PF001P

pythagorean_even_square_has_even_root

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

Exact expanded first-order arithmetic 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)

Constructive proof overview

Generated structural guide

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

The unchanged tactic script uses 3 declared prerequisites and contains 17 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

PF0013 pythagorean_coordinate_parity_choice PF000I pythagorean_parameter_odd_square even_odd_exclusive_pointwise Stable theorem; checked-use authorized

Direct dependents

PF001Q pythagorean_odd_square_sum_opposite_roots

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

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.

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
    have hodd : exists pp_odd_pi_square_contradiction. (a * a) = 2 * pp_odd_pi_square_contradiction + 1
  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 exact 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 : 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