PF002C · theorem body

fermat_four_even_square_root

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

An even square has an explicitly even root, proved by constructive parity cases.

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 k. a * a = 2 * k) -> exists k. a = 2 * k

Proof neighborhood

Direct theorem prerequisites

parity_cases · Stable closed PF000I pythagorean_parameter_odd_square even_odd_exclusive_pointwise · Stable closed

Direct theorem dependents

PF002E fermat_four_odd_double_square_factors

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

20 script commands · 11 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 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 hsquare
02Use earlier factsL3–3

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

  1. L3
    specialize parity_cases a
03Separate the logical casesL4–5

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

  1. L4
    cases parity_cases
  2. L5
    cases parity_cases_witness
04Construct an explicit witnessL6–6

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

  1. L6
    exists x
05Use earlier factsL7–7

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

  1. L7
    exact parity_cases_witness_left
06Separate the logical casesL8–8

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

  1. L8
    exfalso
07Establish hoddL9–10

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

  1. L9
  2. L10
    apply pythagorean_parameter_odd_square
08Construct an explicit witnessL11–11

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

  1. L11
    exists x
09Use earlier factsL12–12

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

  1. L12
    exact parity_cases_witness_right
10Separate the logical casesL13–14

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

  1. L13
    cases hsquare
  2. L14
    cases hodd
11Use earlier factsL15–20

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

  1. L15
    specialize even_odd_exclusive_pointwise (a * a)
  2. L16
    specialize even_odd_exclusive_pointwise (x1)
  3. L17
    specialize even_odd_exclusive_pointwise (x2)
  4. L18
    apply even_odd_exclusive_pointwise
  5. L19
    exact hsquare_witness
  6. L20
    exact hodd_witness

Library-wide reading audit

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