PF0023 · theorem body

pythagorean_positive_primitive_classification

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

Complete constructive classification: a positive ordered triple is primitive Pythagorean if and only if it has positive strictly ordered coprime opposite-parity Euclid parameters 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

∀ a. ∀ b. ∀ c. (PrimitiveTriple(a,b,c) → EuclidParametrization(a,b,c)) ∧ (EuclidParametrization(a,b,c) → PrimitiveTriple(a,b,c))

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

ND0069 PrimitiveTriple ND0074 EuclidParametrization

In local proof propositions

none
Exact expanded first-order statement
forall a b c. ((((~((a) = 0) /\ (~((b) = 0) /\ (~((c) = 0) /\ ((((a) * (a) + (b) * (b) = (c) * (c)) /\ (forall pff_divisor_pi_classification_positive. (exists pff_left_pi_classification_positive. (a) = pff_divisor_pi_classification_positive * pff_left_pi_classification_positive) -> (exists pff_right_pi_classification_positive. (b) = pff_divisor_pi_classification_positive * pff_right_pi_classification_positive) -> pff_divisor_pi_classification_positive = 1))))))) -> (exists pi_larger_classification_parameters pi_smaller_classification_parameters. ((~((pi_smaller_classification_parameters) = 0) /\ ((exists pi_gap_classification_parameters. pi_gap_classification_parameters + S (pi_smaller_classification_parameters) = (pi_larger_classification_parameters)) /\ ((forall pff_divisor_pi_classification_parameters_coprime. (exists pff_left_pi_classification_parameters_coprime. (pi_larger_classification_parameters) = pff_divisor_pi_classification_parameters_coprime * pff_left_pi_classification_parameters_coprime) -> (exists pff_right_pi_classification_parameters_coprime. (pi_smaller_classification_parameters) = pff_divisor_pi_classification_parameters_coprime * pff_right_pi_classification_parameters_coprime) -> pff_divisor_pi_classification_parameters_coprime = 1) /\ (((((exists pp_even_pi_classification_parameters_parity_first_even. (pi_larger_classification_parameters) = 2 * pp_even_pi_classification_parameters_parity_first_even) /\ (exists pp_odd_pi_classification_parameters_parity_second_odd. (pi_smaller_classification_parameters) = 2 * pp_odd_pi_classification_parameters_parity_second_odd + 1)) \/ ((exists pp_odd_pi_classification_parameters_parity_first_odd. (pi_larger_classification_parameters) = 2 * pp_odd_pi_classification_parameters_parity_first_odd + 1) /\ (exists pp_even_pi_classification_parameters_parity_second_even. (pi_smaller_classification_parameters) = 2 * pp_even_pi_classification_parameters_parity_second_even)))) /\ ((c) = (pi_larger_classification_parameters) * (pi_larger_classification_parameters) + (pi_smaller_classification_parameters) * (pi_smaller_classification_parameters) /\ (((pi_larger_classification_parameters) * (pi_larger_classification_parameters) = (pi_smaller_classification_parameters) * (pi_smaller_classification_parameters) + (a) /\ (b) = 2 * ((pi_larger_classification_parameters) * (pi_smaller_classification_parameters))) \/ ((pi_larger_classification_parameters) * (pi_larger_classification_parameters) = (pi_smaller_classification_parameters) * (pi_smaller_classification_parameters) + (b) /\ (a) = 2 * ((pi_larger_classification_parameters) * (pi_smaller_classification_parameters)))))))))))) /\ ((exists pi_larger_classification_parameters pi_smaller_classification_parameters. ((~((pi_smaller_classification_parameters) = 0) /\ ((exists pi_gap_classification_parameters. pi_gap_classification_parameters + S (pi_smaller_classification_parameters) = (pi_larger_classification_parameters)) /\ ((forall pff_divisor_pi_classification_parameters_coprime. (exists pff_left_pi_classification_parameters_coprime. (pi_larger_classification_parameters) = pff_divisor_pi_classification_parameters_coprime * pff_left_pi_classification_parameters_coprime) -> (exists pff_right_pi_classification_parameters_coprime. (pi_smaller_classification_parameters) = pff_divisor_pi_classification_parameters_coprime * pff_right_pi_classification_parameters_coprime) -> pff_divisor_pi_classification_parameters_coprime = 1) /\ (((((exists pp_even_pi_classification_parameters_parity_first_even. (pi_larger_classification_parameters) = 2 * pp_even_pi_classification_parameters_parity_first_even) /\ (exists pp_odd_pi_classification_parameters_parity_second_odd. (pi_smaller_classification_parameters) = 2 * pp_odd_pi_classification_parameters_parity_second_odd + 1)) \/ ((exists pp_odd_pi_classification_parameters_parity_first_odd. (pi_larger_classification_parameters) = 2 * pp_odd_pi_classification_parameters_parity_first_odd + 1) /\ (exists pp_even_pi_classification_parameters_parity_second_even. (pi_smaller_classification_parameters) = 2 * pp_even_pi_classification_parameters_parity_second_even)))) /\ ((c) = (pi_larger_classification_parameters) * (pi_larger_classification_parameters) + (pi_smaller_classification_parameters) * (pi_smaller_classification_parameters) /\ (((pi_larger_classification_parameters) * (pi_larger_classification_parameters) = (pi_smaller_classification_parameters) * (pi_smaller_classification_parameters) + (a) /\ (b) = 2 * ((pi_larger_classification_parameters) * (pi_smaller_classification_parameters))) \/ ((pi_larger_classification_parameters) * (pi_larger_classification_parameters) = (pi_smaller_classification_parameters) * (pi_smaller_classification_parameters) + (b) /\ (a) = 2 * ((pi_larger_classification_parameters) * (pi_smaller_classification_parameters))))))))))) -> ((~((a) = 0) /\ (~((b) = 0) /\ (~((c) = 0) /\ ((((a) * (a) + (b) * (b) = (c) * (c)) /\ (forall pff_divisor_pi_classification_positive. (exists pff_left_pi_classification_positive. (a) = pff_divisor_pi_classification_positive * pff_left_pi_classification_positive) -> (exists pff_right_pi_classification_positive. (b) = pff_divisor_pi_classification_positive * pff_right_pi_classification_positive) -> pff_divisor_pi_classification_positive = 1)))))))))

Proof neighborhood

Direct theorem prerequisites

PF001Z pythagorean_positive_primitive_inverse PF0022 pythagorean_positive_primitive_from_parameters

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

12 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 positive primitive inversepythagorean positive primitive from parameters
01Fix variables and assumptionsL1–3

Work with arbitrary variables or the premises of the current implication.

  1. L1
    intro a
  2. L2
    intro b
  3. L3
    intro c
02Separate the logical casesL4–4

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

  1. L4
    split
03Use earlier factsL5–12

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

  1. L5
    specialize pythagorean_positive_primitive_inverse a
  2. L6
    specialize pythagorean_positive_primitive_inverse b
  3. L7
    specialize pythagorean_positive_primitive_inverse c
  4. L8
    exact pythagorean_positive_primitive_inverse
  5. L9
    specialize pythagorean_positive_primitive_from_parameters a
  6. L10
    specialize pythagorean_positive_primitive_from_parameters b
  7. L11
    specialize pythagorean_positive_primitive_from_parameters c
  8. L12
    exact pythagorean_positive_primitive_from_parameters

Library-wide reading audit

Original defined command ledger · 12 lines
  1. 0001intro a
  2. 0002intro b
  3. 0003intro c
  4. 0004split
  5. 0005specialize pythagorean_positive_primitive_inverse a
  6. 0006specialize pythagorean_positive_primitive_inverse b
  7. 0007specialize pythagorean_positive_primitive_inverse c
  8. 0008exact pythagorean_positive_primitive_inverse
  9. 0009specialize pythagorean_positive_primitive_from_parameters a
  10. 0010specialize pythagorean_positive_primitive_from_parameters b
  11. 0011specialize pythagorean_positive_primitive_from_parameters c
  12. 0012exact pythagorean_positive_primitive_from_parameters