GF007C

gaussian_search_divisor_of_nonzero_nonzero

Alpha v34 independently verified · alpha_closed; checked-use authorized; not Stable

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

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.

Exact expanded first-order arithmetic 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)

Constructive proof overview

Generated structural guide

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

The unchanged tactic script uses 1 declared prerequisite and contains 10 exact native proof lines.

Alpha v34 checked-use · first admitted v30 · independently kernel and Lean verified; not Stable

Proof neighborhood

Direct dependencies

GF0047 gaussian_zero_divides_only_zero

Direct dependents

GF0080 gaussian_irreducible_divisor_bounded_norm

Formal native tactic body

Dependencies are introduced as named hypotheses before line 1. Local theorem links identify exact declared prerequisites. This exact body belongs to a complete independently kernel-checked constructive proof bundle and has Alpha checked-use authority; it does not imply Stable membership.

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.

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 exact 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