GF0017

gaussian_norm_zero_implies_code_zero

Zero actual Gaussian norm forces literal zero canonical code, by constructive equality decision.

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. GNorm(z,0) → z = 0

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

Definition DAG

GNorm(z,n) · 1

Actual proof prerequisites

eq_decidable · checked external prerequisitegaussian_norm_nonzero
Original expanded first-order statement 
forall z. (exists ge_norm_rp_norm_zero_given ge_norm_rn_norm_zero_given ge_norm_ip_norm_zero_given ge_norm_in_norm_zero_given. ((exists ge_representation_real_code_norm_zero_givenrepresentation ge_representation_imaginary_code_norm_zero_givenrepresentation. (((z) = ((ge_representation_real_code_norm_zero_givenrepresentation) + (ge_representation_imaginary_code_norm_zero_givenrepresentation)) * S ((ge_representation_real_code_norm_zero_givenrepresentation) + (ge_representation_imaginary_code_norm_zero_givenrepresentation)) + ((ge_representation_imaginary_code_norm_zero_givenrepresentation) + (ge_representation_imaginary_code_norm_zero_givenrepresentation))) /\ ((exists ge_balance_positive_norm_zero_givenrepresentationreal ge_balance_negative_norm_zero_givenrepresentationreal. (((((ge_representation_real_code_norm_zero_givenrepresentation) = 2 * (ge_balance_positive_norm_zero_givenrepresentationreal) /\ (ge_balance_negative_norm_zero_givenrepresentationreal) = 0) \/ exists ge_signed_half_norm_zero_givenrepresentationrealdecode. (((ge_representation_real_code_norm_zero_givenrepresentation) = 2 * ge_signed_half_norm_zero_givenrepresentationrealdecode + 1 /\ (ge_balance_positive_norm_zero_givenrepresentationreal) = 0) /\ (ge_balance_negative_norm_zero_givenrepresentationreal) = S ge_signed_half_norm_zero_givenrepresentationrealdecode))) /\ ((ge_norm_rp_norm_zero_given) + ge_balance_negative_norm_zero_givenrepresentationreal = (ge_norm_rn_norm_zero_given) + ge_balance_positive_norm_zero_givenrepresentationreal))) /\ (exists ge_balance_positive_norm_zero_givenrepresentationimaginary ge_balance_negative_norm_zero_givenrepresentationimaginary. (((((ge_representation_imaginary_code_norm_zero_givenrepresentation) = 2 * (ge_balance_positive_norm_zero_givenrepresentationimaginary) /\ (ge_balance_negative_norm_zero_givenrepresentationimaginary) = 0) \/ exists ge_signed_half_norm_zero_givenrepresentationimaginarydecode. (((ge_representation_imaginary_code_norm_zero_givenrepresentation) = 2 * ge_signed_half_norm_zero_givenrepresentationimaginarydecode + 1 /\ (ge_balance_positive_norm_zero_givenrepresentationimaginary) = 0) /\ (ge_balance_negative_norm_zero_givenrepresentationimaginary) = S ge_signed_half_norm_zero_givenrepresentationimaginarydecode))) /\ ((ge_norm_ip_norm_zero_given) + ge_balance_negative_norm_zero_givenrepresentationimaginary = (ge_norm_in_norm_zero_given) + ge_balance_positive_norm_zero_givenrepresentationimaginary)))))) /\ (exists ge_real_square_norm_zero_givensquare ge_imaginary_square_norm_zero_givensquare. ((((((ge_norm_rp_norm_zero_given) * (ge_norm_rp_norm_zero_given))) + (((ge_norm_rn_norm_zero_given) * (ge_norm_rn_norm_zero_given)))) = ((ge_real_square_norm_zero_givensquare) + (((((ge_norm_rp_norm_zero_given) * (ge_norm_rn_norm_zero_given))) + (((ge_norm_rn_norm_zero_given) * (ge_norm_rp_norm_zero_given))))))) /\ ((((((ge_norm_ip_norm_zero_given) * (ge_norm_ip_norm_zero_given))) + (((ge_norm_in_norm_zero_given) * (ge_norm_in_norm_zero_given)))) = ((ge_imaginary_square_norm_zero_givensquare) + (((((ge_norm_ip_norm_zero_given) * (ge_norm_in_norm_zero_given))) + (((ge_norm_in_norm_zero_given) * (ge_norm_ip_norm_zero_given))))))) /\ ((0) = ge_real_square_norm_zero_givensquare + ge_imaginary_square_norm_zero_givensquare)))))) -> z=0

Complete tactic proof in conservative notation

All 15 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

15 script commands · 7 reading checkpoints · 1 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.

Named ingredients (1)
gaussian norm nonzero
01Fix variables and assumptionsL1–2

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

  1. L1
    intro z
  2. L2
    intro h
02Establish hzL3–6

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply eq decidable.

  1. L3
    have hz : z=0 \/ ~(z=0)
  2. L4
    specialize eq_decidable (z)
  3. L5
    specialize eq_decidable (0)
  4. L6
    apply eq_decidable
03Separate the logical casesL7–7

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

  1. L7
    cases hz
04Use earlier factsL8–8

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

  1. L8
    exact hz_left
05Separate the logical casesL9–9

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

  1. L9
    exfalso
06Use earlier factsL10–14

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

  1. L10
    specialize gaussian_norm_nonzero (z)
  2. L11
    specialize gaussian_norm_nonzero (0)
  3. L12
    apply gaussian_norm_nonzero
  4. L13
    exact h
  5. L14
    exact hz_right
07Calculate and transport equalitiesL15–15

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

  1. L15
    refl

Library-wide reading audit

Original defined command ledger · 15 lines
  1. 0001intro z
  2. 0002intro h
  3. 0003have hz : z=0 \/ ~(z=0)
  4. 0004specialize eq_decidable (z)
  5. 0005specialize eq_decidable (0)
  6. 0006apply eq_decidable
  7. 0007cases hz
  8. 0008exact hz_left
  9. 0009exfalso
  10. 0010specialize gaussian_norm_nonzero (z)
  11. 0011specialize gaussian_norm_nonzero (0)
  12. 0012apply gaussian_norm_nonzero
  13. 0013exact h
  14. 0014exact hz_right
  15. 0015refl