TS0001

odd_square_is_four_multiple_plus_one

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

The square of an odd natural has an explicit residue-one witness modulo four.

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 z k. z = 2 * k + 1 -> exists q. z * z = 4 * q + 1

Constructive proof overview

Generated structural guide

The square of an odd natural has an explicit residue-one witness modulo four.

The unchanged tactic script uses 12 declared prerequisites and contains 29 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_add Stable theorem; checked-use authorized add_mul Stable theorem; checked-use authorized mul_assoc Stable theorem; checked-use authorized mul_comm Stable theorem; checked-use authorized add_assoc Stable theorem; checked-use authorized add_comm Stable theorem; checked-use authorized zero_add Stable theorem; checked-use authorized mul_succ_left Stable theorem; checked-use authorized one_mul Stable theorem; checked-use authorized mul_zero_left Stable theorem; checked-use authorized mul_double_right Stable theorem; checked-use authorized four_mul_eq_double_double Stable theorem; checked-use authorized

Direct dependents

TS0002 square_mod_four_zero_or_one

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

29 script commands · 10 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 assumptionsL1–3

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

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

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

  1. L4
    exists k * k + k
03Calculate and transport equalitiesL5–10

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

  1. L5
    rewrite hodd
  2. L6
    rewrite hodd
  3. L7
    simp [mul_add, add_mul, mul_assoc, mul_comm, add_assoc, add_comm, zero_add]
  4. L8
    congr
  5. L9
    trans (2 * k + 2 * k) + 2 * (k * (2 * k))
  6. L10
    symm
04Use earlier factsL11–11

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

  1. L11
    apply add_assoc
05Calculate and transport equalitiesL12–19

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

  1. L12
    trans (2 * (2 * k)) + 2 * (k * (2 * k))
  2. L13
    congr
  3. L14
    symm
  4. L15
    simp [mul_succ_left, one_mul, mul_zero_left, zero_add]
  5. L16
    refl
  6. L17
    trans (4 * k) + 2 * (k * (2 * k))
  7. L18
    congr
  8. L19
    symm
06Use earlier factsL20–20

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

  1. L20
    apply four_mul_eq_double_double
07Calculate and transport equalitiesL21–26

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

  1. L21
    refl
  2. L22
    congr
  3. L23
    refl
  4. L24
    trans 2 * (2 * (k * k))
  5. L25
    congr
  6. L26
    refl
08Use earlier factsL27–27

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

  1. L27
    apply mul_double_right
09Calculate and transport equalitiesL28–28

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

  1. L28
    symm
10Use earlier factsL29–29

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

  1. L29
    apply four_mul_eq_double_double

Library-wide reading audit

Original exact command ledger · 29 lines
  1. 0001intro z
  2. 0002intro k
  3. 0003intro hodd
  4. 0004exists k * k + k
  5. 0005rewrite hodd
  6. 0006rewrite hodd
  7. 0007simp [mul_add, add_mul, mul_assoc, mul_comm, add_assoc, add_comm, zero_add]
  8. 0008congr
  9. 0009trans (2 * k + 2 * k) + 2 * (k * (2 * k))
  10. 0010symm
  11. 0011apply add_assoc
  12. 0012trans (2 * (2 * k)) + 2 * (k * (2 * k))
  13. 0013congr
  14. 0014symm
  15. 0015simp [mul_succ_left, one_mul, mul_zero_left, zero_add]
  16. 0016refl
  17. 0017trans (4 * k) + 2 * (k * (2 * k))
  18. 0018congr
  19. 0019symm
  20. 0020apply four_mul_eq_double_double
  21. 0021refl
  22. 0022congr
  23. 0023refl
  24. 0024trans 2 * (2 * (k * k))
  25. 0025congr
  26. 0026refl
  27. 0027apply mul_double_right
  28. 0028symm
  29. 0029apply four_mul_eq_double_double