PF0015 · theorem body

pythagorean_odd_square_has_odd_root

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

An explicitly odd natural square has an explicitly odd natural root.

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

∀ z. Odd(z · 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

PD0010 Odd

In local proof propositions

PD0009 Even
Exact expanded first-order statement
forall z. (exists q. z * z = 2 * q + 1) -> exists q. z = 2 * q + 1

Proof neighborhood

Direct theorem prerequisites

parity_cases · Stable closed PF000H pythagorean_parameter_even_square even_odd_exclusive_pointwise · Stable closed

Direct theorem dependents

PF0016 pythagorean_primitive_hypotenuse_odd PF001Q pythagorean_odd_square_sum_opposite_roots 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

21 script commands · 10 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 (1)
pythagorean parameter even square
01Fix variables and assumptionsL1–2

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

  1. L1
    intro z
  2. L2
    intro hodd
02Use earlier factsL3–3

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

  1. L3
    specialize parity_cases z
03Separate the logical casesL4–6

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

  1. L4
    cases parity_cases
  2. L5
    cases parity_cases_witness
  3. L6
    exfalso
04Establish hevenL7–9

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

  1. L7
    have heven : Even(z · z)Definitions: Even(z · z)Original native command in the exact edition
  2. L8
    specialize pythagorean_parameter_even_square z
  3. L9
    apply pythagorean_parameter_even_square
05Construct an explicit witnessL10–10

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

  1. L10
    exists x
06Use earlier factsL11–11

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

  1. L11
    exact parity_cases_witness_left
07Separate the logical casesL12–13

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

  1. L12
    cases heven
  2. L13
    cases hodd
08Use earlier factsL14–19

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

  1. L14
    specialize even_odd_exclusive_pointwise (z * z)
  2. L15
    specialize even_odd_exclusive_pointwise x1
  3. L16
    specialize even_odd_exclusive_pointwise x2
  4. L17
    apply even_odd_exclusive_pointwise
  5. L18
    exact heven_witness
  6. L19
    exact hodd_witness
09Construct an explicit witnessL20–20

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

  1. L20
    exists x
10Use earlier factsL21–21

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

  1. L21
    exact parity_cases_witness_right

Library-wide reading audit

Original defined command ledger · 21 lines
  1. 0001intro z
  2. 0002intro hodd
  3. 0003specialize parity_cases z
  4. 0004cases parity_cases
  5. 0005cases parity_cases_witness
  6. 0006exfalso
  7. 0007have heven : Even(z · z)
    Exact native replay linehave heven : exists q. z * z = 2 * q
  8. 0008specialize pythagorean_parameter_even_square z
  9. 0009apply pythagorean_parameter_even_square
  10. 0010exists x
  11. 0011exact parity_cases_witness_left
  12. 0012cases heven
  13. 0013cases hodd
  14. 0014specialize even_odd_exclusive_pointwise (z * z)
  15. 0015specialize even_odd_exclusive_pointwise x1
  16. 0016specialize even_odd_exclusive_pointwise x2
  17. 0017apply even_odd_exclusive_pointwise
  18. 0018exact heven_witness
  19. 0019exact hodd_witness
  20. 0020exists x
  21. 0021exact parity_cases_witness_right