean_primitive_leg_swap d
  8. 0008specialize pythagorean_primitive_leg_swap (2 * (m * n))
  9. 0009specialize pythagorean_primitive_leg_swap (m * m + n * n)
  10. 0010apply pythagorean_primitive_leg_swap
  11. 0011specialize pythagorean_primitive_euclidean_constructor m
  12. 0012specialize pythagorean_primitive_euclidean_constructor n
  13. 0013specialize pythagorean_primitive_euclidean_constructor d
  14. 0014apply pythagorean_primitive_euclidean_constructor
  15. 0015exact hgap
  16. 0016exact hcoprime
  17. 0017exact hopposite