GF0047

gaussian_zero_divides_only_zero

A zero Gaussian divisor can divide only the actual zero code.

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. GDvd(0,z) → z = 0

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

Definition DAG

GDvd(d,z) · 1

Actual proof prerequisites

gaussian_multiply_functional · checked external prerequisitegaussian_multiply_zero_leftgaussian_multiply_input_right_valid
Original expanded first-order statement 
forall z. (exists gr_quotient_zero_divisor. (exists ge_first_rp_zero_divisorproduct ge_first_rn_zero_divisorproduct ge_first_ip_zero_divisorproduct ge_first_in_zero_divisorproduct ge_second_rp_zero_divisorproduct ge_second_rn_zero_divisorproduct ge_second_ip_zero_divisorproduct ge_second_in_zero_divisorproduct. ((exists ge_representation_real_code_zero_divisorproductfirst ge_representation_imaginary_code_zero_divisorproductfirst. (((0) = ((ge_representation_real_code_zero_divisorproductfirst) + (ge_representation_imaginary_code_zero_divisorproductfirst)) * S ((ge_representation_real_code_zero_divisorproductfirst) + (ge_representation_imaginary_code_zero_divisorproductfirst)) + ((ge_representation_imaginary_code_zero_divisorproductfirst) + (ge_representation_imaginary_code_zero_divisorproductfirst))) /\ ((exists ge_balance_positive_zero_divisorproductfirstreal ge_balance_negative_zero_divisorproductfirstreal. (((((ge_representation_real_code_zero_divisorproductfirst) = 2 * (ge_balance_positive_zero_divisorproductfirstreal) /\ (ge_balance_negative_zero_divisorproductfirstreal) = 0) \/ exists ge_signed_half_zero_divisorproductfirstrealdecode. (((ge_representation_real_code_zero_divisorproductfirst) = 2 * ge_signed_half_zero_divisorproductfirstrealdecode + 1 /\ (ge_balance_positive_zero_divisorproductfirstreal) = 0) /\ (ge_balance_negative_zero_divisorproductfirstreal) = S ge_signed_half_zero_divisorproductfirstrealdecode))) /\ ((ge_first_rp_zero_divisorproduct) + ge_balance_negative_zero_divisorproductfirstreal = (ge_first_rn_zero_divisorproduct) + ge_balance_positive_zero_divisorproductfirstreal))) /\ (exists ge_balance_positive_zero_divisorproductfirstimaginary ge_balance_negative_zero_divisorproductfirstimaginary. (((((ge_representation_imaginary_code_zero_divisorproductfirst) = 2 * (ge_balance_positive_zero_divisorproductfirstimaginary) /\ (ge_balance_negative_zero_divisorproductfirstimaginary) = 0) \/ exists ge_signed_half_zero_divisorproductfirstimaginarydecode. (((ge_representation_imaginary_code_zero_divisorproductfirst) = 2 * ge_signed_half_zero_divisorproductfirstimaginarydecode + 1 /\ (ge_balance_positive_zero_divisorproductfirstimaginary) = 0) /\ (ge_balance_negative_zero_divisorproductfirstimaginary) = S ge_signed_half_zero_divisorproductfirstimaginarydecode))) /\ ((ge_first_ip_zero_divisorproduct) + ge_balance_negative_zero_divisorproductfirstimaginary = (ge_first_in_zero_divisorproduct) + ge_balance_positive_zero_divisorproductfirstimaginary)))))) /\ ((exists ge_representation_real_code_zero_divisorproductsecond ge_representation_imaginary_code_zero_divisorproductsecond. (((gr_quotient_zero_divisor) = ((ge_representation_real_code_zero_divisorproductsecond) + (ge_representation_imaginary_code_zero_divisorproductsecond)) * S ((ge_representation_real_code_zero_divisorproductsecond) + (ge_representation_imaginary_code_zero_divisorproductsecond)) + ((ge_representation_imaginary_code_zero_divisorproductsecond) + (ge_representation_imaginary_code_zero_divisorproductsecond))) /\ ((exists ge_balance_positive_zero_divisorproductsecondreal ge_balance_negative_zero_divisorproductsecondreal. (((((ge_representation_real_code_zero_divisorproductsecond) = 2 * (ge_balance_positive_zero_divisorproductsecondreal) /\ (ge_balance_negative_zero_divisorproductsecondreal) = 0) \/ exists ge_signed_half_zero_divisorproductsecondrealdecode. (((ge_representation_real_code_zero_divisorproductsecond) = 2 * ge_signed_half_zero_divisorproductsecondrealdecode + 1 /\ (ge_balance_positive_zero_divisorproductsecondreal) = 0) /\ (ge_balance_negative_zero_divisorproductsecondreal) = S ge_signed_half_zero_divisorproductsecondrealdecode))) /\ ((ge_second_rp_zero_divisorproduct) + ge_balance_negative_zero_divisorproductsecondreal = (ge_second_rn_zero_divisorproduct) + ge_balance_positive_zero_divisorproductsecondreal))) /\ (exists ge_balance_positive_zero_divisorproductsecondimaginary ge_balance_negative_zero_divisorproductsecondimaginary. (((((ge_representation_imaginary_code_zero_divisorproductsecond) = 2 * (ge_balance_positive_zero_divisorproductsecondimaginary) /\ (ge_balance_negative_zero_divisorproductsecondimaginary) = 0) \/ exists ge_signed_half_zero_divisorproductsecondimaginarydecode. (((ge_representation_imaginary_code_zero_divisorproductsecond) = 2 * ge_signed_half_zero_divisorproductsecondimaginarydecode + 1 /\ (ge_balance_positive_zero_divisorproductsecondimaginary) = 0) /\ (ge_balance_negative_zero_divisorproductsecondimaginary) = S ge_signed_half_zero_divisorproductsecondimaginarydecode))) /\ ((ge_second_ip_zero_divisorproduct) + ge_balance_negative_zero_divisorproductsecondimaginary = (ge_second_in_zero_divisorproduct) + ge_balance_positive_zero_divisorproductsecondimaginary)))))) /\ (exists ge_representation_real_code_zero_divisorproductoutput ge_representation_imaginary_code_zero_divisorproductoutput. (((z) = ((ge_representation_real_code_zero_divisorproductoutput) + (ge_representation_imaginary_code_zero_divisorproductoutput)) * S ((ge_representation_real_code_zero_divisorproductoutput) + (ge_representation_imaginary_code_zero_divisorproductoutput)) + ((ge_representation_imaginary_code_zero_divisorproductoutput) + (ge_representation_imaginary_code_zero_divisorproductoutput))) /\ ((exists ge_balance_positive_zero_divisorproductoutputreal ge_balance_negative_zero_divisorproductoutputreal. (((((ge_representation_real_code_zero_divisorproductoutput) = 2 * (ge_balance_positive_zero_divisorproductoutputreal) /\ (ge_balance_negative_zero_divisorproductoutputreal) = 0) \/ exists ge_signed_half_zero_divisorproductoutputrealdecode. (((ge_representation_real_code_zero_divisorproductoutput) = 2 * ge_signed_half_zero_divisorproductoutputrealdecode + 1 /\ (ge_balance_positive_zero_divisorproductoutputreal) = 0) /\ (ge_balance_negative_zero_divisorproductoutputreal) = S ge_signed_half_zero_divisorproductoutputrealdecode))) /\ ((((((((ge_first_rp_zero_divisorproduct) * (ge_second_rp_zero_divisorproduct))) + (((ge_first_rn_zero_divisorproduct) * (ge_second_rn_zero_divisorproduct))))) + (((((ge_first_ip_zero_divisorproduct) * (ge_second_in_zero_divisorproduct))) + (((ge_first_in_zero_divisorproduct) * (ge_second_ip_zero_divisorproduct))))))) + ge_balance_negative_zero_divisorproductoutputreal = (((((((ge_first_rp_zero_divisorproduct) * (ge_second_rn_zero_divisorproduct))) + (((ge_first_rn_zero_divisorproduct) * (ge_second_rp_zero_divisorproduct))))) + (((((ge_first_ip_zero_divisorproduct) * (ge_second_ip_zero_divisorproduct))) + (((ge_first_in_zero_divisorproduct) * (ge_second_in_zero_divisorproduct))))))) + ge_balance_positive_zero_divisorproductoutputreal))) /\ (exists ge_balance_positive_zero_divisorproductoutputimaginary ge_balance_negative_zero_divisorproductoutputimaginary. (((((ge_representation_imaginary_code_zero_divisorproductoutput) = 2 * (ge_balance_positive_zero_divisorproductoutputimaginary) /\ (ge_balance_negative_zero_divisorproductoutputimaginary) = 0) \/ exists ge_signed_half_zero_divisorproductoutputimaginarydecode. (((ge_representation_imaginary_code_zero_divisorproductoutput) = 2 * ge_signed_half_zero_divisorproductoutputimaginarydecode + 1 /\ (ge_balance_positive_zero_divisorproductoutputimaginary) = 0) /\ (ge_balance_negative_zero_divisorproductoutputimaginary) = S ge_signed_half_zero_divisorproductoutputimaginarydecode))) /\ ((((((((ge_first_rp_zero_divisorproduct) * (ge_second_ip_zero_divisorproduct))) + (((ge_first_rn_zero_divisorproduct) * (ge_second_in_zero_divisorproduct))))) + (((((ge_first_ip_zero_divisorproduct) * (ge_second_rp_zero_divisorproduct))) + (((ge_first_in_zero_divisorproduct) * (ge_second_rn_zero_divisorproduct))))))) + ge_balance_negative_zero_divisorproductoutputimaginary = (((((((ge_first_rp_zero_divisorproduct) * (ge_second_in_zero_divisorproduct))) + (((ge_first_rn_zero_divisorproduct) * (ge_second_ip_zero_divisorproduct))))) + (((((ge_first_ip_zero_divisorproduct) * (ge_second_rn_zero_divisorproduct))) + (((ge_first_in_zero_divisorproduct) * (ge_second_rp_zero_divisorproduct))))))) + ge_balance_positive_zero_divisorproductoutputimaginary)))))))))) -> z=0

Complete tactic proof in conservative notation

All 16 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

16 script commands · 4 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.

Named ingredients (2)
gaussian multiply zero leftgaussian multiply input right valid
01Fix variables and assumptionsL1–2

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

  1. L1
    intro z
  2. L2
    intro h
02Separate the logical casesL3–3

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

  1. L3
    cases h
03Use earlier factsL4–13

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

  1. L4
    specialize gaussian_multiply_functional (0)
  2. L5
    specialize gaussian_multiply_functional (x)
  3. L6
    specialize gaussian_multiply_functional (z)
  4. L7
    specialize gaussian_multiply_functional (0)
  5. L8
    apply gaussian_multiply_functional
  6. L9
    exact h_witness
  7. L10
    specialize gaussian_multiply_zero_left (x)
  8. L11
    apply gaussian_multiply_zero_left
  9. L12
    specialize gaussian_multiply_input_right_valid (0)
  10. L13
    specialize gaussian_multiply_input_right_valid (x)
04Use earlier factsL14–16

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

  1. L14
    specialize gaussian_multiply_input_right_valid (z)
  2. L15
    apply gaussian_multiply_input_right_valid
  3. L16
    exact h_witness

Library-wide reading audit

Original defined command ledger · 16 lines
  1. 0001intro z
  2. 0002intro h
  3. 0003cases h
  4. 0004specialize gaussian_multiply_functional (0)
  5. 0005specialize gaussian_multiply_functional (x)
  6. 0006specialize gaussian_multiply_functional (z)
  7. 0007specialize gaussian_multiply_functional (0)
  8. 0008apply gaussian_multiply_functional
  9. 0009exact h_witness
  10. 0010specialize gaussian_multiply_zero_left (x)
  11. 0011apply gaussian_multiply_zero_left
  12. 0012specialize gaussian_multiply_input_right_valid (0)
  13. 0013specialize gaussian_multiply_input_right_valid (x)
  14. 0014specialize gaussian_multiply_input_right_valid (z)
  15. 0015apply gaussian_multiply_input_right_valid
  16. 0016exact h_witness