PF002C

fermat_four_even_square_root

Alpha v34 independently verified · alpha_closed; checked-use authorized; 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.

Exact expanded first-order arithmetic statement

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

Constructive proof overview

Generated structural guide

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

The unchanged tactic script uses 3 declared prerequisites and contains 20 exact native proof lines.

dependency-curried kernel-checked theorem body; Alpha enrollment and checked-use authority follow separately sealed release evidence; Stable membership remains unchanged

Proof neighborhood

Direct dependencies

parity_cases Stable theorem; checked-use authorized PF000I pythagorean_parameter_odd_square even_odd_exclusive_pointwise Stable theorem; checked-use authorized

Direct dependents

PF002E fermat_four_odd_double_square_factors

Formal native tactic body

Dependencies are introduced as named hypotheses before line 1. Local theorem links identify exact declared prerequisites. This exact body belongs to a complete independently kernel-checked constructive proof bundle and has Alpha checked-use authority; it does not imply Stable membership.

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.

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
    have hodd : exists k. a * a = 2 * k + 1
  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 exact 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 : 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