PF001P · theorem body

pythagorean_even_square_has_even_root

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

An explicitly even natural square has an explicitly even natural square 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

∀ a. Even(a · a) → Even(a)

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

PD0009 Even

In local proof propositions

PD0010 Odd
Exact expanded first-order statement
forall a. (exists pp_even_pi_even_square. (a * a) = 2 * pp_even_pi_even_square) -> (exists pp_even_pi_even_root. (a) = 2 * pp_even_pi_even_root)

Proof neighborhood

Direct theorem prerequisites

PF0013 pythagorean_coordinate_parity_choice PF000I pythagorean_parameter_odd_square even_odd_exclusive_pointwise · Stable closed

Direct theorem dependents

PF001Q pythagorean_odd_square_sum_opposite_roots

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

17 script commands · 8 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 coordinate parity choicepythagorean parameter odd square
01Fix variables and assumptionsL1–2

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

  1. L1
    intro a
  2. L2
    intro heven
02Use earlier factsL3–3

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

  1. L3
    specialize pythagorean_coordinate_parity_choice a
03Separate the logical casesL4–4

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

  1. L4
    cases pythagorean_coordinate_parity_choice
04Use earlier factsL5–5

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

  1. L5
    exact pythagorean_coordinate_parity_choice_left
05Separate the logical casesL6–6

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

  1. L6
    exfalso
06Establish hoddL7–9

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

  1. L7
  2. L8
    apply pythagorean_parameter_odd_square
  3. L9
    exact pythagorean_coordinate_parity_choice_right
07Separate the logical casesL10–11

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

  1. L10
    cases heven
  2. L11
    cases hodd
08Use earlier factsL12–17

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

  1. L12
    specialize even_odd_exclusive_pointwise (a * a)
  2. L13
    specialize even_odd_exclusive_pointwise x
  3. L14
    specialize even_odd_exclusive_pointwise x1
  4. L15
    apply even_odd_exclusive_pointwise
  5. L16
    exact heven_witness
  6. L17
    exact hodd_witness

Library-wide reading audit

Original defined command ledger · 17 lines
  1. 0001intro a
  2. 0002intro heven
  3. 0003specialize pythagorean_coordinate_parity_choice a
  4. 0004cases pythagorean_coordinate_parity_choice
  5. 0005exact pythagorean_coordinate_parity_choice_left
  6. 0006exfalso
  7. 0007have hodd : Odd(a · a)
    Exact native replay linehave hodd : exists pp_odd_pi_square_contradiction. (a * a) = 2 * pp_odd_pi_square_contradiction + 1
  8. 0008apply pythagorean_parameter_odd_square
  9. 0009exact pythagorean_coordinate_parity_choice_right
  10. 0010cases heven
  11. 0011cases hodd
  12. 0012specialize even_odd_exclusive_pointwise (a * a)
  13. 0013specialize even_odd_exclusive_pointwise x
  14. 0014specialize even_odd_exclusive_pointwise x1
  15. 0015apply even_odd_exclusive_pointwise
  16. 0016exact heven_witness
  17. 0017exact hodd_witness