GF007C

gaussian_search_divisor_of_nonzero_nonzero

An actual divisor of a nonzero Gaussian value is nonzero, including the canonical zero boundary.

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

∀ d. ∀ z. GDvd(d,z) → ¬z = 0 → ¬d = 0

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

Definition DAG

GDvd(d,z) · 1

Actual proof prerequisites

gaussian_zero_divides_only_zero
Original expanded first-order statement 
forall d z. (exists gr_quotient_nonzero_divisor. (exists ge_first_rp_nonzero_divisorproduct ge_first_rn_nonzero_divisorproduct ge_first_ip_nonzero_divisorproduct ge_first_in_nonzero_divisorproduct ge_second_rp_nonzero_divisorproduct ge_second_rn_nonzero_divisorproduct ge_second_ip_nonzero_divisorproduct ge_second_in_nonzero_divisorproduct. ((exists ge_representation_real_code_nonzero_divisorproductfirst ge_representation_imaginary_code_nonzero_divisorproductfirst. (((d) = ((ge_representation_real_code_nonzero_divisorproductfirst) + (ge_representation_imaginary_code_nonzero_divisorproductfirst)) * S ((ge_representation_real_code_nonzero_divisorproductfirst) + (ge_representation_imaginary_code_nonzero_divisorproductfirst)) + ((ge_representation_imaginary_code_nonzero_divisorproductfirst) + (ge_representation_imaginary_code_nonzero_divisorproductfirst))) /\ ((exists ge_balance_positive_nonzero_divisorproductfirstreal ge_balance_negative_nonzero_divisorproductfirstreal. (((((ge_representation_real_code_nonzero_divisorproductfirst) = 2 * (ge_balance_positive_nonzero_divisorproductfirstreal) /\ (ge_balance_negative_nonzero_divisorproductfirstreal) = 0) \/ exists ge_signed_half_nonzero_divisorproductfirstrealdecode. (((ge_representation_real_code_nonzero_divisorproductfirst) = 2 * ge_signed_half_nonzero_divisorproductfirstrealdecode + 1 /\ (ge_balance_positive_nonzero_divisorproductfirstreal) = 0) /\ (ge_balance_negative_nonzero_divisorproductfirstreal) = S ge_signed_half_nonzero_divisorproductfirstrealdecode))) /\ ((ge_first_rp_nonzero_divisorproduct) + ge_balance_negative_nonzero_divisorproductfirstreal = (ge_first_rn_nonzero_divisorproduct) + ge_balance_positive_nonzero_divisorproductfirstreal))) /\ (exists ge_balance_positive_nonzero_divisorproductfirstimaginary ge_balance_negative_nonzero_divisorproductfirstimaginary. (((((ge_representation_imaginary_code_nonzero_divisorproductfirst) = 2 * (ge_balance_positive_nonzero_divisorproductfirstimaginary) /\ (ge_balance_negative_nonzero_divisorproductfirstimaginary) = 0) \/ exists ge_signed_half_nonzero_divisorproductfirstimaginarydecode. (((ge_representation_imaginary_code_nonzero_divisorproductfirst) = 2 * ge_signed_half_nonzero_divisorproductfirstimaginarydecode + 1 /\ (ge_balance_positive_nonzero_divisorproductfirstimaginary) = 0) /\ (ge_balance_negative_nonzero_divisorproductfirstimaginary) = S ge_signed_half_nonzero_divisorproductfirstimaginarydecode))) /\ ((ge_first_ip_nonzero_divisorproduct) + ge_balance_negative_nonzero_divisorproductfirstimaginary = (ge_first_in_nonzero_divisorproduct) + ge_balance_positive_nonzero_divisorproductfirstimaginary)))))) /\ ((exists ge_representation_real_code_nonzero_divisorproductsecond ge_representation_imaginary_code_nonzero_divisorproductsecond. (((gr_quotient_nonzero_divisor) = ((ge_representation_real_code_nonzero_divisorproductsecond) + (ge_representation_imaginary_code_nonzero_divisorproductsecond)) * S ((ge_representation_real_code_nonzero_divisorproductsecond) + (ge_representation_imaginary_code_nonzero_divisorproductsecond)) + ((ge_representation_imaginary_code_nonzero_divisorproductsecond) + (ge_representation_imaginary_code_nonzero_divisorproductsecond))) /\ ((exists ge_balance_positive_nonzero_divisorproductsecondreal ge_balance_negative_nonzero_divisorproductsecondreal. (((((ge_representation_real_code_nonzero_divisorproductsecond) = 2 * (ge_balance_positive_nonzero_divisorproductsecondreal) /\ (ge_balance_negative_nonzero_divisorproductsecondreal) = 0) \/ exists ge_signed_half_nonzero_divisorproductsecondrealdecode. (((ge_representation_real_code_nonzero_divisorproductsecond) = 2 * ge_signed_half_nonzero_divisorproductsecondrealdecode + 1 /\ (ge_balance_positive_nonzero_divisorproductsecondreal) = 0) /\ (ge_balance_negative_nonzero_divisorproductsecondreal) = S ge_signed_half_nonzero_divisorproductsecondrealdecode))) /\ ((ge_second_rp_nonzero_divisorproduct) + ge_balance_negative_nonzero_divisorproductsecondreal = (ge_second_rn_nonzero_divisorproduct) + ge_balance_positive_nonzero_divisorproductsecondreal))) /\ (exists ge_balance_positive_nonzero_divisorproductsecondimaginary ge_balance_negative_nonzero_divisorproductsecondimaginary. (((((ge_representation_imaginary_code_nonzero_divisorproductsecond) = 2 * (ge_balance_positive_nonzero_divisorproductsecondimaginary) /\ (ge_balance_negative_nonzero_divisorproductsecondimaginary) = 0) \/ exists ge_signed_half_nonzero_divisorproductsecondimaginarydecode. (((ge_representation_imaginary_code_nonzero_divisorproductsecond) = 2 * ge_signed_half_nonzero_divisorproductsecondimaginarydecode + 1 /\ (ge_balance_positive_nonzero_divisorproductsecondimaginary) = 0) /\ (ge_balance_negative_nonzero_divisorproductsecondimaginary) = S ge_signed_half_nonzero_divisorproductsecondimaginarydecode))) /\ ((ge_second_ip_nonzero_divisorproduct) + ge_balance_negative_nonzero_divisorproductsecondimaginary = (ge_second_in_nonzero_divisorproduct) + ge_balance_positive_nonzero_divisorproductsecondimaginary)))))) /\ (exists ge_representation_real_code_nonzero_divisorproductoutput ge_representation_imaginary_code_nonzero_divisorproductoutput. (((z) = ((ge_representation_real_code_nonzero_divisorproductoutput) + (ge_representation_imaginary_code_nonzero_divisorproductoutput)) * S ((ge_representation_real_code_nonzero_divisorproductoutput) + (ge_representation_imaginary_code_nonzero_divisorproductoutput)) + ((ge_representation_imaginary_code_nonzero_divisorproductoutput) + (ge_representation_imaginary_code_nonzero_divisorproductoutput))) /\ ((exists ge_balance_positive_nonzero_divisorproductoutputreal ge_balance_negative_nonzero_divisorproductoutputreal. (((((ge_representation_real_code_nonzero_divisorproductoutput) = 2 * (ge_balance_positive_nonzero_divisorproductoutputreal) /\ (ge_balance_negative_nonzero_divisorproductoutputreal) = 0) \/ exists ge_signed_half_nonzero_divisorproductoutputrealdecode. (((ge_representation_real_code_nonzero_divisorproductoutput) = 2 * ge_signed_half_nonzero_divisorproductoutputrealdecode + 1 /\ (ge_balance_positive_nonzero_divisorproductoutputreal) = 0) /\ (ge_balance_negative_nonzero_divisorproductoutputreal) = S ge_signed_half_nonzero_divisorproductoutputrealdecode))) /\ ((((((((ge_first_rp_nonzero_divisorproduct) * (ge_second_rp_nonzero_divisorproduct))) + (((ge_first_rn_nonzero_divisorproduct) * (ge_second_rn_nonzero_divisorproduct))))) + (((((ge_first_ip_nonzero_divisorproduct) * (ge_second_in_nonzero_divisorproduct))) + (((ge_first_in_nonzero_divisorproduct) * (ge_second_ip_nonzero_divisorproduct))))))) + ge_balance_negative_nonzero_divisorproductoutputreal = (((((((ge_first_rp_nonzero_divisorproduct) * (ge_second_rn_nonzero_divisorproduct))) + (((ge_first_rn_nonzero_divisorproduct) * (ge_second_rp_nonzero_divisorproduct))))) + (((((ge_first_ip_nonzero_divisorproduct) * (ge_second_ip_nonzero_divisorproduct))) + (((ge_first_in_nonzero_divisorproduct) * (ge_second_in_nonzero_divisorproduct))))))) + ge_balance_positive_nonzero_divisorproductoutputreal))) /\ (exists ge_balance_positive_nonzero_divisorproductoutputimaginary ge_balance_negative_nonzero_divisorproductoutputimaginary. (((((ge_representation_imaginary_code_nonzero_divisorproductoutput) = 2 * (ge_balance_positive_nonzero_divisorproductoutputimaginary) /\ (ge_balance_negative_nonzero_divisorproductoutputimaginary) = 0) \/ exists ge_signed_half_nonzero_divisorproductoutputimaginarydecode. (((ge_representation_imaginary_code_nonzero_divisorproductoutput) = 2 * ge_signed_half_nonzero_divisorproductoutputimaginarydecode + 1 /\ (ge_balance_positive_nonzero_divisorproductoutputimaginary) = 0) /\ (ge_balance_negative_nonzero_divisorproductoutputimaginary) = S ge_signed_half_nonzero_divisorproductoutputimaginarydecode))) /\ ((((((((ge_first_rp_nonzero_divisorproduct) * (ge_second_ip_nonzero_divisorproduct))) + (((ge_first_rn_nonzero_divisorproduct) * (ge_second_in_nonzero_divisorproduct))))) + (((((ge_first_ip_nonzero_divisorproduct) * (ge_second_rp_nonzero_divisorproduct))) + (((ge_first_in_nonzero_divisorproduct) * (ge_second_rn_nonzero_divisorproduct))))))) + ge_balance_negative_nonzero_divisorproductoutputimaginary = (((((((ge_first_rp_nonzero_divisorproduct) * (ge_second_in_nonzero_divisorproduct))) + (((ge_first_rn_nonzero_divisorproduct) * (ge_second_ip_nonzero_divisorproduct))))) + (((((ge_first_ip_nonzero_divisorproduct) * (ge_second_rn_nonzero_divisorproduct))) + (((ge_first_in_nonzero_divisorproduct) * (ge_second_rp_nonzero_divisorproduct))))))) + ge_balance_positive_nonzero_divisorproductoutputimaginary)))))))))) -> ~(z=0) -> ~(d=0)

Complete tactic proof in conservative notation

All 10 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

10 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 (1)
gaussian zero divides only zero
01Fix variables and assumptionsL1–5

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

  1. L1
    intro d
  2. L2
    intro z
  3. L3
    intro hd
  4. L4
    intro hz
  5. L5
    intro hdzero
02Use earlier factsL6–8

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

  1. L6
    apply hz
  2. L7
    specialize gaussian_zero_divides_only_zero (z)
  3. L8
    apply gaussian_zero_divides_only_zero
03Calculate and transport equalitiesL9–9

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

  1. L9
    rewrite hdzero at hd
04Use earlier factsL10–10

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

  1. L10
    exact hd

Library-wide reading audit

Original defined command ledger · 10 lines
  1. 0001intro d
  2. 0002intro z
  3. 0003intro hd
  4. 0004intro hz
  5. 0005intro hdzero
  6. 0006apply hz
  7. 0007specialize gaussian_zero_divides_only_zero (z)
  8. 0008apply gaussian_zero_divides_only_zero
  9. 0009rewrite hdzero at hd
  10. 0010exact hd