PF0006 · theorem body

pythagorean_euclidean_even_leg_not_odd

Alpha v34 checked-use · independently kernel and Lean verified; not Stable

The explicitly even Euclidean cross-product coordinate cannot simultaneously have an odd witness.

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 m n q. 2 * (m * n) = 2 * q + 1 -> false

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 m n q. 2 * (m * n) = 2 * q + 1 -> false

Proof neighborhood

Direct theorem prerequisites

even_odd_exclusive_k1 · Alpha closed

Direct theorem dependents

none

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

10 script commands · 4 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.

Definition notation is shown below. Open the paired exact edition for the original native formulas. Source pairing is not a new equivalence certificate.

01Fix variables and assumptionsL1–4

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

  1. L1
    intro m
  2. L2
    intro n
  3. L3
    intro q
  4. L4
    intro hodd
02Use earlier factsL5–8

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

  1. L5
    specialize even_odd_exclusive_k1 (2 * (m * n))
  2. L6
    specialize even_odd_exclusive_k1 (m * n)
  3. L7
    specialize even_odd_exclusive_k1 q
  4. L8
    apply even_odd_exclusive_k1
03Calculate and transport equalitiesL9–9

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

  1. L9
    refl
04Use earlier factsL10–10

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

  1. L10
    exact hodd

Library-wide reading audit

Original defined command ledger · 10 lines
  1. 0001intro m
  2. 0002intro n
  3. 0003intro q
  4. 0004intro hodd
  5. 0005specialize even_odd_exclusive_k1 (2 * (m * n))
  6. 0006specialize even_odd_exclusive_k1 (m * n)
  7. 0007specialize even_odd_exclusive_k1 q
  8. 0008apply even_odd_exclusive_k1
  9. 0009refl
  10. 0010exact hodd