PF0016 · theorem body

pythagorean_primitive_hypotenuse_odd

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

Every primitive natural Pythagorean triple has an explicitly witnessed odd hypotenuse.

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) → Odd(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 PD0010 Odd

In local proof propositions

CF0016 OppositeParity PD0010 Odd
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 q. z = 2 * q + 1

Proof neighborhood

Direct theorem prerequisites

PF0014 pythagorean_primitive_legs_opposite_parity PF000M pythagorean_opposite_parity_hypotenuse_odd PF0015 pythagorean_odd_square_has_odd_root

Direct theorem dependents

PF0017 pythagorean_primitive_normal_form PF001V pythagorean_odd_even_half_factors PF001X pythagorean_primitive_odd_even_inverse PF002K fermat_four_primitive_odd_even_descent

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

24 script commands · 9 reading checkpoints · 2 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 opposite parity hypotenuse oddpythagorean odd square has odd root
01Fix variables and assumptionsL1–4

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

  1. L1
    intro a
  2. L2
    intro b
  3. L3
    intro c
  4. L4
    intro hprimitive
02Establish hparityL5–10

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply pythagorean primitive legs opposite parity.

  1. L5
    have hparity : OppositeParity(a,b)Definitions: OppositeParity(a,b)Original native command in the exact edition
  2. L6
    specialize pythagorean_primitive_legs_opposite_parity a
  3. L7
    specialize pythagorean_primitive_legs_opposite_parity b
  4. L8
    specialize pythagorean_primitive_legs_opposite_parity c
  5. L9
    apply pythagorean_primitive_legs_opposite_parity
  6. L10
    exact hprimitive
03Establish hsumL11–17

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply pythagorean opposite parity hypotenuse odd.

  1. L11
    have hsum : Odd(a · a + b · b)Definitions: Odd(a · a + b · b)Original native command in the exact edition
  2. L12
    specialize pythagorean_opposite_parity_hypotenuse_odd a
  3. L13
    specialize pythagorean_opposite_parity_hypotenuse_odd b
  4. L14
    apply pythagorean_opposite_parity_hypotenuse_odd
  5. L15
    exact hparity
  6. L16
    specialize pythagorean_odd_square_has_odd_root c
  7. L17
    apply pythagorean_odd_square_has_odd_root
04Separate the logical casesL18–18

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

  1. L18
    cases hsum
05Construct an explicit witnessL19–19

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

  1. L19
    exists x
06Calculate and transport equalitiesL20–20

Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.

  1. L20
    trans a * a + b * b
07Separate the logical casesL21–21

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

  1. L21
    cases hprimitive
08Calculate and transport equalitiesL22–22

Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.

  1. L22
    symm
09Use earlier factsL23–24

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

  1. L23
    exact hprimitive_left
  2. L24
    exact hsum_witness

Library-wide reading audit

Original defined command ledger · 24 lines
  1. 0001intro a
  2. 0002intro b
  3. 0003intro c
  4. 0004intro hprimitive
  5. 0005have hparity : OppositeParity(a,b)
    Exact native replay linehave hparity : (((exists pp_even_hypotenuse_local_first_even. (a) = 2 * pp_even_hypotenuse_local_first_even) /\ (exists pp_odd_hypotenuse_local_second_odd. (b) = 2 * pp_odd_hypotenuse_local_second_odd + 1)) \/ ((exists pp_odd_hypotenuse_local_first_odd. (a) = 2 * pp_odd_hypotenuse_local_first_odd + 1) /\ (exists pp_even_hypotenuse_local_second_even. (b) = 2 * pp_even_hypotenuse_local_second_even)))
  6. 0006specialize pythagorean_primitive_legs_opposite_parity a
  7. 0007specialize pythagorean_primitive_legs_opposite_parity b
  8. 0008specialize pythagorean_primitive_legs_opposite_parity c
  9. 0009apply pythagorean_primitive_legs_opposite_parity
  10. 0010exact hprimitive
  11. 0011have hsum : Odd(a · a + b · b)
    Exact native replay linehave hsum : exists q. a * a + b * b = 2 * q + 1
  12. 0012specialize pythagorean_opposite_parity_hypotenuse_odd a
  13. 0013specialize pythagorean_opposite_parity_hypotenuse_odd b
  14. 0014apply pythagorean_opposite_parity_hypotenuse_odd
  15. 0015exact hparity
  16. 0016specialize pythagorean_odd_square_has_odd_root c
  17. 0017apply pythagorean_odd_square_has_odd_root
  18. 0018cases hsum
  19. 0019exists x
  20. 0020trans a * a + b * b
  21. 0021cases hprimitive
  22. 0022symm
  23. 0023exact hprimitive_left
  24. 0024exact hsum_witness