GI003F

gaussian_representation_exists

Every pair of arbitrary represented integers has an actually constructed canonical natural Gaussian code.

Alpha v34 checked-use · first admitted v28 · 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.

The natural-code carrier consists of genuine pairs of the existing signed integers; no new primitive arithmetic is trusted. The theorem constructs quotient, remainder, and actual norm witnesses. Gaussian gcd, unique factorization, and prime classification are separate targets.

Exact theorem in conservative defined notation

∀ a. ∀ b. ∀ c. ∀ d. ∃ z. ZPairRep(z,a,b,c,d)

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

Definition DAG

SignedBalance(z,p,n) · 2ZPairRep(z,ap,an,bp,bn) · 1

Actual proof prerequisites

signed_balance_total · checked external prerequisite
Original expanded first-order statement 
forall a b c d. exists z. (exists ge_representation_real_code_representation_total ge_representation_imaginary_code_representation_total. (((z) = ((ge_representation_real_code_representation_total) + (ge_representation_imaginary_code_representation_total)) * S ((ge_representation_real_code_representation_total) + (ge_representation_imaginary_code_representation_total)) + ((ge_representation_imaginary_code_representation_total) + (ge_representation_imaginary_code_representation_total))) /\ ((exists ge_balance_positive_representation_totalreal ge_balance_negative_representation_totalreal. (((((ge_representation_real_code_representation_total) = 2 * (ge_balance_positive_representation_totalreal) /\ (ge_balance_negative_representation_totalreal) = 0) \/ exists ge_signed_half_representation_totalrealdecode. (((ge_representation_real_code_representation_total) = 2 * ge_signed_half_representation_totalrealdecode + 1 /\ (ge_balance_positive_representation_totalreal) = 0) /\ (ge_balance_negative_representation_totalreal) = S ge_signed_half_representation_totalrealdecode))) /\ ((a) + ge_balance_negative_representation_totalreal = (b) + ge_balance_positive_representation_totalreal))) /\ (exists ge_balance_positive_representation_totalimaginary ge_balance_negative_representation_totalimaginary. (((((ge_representation_imaginary_code_representation_total) = 2 * (ge_balance_positive_representation_totalimaginary) /\ (ge_balance_negative_representation_totalimaginary) = 0) \/ exists ge_signed_half_representation_totalimaginarydecode. (((ge_representation_imaginary_code_representation_total) = 2 * ge_signed_half_representation_totalimaginarydecode + 1 /\ (ge_balance_positive_representation_totalimaginary) = 0) /\ (ge_balance_negative_representation_totalimaginary) = S ge_signed_half_representation_totalimaginarydecode))) /\ ((c) + ge_balance_negative_representation_totalimaginary = (d) + ge_balance_positive_representation_totalimaginary))))))

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 · 10 reading checkpoints · 2 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 a
  2. L2
    intro b
  3. L3
    intro c
  4. L4
    intro d
02Establish hrealL5–8

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply signed balance total.

  1. L5
    have hreal : ∃ rc. SignedBalance(rc,a,b)Definitions: SignedBalance(rc,a,b)Original native command in the exact edition
  2. L6
    specialize signed_balance_total a
  3. L7
    specialize signed_balance_total b
  4. L8
    apply signed_balance_total
03Separate the logical casesL9–9

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

  1. L9
    cases hreal
04Establish himaginaryL10–13

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply signed balance total.

  1. L10
    have himaginary : ∃ ic. SignedBalance(ic,c,d)Definitions: SignedBalance(ic,c,d)Original native command in the exact edition
  2. L11
    specialize signed_balance_total c
  3. L12
    specialize signed_balance_total d
  4. L13
    apply signed_balance_total
05Separate the logical casesL14–14

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

  1. L14
    cases himaginary
06Construct an explicit witnessL15–17

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

  1. L15
    exists ((x) + (x1)) * S ((x) + (x1)) + ((x1) + (x1))
  2. L16
    exists x
  3. L17
    exists x1
07Separate the logical casesL18–18

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

  1. L18
    split
08Calculate and transport equalitiesL19–19

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

  1. L19
    refl
09Separate the logical casesL20–20

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

  1. L20
    split
10Use earlier factsL21–22

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

  1. L21
    exact hreal_witness
  2. L22
    exact himaginary_witness

Library-wide reading audit

Original defined command ledger · 22 lines
  1. 0001intro a
  2. 0002intro b
  3. 0003intro c
  4. 0004intro d
  5. 0005have hreal : ∃ rc. SignedBalance(rc,a,b)
  6. 0006specialize signed_balance_total a
  7. 0007specialize signed_balance_total b
  8. 0008apply signed_balance_total
  9. 0009cases hreal
  10. 0010have himaginary : ∃ ic. SignedBalance(ic,c,d)
  11. 0011specialize signed_balance_total c
  12. 0012specialize signed_balance_total d
  13. 0013apply signed_balance_total
  14. 0014cases himaginary
  15. 0015exists ((x) + (x1)) * S ((x) + (x1)) + ((x1) + (x1))
  16. 0016exists x
  17. 0017exists x1
  18. 0018split
  19. 0019refl
  20. 0020split
  21. 0021exact hreal_witness
  22. 0022exact himaginary_witness