TS0000 · theorem body

even_square_is_four_multiple

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

The square of an even natural has an explicit multiple-of-four 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 z k. z = 2 * k -> exists q. z * z = 4 * q + 0

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 z k. z = 2 * k -> exists q. z * z = 4 * q + 0

Proof neighborhood

Direct theorem prerequisites

mul_assoc · Stable closed mul_comm · Stable closed

Direct theorem dependents

TS0002 square_mod_four_zero_or_one

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

28 script commands · 11 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 assumptionsL1–3

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

  1. L1
    intro z
  2. L2
    intro k
  3. L3
    intro heven
02Construct an explicit witnessL4–4

Supply the displayed value, then prove that it has the required property.

  1. L4
    exists k * k
03Calculate and transport equalitiesL5–11

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

  1. L5
    rewrite heven
  2. L6
    rewrite heven
  3. L7
    simp [mul_assoc, mul_comm]
  4. L8
    trans 2 * ((k * 2) * k)
  5. L9
    congr
  6. L10
    refl
  7. L11
    symm
04Use earlier factsL12–12

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

  1. L12
    apply mul_assoc
05Calculate and transport equalitiesL13–16

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

  1. L13
    trans 2 * ((2 * k) * k)
  2. L14
    congr
  3. L15
    refl
  4. L16
    congr
06Use earlier factsL17–17

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

  1. L17
    apply mul_comm
07Calculate and transport equalitiesL18–21

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

  1. L18
    refl
  2. L19
    trans 2 * (2 * (k * k))
  3. L20
    congr
  4. L21
    refl
08Use earlier factsL22–22

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

  1. L22
    apply mul_assoc
09Calculate and transport equalitiesL23–24

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

  1. L23
    trans (2 * 2) * (k * k)
  2. L24
    symm
10Use earlier factsL25–25

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

  1. L25
    apply mul_assoc
11Calculate and transport equalitiesL26–28

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

  1. L26
    congr
  2. L27
    norm_num
  3. L28
    refl

Library-wide reading audit

Original defined command ledger · 28 lines
  1. 0001intro z
  2. 0002intro k
  3. 0003intro heven
  4. 0004exists k * k
  5. 0005rewrite heven
  6. 0006rewrite heven
  7. 0007simp [mul_assoc, mul_comm]
  8. 0008trans 2 * ((k * 2) * k)
  9. 0009congr
  10. 0010refl
  11. 0011symm
  12. 0012apply mul_assoc
  13. 0013trans 2 * ((2 * k) * k)
  14. 0014congr
  15. 0015refl
  16. 0016congr
  17. 0017apply mul_comm
  18. 0018refl
  19. 0019trans 2 * (2 * (k * k))
  20. 0020congr
  21. 0021refl
  22. 0022apply mul_assoc
  23. 0023trans (2 * 2) * (k * k)
  24. 0024symm
  25. 0025apply mul_assoc
  26. 0026congr
  27. 0027norm_num
  28. 0028refl