PF000V · theorem body

pythagorean_primitive_euclidean_swapped_constructor

Alpha v34 checked-use · independently kernel and Lean verified; 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.

Statement with defined notation

∀ m. ∀ n. ∀ d. m · m = n · n + d → Coprime(m,n) → OppositeParity(m,n) → PrimitivePythagorean(2 · (m · n),d,m · m + n · 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

CF0013 PrimitivePythagorean CF0016 OppositeParity 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) -> ((((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)))

Proof neighborhood

Direct theorem prerequisites

PF000T pythagorean_primitive_euclidean_constructor PF000C pythagorean_primitive_leg_swap

Direct theorem dependents

none

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

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.

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 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 defined 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