GF0020

gaussian_codes_equal_of_representations

Equal represented Gaussian integers have literally equal canonical natural codes.

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

∀ z. ∀ w. ∀ ap. ∀ an. ∀ bp. ∀ bn. ∀ cp. ∀ cn. ∀ dp. ∀ dn. ZPairRep(z,ap,an,bp,bn) → ZPairRep(w,cp,cn,dp,dn) → ap + cn = cp + an ∧ bp + dn = dp + bn → z = w

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

Definition DAG

ZPairRep(z,ap,an,bp,bn) · 2

Actual proof prerequisites

gaussian_representation_functional · checked external prerequisitegaussian_representation_integer_transport · checked external prerequisitegaussian_equal_symmetric · checked external prerequisite
Original expanded first-order statement 
forall z w ap an bp bn cp cn dp dn. (exists ge_representation_real_code_equal_code_first ge_representation_imaginary_code_equal_code_first. (((z) = ((ge_representation_real_code_equal_code_first) + (ge_representation_imaginary_code_equal_code_first)) * S ((ge_representation_real_code_equal_code_first) + (ge_representation_imaginary_code_equal_code_first)) + ((ge_representation_imaginary_code_equal_code_first) + (ge_representation_imaginary_code_equal_code_first))) /\ ((exists ge_balance_positive_equal_code_firstreal ge_balance_negative_equal_code_firstreal. (((((ge_representation_real_code_equal_code_first) = 2 * (ge_balance_positive_equal_code_firstreal) /\ (ge_balance_negative_equal_code_firstreal) = 0) \/ exists ge_signed_half_equal_code_firstrealdecode. (((ge_representation_real_code_equal_code_first) = 2 * ge_signed_half_equal_code_firstrealdecode + 1 /\ (ge_balance_positive_equal_code_firstreal) = 0) /\ (ge_balance_negative_equal_code_firstreal) = S ge_signed_half_equal_code_firstrealdecode))) /\ ((ap) + ge_balance_negative_equal_code_firstreal = (an) + ge_balance_positive_equal_code_firstreal))) /\ (exists ge_balance_positive_equal_code_firstimaginary ge_balance_negative_equal_code_firstimaginary. (((((ge_representation_imaginary_code_equal_code_first) = 2 * (ge_balance_positive_equal_code_firstimaginary) /\ (ge_balance_negative_equal_code_firstimaginary) = 0) \/ exists ge_signed_half_equal_code_firstimaginarydecode. (((ge_representation_imaginary_code_equal_code_first) = 2 * ge_signed_half_equal_code_firstimaginarydecode + 1 /\ (ge_balance_positive_equal_code_firstimaginary) = 0) /\ (ge_balance_negative_equal_code_firstimaginary) = S ge_signed_half_equal_code_firstimaginarydecode))) /\ ((bp) + ge_balance_negative_equal_code_firstimaginary = (bn) + ge_balance_positive_equal_code_firstimaginary)))))) -> (exists ge_representation_real_code_equal_code_second ge_representation_imaginary_code_equal_code_second. (((w) = ((ge_representation_real_code_equal_code_second) + (ge_representation_imaginary_code_equal_code_second)) * S ((ge_representation_real_code_equal_code_second) + (ge_representation_imaginary_code_equal_code_second)) + ((ge_representation_imaginary_code_equal_code_second) + (ge_representation_imaginary_code_equal_code_second))) /\ ((exists ge_balance_positive_equal_code_secondreal ge_balance_negative_equal_code_secondreal. (((((ge_representation_real_code_equal_code_second) = 2 * (ge_balance_positive_equal_code_secondreal) /\ (ge_balance_negative_equal_code_secondreal) = 0) \/ exists ge_signed_half_equal_code_secondrealdecode. (((ge_representation_real_code_equal_code_second) = 2 * ge_signed_half_equal_code_secondrealdecode + 1 /\ (ge_balance_positive_equal_code_secondreal) = 0) /\ (ge_balance_negative_equal_code_secondreal) = S ge_signed_half_equal_code_secondrealdecode))) /\ ((cp) + ge_balance_negative_equal_code_secondreal = (cn) + ge_balance_positive_equal_code_secondreal))) /\ (exists ge_balance_positive_equal_code_secondimaginary ge_balance_negative_equal_code_secondimaginary. (((((ge_representation_imaginary_code_equal_code_second) = 2 * (ge_balance_positive_equal_code_secondimaginary) /\ (ge_balance_negative_equal_code_secondimaginary) = 0) \/ exists ge_signed_half_equal_code_secondimaginarydecode. (((ge_representation_imaginary_code_equal_code_second) = 2 * ge_signed_half_equal_code_secondimaginarydecode + 1 /\ (ge_balance_positive_equal_code_secondimaginary) = 0) /\ (ge_balance_negative_equal_code_secondimaginary) = S ge_signed_half_equal_code_secondimaginarydecode))) /\ ((dp) + ge_balance_negative_equal_code_secondimaginary = (dn) + ge_balance_positive_equal_code_secondimaginary)))))) -> (((((ap) + (cn)) = ((cp) + (an))) /\ (((bp) + (dn)) = ((dp) + (bn))))) -> z=w

Complete tactic proof in conservative notation

All 42 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

42 script commands · 5 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 assumptionsL1–10

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

  1. L1
    intro z
  2. L2
    intro w
  3. L3
    intro ap
  4. L4
    intro an
  5. L5
    intro bp
  6. L6
    intro bn
  7. L7
    intro cp
  8. L8
    intro cn
  9. L9
    intro dp
  10. L10
    intro dn
02Fix variables and assumptionsL11–13

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

  1. L11
    intro hz
  2. L12
    intro hw
  3. L13
    intro heq
03Use earlier factsL14–23

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

  1. L14
    specialize gaussian_representation_functional (z)
  2. L15
    specialize gaussian_representation_functional (w)
  3. L16
    specialize gaussian_representation_functional (ap)
  4. L17
    specialize gaussian_representation_functional (an)
  5. L18
    specialize gaussian_representation_functional (bp)
  6. L19
    specialize gaussian_representation_functional (bn)
  7. L20
    apply gaussian_representation_functional
  8. L21
    exact hz
  9. L22
    specialize gaussian_representation_integer_transport (w)
  10. L23
    specialize gaussian_representation_integer_transport (cp)
04Use earlier factsL24–33

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

  1. L24
    specialize gaussian_representation_integer_transport (cn)
  2. L25
    specialize gaussian_representation_integer_transport (dp)
  3. L26
    specialize gaussian_representation_integer_transport (dn)
  4. L27
    specialize gaussian_representation_integer_transport (ap)
  5. L28
    specialize gaussian_representation_integer_transport (an)
  6. L29
    specialize gaussian_representation_integer_transport (bp)
  7. L30
    specialize gaussian_representation_integer_transport (bn)
  8. L31
    apply gaussian_representation_integer_transport
  9. L32
    specialize gaussian_equal_symmetric (ap)
  10. L33
    specialize gaussian_equal_symmetric (an)
05Use earlier factsL34–42

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

  1. L34
    specialize gaussian_equal_symmetric (bp)
  2. L35
    specialize gaussian_equal_symmetric (bn)
  3. L36
    specialize gaussian_equal_symmetric (cp)
  4. L37
    specialize gaussian_equal_symmetric (cn)
  5. L38
    specialize gaussian_equal_symmetric (dp)
  6. L39
    specialize gaussian_equal_symmetric (dn)
  7. L40
    apply gaussian_equal_symmetric
  8. L41
    exact heq
  9. L42
    exact hw

Library-wide reading audit

Original defined command ledger · 42 lines
  1. 0001intro z
  2. 0002intro w
  3. 0003intro ap
  4. 0004intro an
  5. 0005intro bp
  6. 0006intro bn
  7. 0007intro cp
  8. 0008intro cn
  9. 0009intro dp
  10. 0010intro dn
  11. 0011intro hz
  12. 0012intro hw
  13. 0013intro heq
  14. 0014specialize gaussian_representation_functional (z)
  15. 0015specialize gaussian_representation_functional (w)
  16. 0016specialize gaussian_representation_functional (ap)
  17. 0017specialize gaussian_representation_functional (an)
  18. 0018specialize gaussian_representation_functional (bp)
  19. 0019specialize gaussian_representation_functional (bn)
  20. 0020apply gaussian_representation_functional
  21. 0021exact hz
  22. 0022specialize gaussian_representation_integer_transport (w)
  23. 0023specialize gaussian_representation_integer_transport (cp)
  24. 0024specialize gaussian_representation_integer_transport (cn)
  25. 0025specialize gaussian_representation_integer_transport (dp)
  26. 0026specialize gaussian_representation_integer_transport (dn)
  27. 0027specialize gaussian_representation_integer_transport (ap)
  28. 0028specialize gaussian_representation_integer_transport (an)
  29. 0029specialize gaussian_representation_integer_transport (bp)
  30. 0030specialize gaussian_representation_integer_transport (bn)
  31. 0031apply gaussian_representation_integer_transport
  32. 0032specialize gaussian_equal_symmetric (ap)
  33. 0033specialize gaussian_equal_symmetric (an)
  34. 0034specialize gaussian_equal_symmetric (bp)
  35. 0035specialize gaussian_equal_symmetric (bn)
  36. 0036specialize gaussian_equal_symmetric (cp)
  37. 0037specialize gaussian_equal_symmetric (cn)
  38. 0038specialize gaussian_equal_symmetric (dp)
  39. 0039specialize gaussian_equal_symmetric (dn)
  40. 0040apply gaussian_equal_symmetric
  41. 0041exact heq
  42. 0042exact hw