PF001K

pythagorean_half_sum_reassociation

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

The sum of an odd leg and its even-shifted hypotenuse is twice the upper half-factor.

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. (a + 2 * t) + a = 2 * (a + t)

Constructive proof overview

Generated structural guide

The sum of an odd leg and its even-shifted hypotenuse is twice the upper half-factor.

The unchanged tactic script uses 3 declared prerequisites and contains 3 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

two_mul_eq_add_self Alpha theorem; checked-use authorized add_assoc Stable theorem; checked-use authorized add_comm Stable theorem; checked-use authorized

Direct dependents

PF001M pythagorean_half_product_is_square

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

3 script commands · 2 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 a
  2. L2
    intro t
02Calculate and transport equalitiesL3–3

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

  1. L3
    simp [two_mul_eq_add_self, add_assoc, add_comm]

Library-wide reading audit

Original exact command ledger · 3 lines
  1. 0001intro a
  2. 0002intro t
  3. 0003simp [two_mul_eq_add_self, add_assoc, add_comm]