PF001S

pythagorean_half_roots_even_leg

Alpha v34 independently verified · alpha_closed; checked-use authorized; not Stable

Natural square-root injectivity identifies the even leg with twice the product of the constructed parameters.

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 t h m n. (a + t) * t = h * h -> a + t = m * m -> t = n * n -> 2 * h = 2 * (m * n)

Constructive proof overview

Generated structural guide

Natural square-root injectivity identifies the even leg with twice the product of the constructed parameters.

The unchanged tactic script uses 2 declared prerequisites and contains 22 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

four_square_product_square Alpha theorem; checked-use authorized PF001B square_eq_injective

Direct dependents

PF001W pythagorean_half_factors_extract_parameters

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

22 script commands · 4 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)
square eq injective
01Fix variables and assumptionsL1–8

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

  1. L1
    intro a
  2. L2
    intro t
  3. L3
    intro h
  4. L4
    intro m
  5. L5
    intro n
  6. L6
    intro hproduct
  7. L7
    intro hm
  8. L8
    intro hn
02Establish hsquaresL9–18

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

  1. L9
    have hsquares : h * h = (m * n) * (m * n)
  2. L10
    trans (a + t) * t
  3. L11
    symm
  4. L12
    exact hproduct
  5. L13
    trans (m * m) * (n * n)
  6. L14
    congr
  7. L15
    exact hm
  8. L16
    exact hn
  9. L17
    symm
  10. L18
    apply four_square_product_square
03Calculate and transport equalitiesL19–20

Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.

  1. L19
    congr
  2. L20
    refl
04Use earlier factsL21–22

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

  1. L21
    apply square_eq_injective
  2. L22
    exact hsquares

Library-wide reading audit

Original exact command ledger · 22 lines
  1. 0001intro a
  2. 0002intro t
  3. 0003intro h
  4. 0004intro m
  5. 0005intro n
  6. 0006intro hproduct
  7. 0007intro hm
  8. 0008intro hn
  9. 0009have hsquares : h * h = (m * n) * (m * n)
  10. 0010trans (a + t) * t
  11. 0011symm
  12. 0012exact hproduct
  13. 0013trans (m * m) * (n * n)
  14. 0014congr
  15. 0015exact hm
  16. 0016exact hn
  17. 0017symm
  18. 0018apply four_square_product_square
  19. 0019congr
  20. 0020refl
  21. 0021apply square_eq_injective
  22. 0022exact hsquares