TS0001 · theorem body

odd_square_is_four_multiple_plus_one

Alpha v34 checked-use · independently kernel and Lean verified; 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.

Statement with defined notation

∀ z. ∀ k. z = 2 · k + 1 → Mod4One(z · z)

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

PD0011 Mod4One

In local proof propositions

none
Exact expanded first-order statement
forall z k. z = 2 * k + 1 -> exists q. z * z = 4 * q + 1

Proof neighborhood

Direct theorem prerequisites

mul_add · Stable closed add_mul · Stable closed mul_assoc · Stable closed mul_comm · Stable closed add_assoc · Stable closed add_comm · Stable closed zero_add · Stable closed mul_succ_left · Stable closed one_mul · Stable closed mul_zero_left · Stable closed mul_double_right · Stable closed four_mul_eq_double_double · 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

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.

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 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 defined 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