PF0000

pythagorean_double_product

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

The two symmetric Euclidean cross products are exactly twice their natural product.

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 m n. m * n + n * m = 2 * (m * n)

Constructive proof overview

Generated structural guide

The two symmetric Euclidean cross products are exactly twice their natural product.

The unchanged tactic script uses 3 declared prerequisites and contains 12 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

mul_comm Stable theorem; checked-use authorized mul_succ_left Stable theorem; checked-use authorized one_mul Stable theorem; checked-use authorized

Direct dependents

PF0001 pythagorean_euclidean_identity

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

12 script commands · 8 reading checkpoints · 0 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.

01Fix variables and assumptionsL1–2

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

  1. L1
    intro m
  2. L2
    intro n
02Calculate and transport equalitiesL3–5

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

  1. L3
    trans m * n + m * n
  2. L4
    congr
  3. L5
    refl
03Use earlier factsL6–6

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

  1. L6
    apply mul_comm
04Calculate and transport equalitiesL7–8

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

  1. L7
    symm
  2. L8
    trans 1 * (m * n) + m * n
05Use earlier factsL9–9

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

  1. L9
    apply mul_succ_left
06Calculate and transport equalitiesL10–10

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

  1. L10
    congr
07Use earlier factsL11–11

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

  1. L11
    apply one_mul
08Calculate and transport equalitiesL12–12

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

  1. L12
    refl

Library-wide reading audit

Original exact command ledger · 12 lines
  1. 0001intro m
  2. 0002intro n
  3. 0003trans m * n + m * n
  4. 0004congr
  5. 0005refl
  6. 0006apply mul_comm
  7. 0007symm
  8. 0008trans 1 * (m * n) + m * n
  9. 0009apply mul_succ_left
  10. 0010congr
  11. 0011apply one_mul
  12. 0012refl