PF000U · theorem body

pythagorean_primitive_euclidean_from_order

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

Every witnessed ordered pair of coprime opposite-parity natural parameters constructs an actual primitive Euclidean Pythagorean triple.

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. Le(n,m) → Coprime(m,n) → OppositeParity(m,n) → ∃ x. PrimitivePythagorean(x,2 · (m · n),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 PD0001 Le PD0005 Coprime

In local proof propositions

none
Exact expanded first-order statement
forall m n. (exists gap. gap + n = m) -> (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)))) -> exists d. ((((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)))

Proof neighborhood

Direct theorem prerequisites

PF0008 pythagorean_square_gap_from_order PF000T pythagorean_primitive_euclidean_constructor

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

19 script commands · 5 reading checkpoints · 1 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 from orderpythagorean primitive euclidean constructor
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 horder
  4. L4
    intro hcoprime
  5. L5
    intro hopposite
02Establish hgapL6–10

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply pythagorean square gap from order.

  1. L6
    have hgap : exists d. m * m = n * n + d
  2. L7
    specialize pythagorean_square_gap_from_order m
  3. L8
    specialize pythagorean_square_gap_from_order n
  4. L9
    apply pythagorean_square_gap_from_order
  5. L10
    exact horder
03Separate the logical casesL11–11

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

  1. L11
    cases hgap
04Construct an explicit witnessL12–12

Supply the displayed value, then prove that it has the required property.

  1. L12
    exists x
05Use earlier factsL13–19

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

  1. L13
    specialize pythagorean_primitive_euclidean_constructor m
  2. L14
    specialize pythagorean_primitive_euclidean_constructor n
  3. L15
    specialize pythagorean_primitive_euclidean_constructor x
  4. L16
    apply pythagorean_primitive_euclidean_constructor
  5. L17
    exact hgap_witness
  6. L18
    exact hcoprime
  7. L19
    exact hopposite

Library-wide reading audit

Original defined command ledger · 19 lines
  1. 0001intro m
  2. 0002intro n
  3. 0003intro horder
  4. 0004intro hcoprime
  5. 0005intro hopposite
  6. 0006have hgap : exists d. m * m = n * n + d
  7. 0007specialize pythagorean_square_gap_from_order m
  8. 0008specialize pythagorean_square_gap_from_order n
  9. 0009apply pythagorean_square_gap_from_order
  10. 0010exact horder
  11. 0011cases hgap
  12. 0012exists x
  13. 0013specialize pythagorean_primitive_euclidean_constructor m
  14. 0014specialize pythagorean_primitive_euclidean_constructor n
  15. 0015specialize pythagorean_primitive_euclidean_constructor x
  16. 0016apply pythagorean_primitive_euclidean_constructor
  17. 0017exact hgap_witness
  18. 0018exact hcoprime
  19. 0019exact hopposite