PF0017 · theorem body

pythagorean_primitive_normal_form

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

Every primitive Pythagorean triple admits its complete constructive parity and pairwise-coprimality normal form.

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

∀ x. ∀ y. ∀ z. PrimitivePythagorean(x,y,z) → OppositeParity(x,y) ∧ (Odd(z) ∧ (Coprime(x,y) ∧ (Coprime(x,z) ∧ Coprime(y,z))))

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 PD0010 Odd

In local proof propositions

none
Exact expanded first-order statement
forall x y z. ((((x) * (x) + (y) * (y) = (z) * (z)) /\ (forall pff_divisor_pairwise_source. (exists pff_left_pairwise_source. (x) = pff_divisor_pairwise_source * pff_left_pairwise_source) -> (exists pff_right_pairwise_source. (y) = pff_divisor_pairwise_source * pff_right_pairwise_source) -> pff_divisor_pairwise_source = 1))) -> (((((exists pp_even_normal_parity_first_even. (x) = 2 * pp_even_normal_parity_first_even) /\ (exists pp_odd_normal_parity_second_odd. (y) = 2 * pp_odd_normal_parity_second_odd + 1)) \/ ((exists pp_odd_normal_parity_first_odd. (x) = 2 * pp_odd_normal_parity_first_odd + 1) /\ (exists pp_even_normal_parity_second_even. (y) = 2 * pp_even_normal_parity_second_even)))) /\ ((exists pp_odd_normal_hypotenuse. (z) = 2 * pp_odd_normal_hypotenuse + 1) /\ ((forall pff_divisor_normal_legs. (exists pff_left_normal_legs. (x) = pff_divisor_normal_legs * pff_left_normal_legs) -> (exists pff_right_normal_legs. (y) = pff_divisor_normal_legs * pff_right_normal_legs) -> pff_divisor_normal_legs = 1) /\ ((forall pff_divisor_first_hypotenuse_result. (exists pff_left_first_hypotenuse_result. (x) = pff_divisor_first_hypotenuse_result * pff_left_first_hypotenuse_result) -> (exists pff_right_first_hypotenuse_result. (z) = pff_divisor_first_hypotenuse_result * pff_right_first_hypotenuse_result) -> pff_divisor_first_hypotenuse_result = 1) /\ (forall pff_divisor_second_hypotenuse_result. (exists pff_left_second_hypotenuse_result. (y) = pff_divisor_second_hypotenuse_result * pff_left_second_hypotenuse_result) -> (exists pff_right_second_hypotenuse_result. (z) = pff_divisor_second_hypotenuse_result * pff_right_second_hypotenuse_result) -> pff_divisor_second_hypotenuse_result = 1)))))

Proof neighborhood

Direct theorem prerequisites

PF0014 pythagorean_primitive_legs_opposite_parity PF0016 pythagorean_primitive_hypotenuse_odd PF000Y pythagorean_primitive_pairwise_coprime

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

21 script commands · 5 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 (3)
pythagorean primitive legs opposite paritypythagorean primitive hypotenuse oddpythagorean primitive pairwise coprime
01Fix variables and assumptionsL1–4

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

  1. L1
    intro x
  2. L2
    intro y
  3. L3
    intro z
  4. L4
    intro hprimitive
02Separate the logical casesL5–5

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

  1. L5
    split
03Use earlier factsL6–10

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

  1. L6
    specialize pythagorean_primitive_legs_opposite_parity x
  2. L7
    specialize pythagorean_primitive_legs_opposite_parity y
  3. L8
    specialize pythagorean_primitive_legs_opposite_parity z
  4. L9
    apply pythagorean_primitive_legs_opposite_parity
  5. L10
    exact hprimitive
04Separate the logical casesL11–11

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

  1. L11
    split
05Use earlier factsL12–21

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

  1. L12
    specialize pythagorean_primitive_hypotenuse_odd x
  2. L13
    specialize pythagorean_primitive_hypotenuse_odd y
  3. L14
    specialize pythagorean_primitive_hypotenuse_odd z
  4. L15
    apply pythagorean_primitive_hypotenuse_odd
  5. L16
    exact hprimitive
  6. L17
    specialize pythagorean_primitive_pairwise_coprime x
  7. L18
    specialize pythagorean_primitive_pairwise_coprime y
  8. L19
    specialize pythagorean_primitive_pairwise_coprime z
  9. L20
    apply pythagorean_primitive_pairwise_coprime
  10. L21
    exact hprimitive

Library-wide reading audit

Original defined command ledger · 21 lines
  1. 0001intro x
  2. 0002intro y
  3. 0003intro z
  4. 0004intro hprimitive
  5. 0005split
  6. 0006specialize pythagorean_primitive_legs_opposite_parity x
  7. 0007specialize pythagorean_primitive_legs_opposite_parity y
  8. 0008specialize pythagorean_primitive_legs_opposite_parity z
  9. 0009apply pythagorean_primitive_legs_opposite_parity
  10. 0010exact hprimitive
  11. 0011split
  12. 0012specialize pythagorean_primitive_hypotenuse_odd x
  13. 0013specialize pythagorean_primitive_hypotenuse_odd y
  14. 0014specialize pythagorean_primitive_hypotenuse_odd z
  15. 0015apply pythagorean_primitive_hypotenuse_odd
  16. 0016exact hprimitive
  17. 0017specialize pythagorean_primitive_pairwise_coprime x
  18. 0018specialize pythagorean_primitive_pairwise_coprime y
  19. 0019specialize pythagorean_primitive_pairwise_coprime z
  20. 0020apply pythagorean_primitive_pairwise_coprime
  21. 0021exact hprimitive