GF000A

gaussian_natural_real_representation

Embed a natural real coordinate using its actual even signed code and the unchanged Gaussian pair encoding.

Alpha v34 checked-use · first admitted v30 · independently kernel and Lean verified; not Stable

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.

Inputs are genuine canonical signed-pair codes, not arbitrary naturals. Products start at the actual Gaussian identity, whose code is six. The factor list uses the proved prime-divisor property; irreducibility alone is not silently renamed primality. Uniqueness supplies equal lengths, a bounded bijection, and an actual unit at each match, including repeated factors. Units have empty factorizations and zero is excluded. Sorted primary representatives, Gaussian prime classification, and Eisenstein factorization are separate targets.

Exact theorem in conservative defined notation

∀ k. ZPairRep((2 · k + 0) · S (2 · k + 0) + (0 + 0),k,0,0,0)

Every linked abbreviation expands hygienically to the identical original native formula.

Definition DAG

ZPairRep(z,ap,an,bp,bn) · 1

Actual proof prerequisites

zero_add · checked external prerequisite
Original expanded first-order statement 
forall k. (exists ge_representation_real_code_ring_natural_real ge_representation_imaginary_code_ring_natural_real. (((((2*k) + (0)) * S ((2*k) + (0)) + ((0) + (0))) = ((ge_representation_real_code_ring_natural_real) + (ge_representation_imaginary_code_ring_natural_real)) * S ((ge_representation_real_code_ring_natural_real) + (ge_representation_imaginary_code_ring_natural_real)) + ((ge_representation_imaginary_code_ring_natural_real) + (ge_representation_imaginary_code_ring_natural_real))) /\ ((exists ge_balance_positive_ring_natural_realreal ge_balance_negative_ring_natural_realreal. (((((ge_representation_real_code_ring_natural_real) = 2 * (ge_balance_positive_ring_natural_realreal) /\ (ge_balance_negative_ring_natural_realreal) = 0) \/ exists ge_signed_half_ring_natural_realrealdecode. (((ge_representation_real_code_ring_natural_real) = 2 * ge_signed_half_ring_natural_realrealdecode + 1 /\ (ge_balance_positive_ring_natural_realreal) = 0) /\ (ge_balance_negative_ring_natural_realreal) = S ge_signed_half_ring_natural_realrealdecode))) /\ ((k) + ge_balance_negative_ring_natural_realreal = (0) + ge_balance_positive_ring_natural_realreal))) /\ (exists ge_balance_positive_ring_natural_realimaginary ge_balance_negative_ring_natural_realimaginary. (((((ge_representation_imaginary_code_ring_natural_real) = 2 * (ge_balance_positive_ring_natural_realimaginary) /\ (ge_balance_negative_ring_natural_realimaginary) = 0) \/ exists ge_signed_half_ring_natural_realimaginarydecode. (((ge_representation_imaginary_code_ring_natural_real) = 2 * ge_signed_half_ring_natural_realimaginarydecode + 1 /\ (ge_balance_positive_ring_natural_realimaginary) = 0) /\ (ge_balance_negative_ring_natural_realimaginary) = S ge_signed_half_ring_natural_realimaginarydecode))) /\ ((0) + ge_balance_negative_ring_natural_realimaginary = (0) + ge_balance_positive_ring_natural_realimaginary))))))

Complete tactic proof in conservative notation

All 22 original proof lines are preserved. Only local proposition formulas are abbreviated; every abbreviation has an exact binder-safe expansion check. The linked exact edition contains the unchanged replay script.

Read the argument

Proof checkpoints

22 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 assumptionsL1–1

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

  1. L1
    intro k
02Construct an explicit witnessL2–3

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

  1. L2
    exists (2*k)
  2. L3
    exists (0)
03Separate the logical casesL4–4

Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.

  1. L4
    split
04Calculate and transport equalitiesL5–5

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

  1. L5
    refl
05Separate the logical casesL6–6

Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.

  1. L6
    split
06Construct an explicit witnessL7–8

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

  1. L7
    exists (k)
  2. L8
    exists (0)
07Separate the logical casesL9–11

Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.

  1. L9
    split
  2. L10
    left
  3. L11
    split
08Calculate and transport equalitiesL12–14

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

  1. L12
    refl
  2. L13
    refl
  3. L14
    simp [zero_add]
09Construct an explicit witnessL15–16

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

  1. L15
    exists (0)
  2. L16
    exists (0)
10Separate the logical casesL17–19

Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.

  1. L17
    split
  2. L18
    left
  3. L19
    split
11Calculate and transport equalitiesL20–22

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

  1. L20
    simp
  2. L21
    refl
  3. L22
    simp

Library-wide reading audit

Original defined command ledger · 22 lines
  1. 0001intro k
  2. 0002exists (2*k)
  3. 0003exists (0)
  4. 0004split
  5. 0005refl
  6. 0006split
  7. 0007exists (k)
  8. 0008exists (0)
  9. 0009split
  10. 0010left
  11. 0011split
  12. 0012refl
  13. 0013refl
  14. 0014simp [zero_add]
  15. 0015exists (0)
  16. 0016exists (0)
  17. 0017split
  18. 0018left
  19. 0019split
  20. 0020simp
  21. 0021refl
  22. 0022simp