PF001S · theorem body

pythagorean_half_roots_even_leg

Alpha v34 checked-use · independently kernel and Lean verified; 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.

Statement with defined notation

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

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

none

In local proof propositions

none
Exact expanded first-order statement
forall a t h m n. (a + t) * t = h * h -> a + t = m * m -> t = n * n -> 2 * h = 2 * (m * n)

Proof neighborhood

Direct theorem prerequisites

four_square_product_square · Alpha closed PF001B square_eq_injective

Direct theorem dependents

PF001W pythagorean_half_factors_extract_parameters

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

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.

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)
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 defined 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