GF0079

gaussian_search_nonunit_norm_two

The actual nonzero norm of a Gaussian nonunit is at least two; a norm-one exception would construct an inverse.

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. ∀ N. GNorm(z,N) → ¬N = 0 → ¬GUnit(z) → Lt(1,N)

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

Definition DAG

GNorm(z,n) · 1GUnit(z) · 1Lt(a,b) · 1

Actual proof prerequisites

gaussian_search_two_le_nonzero_not_onegaussian_norm_one_is_unitgaussian_norm_value_transport
Original expanded first-order statement 
forall z N. (exists ge_norm_rp_nonunit_actual_norm ge_norm_rn_nonunit_actual_norm ge_norm_ip_nonunit_actual_norm ge_norm_in_nonunit_actual_norm. ((exists ge_representation_real_code_nonunit_actual_normrepresentation ge_representation_imaginary_code_nonunit_actual_normrepresentation. (((z) = ((ge_representation_real_code_nonunit_actual_normrepresentation) + (ge_representation_imaginary_code_nonunit_actual_normrepresentation)) * S ((ge_representation_real_code_nonunit_actual_normrepresentation) + (ge_representation_imaginary_code_nonunit_actual_normrepresentation)) + ((ge_representation_imaginary_code_nonunit_actual_normrepresentation) + (ge_representation_imaginary_code_nonunit_actual_normrepresentation))) /\ ((exists ge_balance_positive_nonunit_actual_normrepresentationreal ge_balance_negative_nonunit_actual_normrepresentationreal. (((((ge_representation_real_code_nonunit_actual_normrepresentation) = 2 * (ge_balance_positive_nonunit_actual_normrepresentationreal) /\ (ge_balance_negative_nonunit_actual_normrepresentationreal) = 0) \/ exists ge_signed_half_nonunit_actual_normrepresentationrealdecode. (((ge_representation_real_code_nonunit_actual_normrepresentation) = 2 * ge_signed_half_nonunit_actual_normrepresentationrealdecode + 1 /\ (ge_balance_positive_nonunit_actual_normrepresentationreal) = 0) /\ (ge_balance_negative_nonunit_actual_normrepresentationreal) = S ge_signed_half_nonunit_actual_normrepresentationrealdecode))) /\ ((ge_norm_rp_nonunit_actual_norm) + ge_balance_negative_nonunit_actual_normrepresentationreal = (ge_norm_rn_nonunit_actual_norm) + ge_balance_positive_nonunit_actual_normrepresentationreal))) /\ (exists ge_balance_positive_nonunit_actual_normrepresentationimaginary ge_balance_negative_nonunit_actual_normrepresentationimaginary. (((((ge_representation_imaginary_code_nonunit_actual_normrepresentation) = 2 * (ge_balance_positive_nonunit_actual_normrepresentationimaginary) /\ (ge_balance_negative_nonunit_actual_normrepresentationimaginary) = 0) \/ exists ge_signed_half_nonunit_actual_normrepresentationimaginarydecode. (((ge_representation_imaginary_code_nonunit_actual_normrepresentation) = 2 * ge_signed_half_nonunit_actual_normrepresentationimaginarydecode + 1 /\ (ge_balance_positive_nonunit_actual_normrepresentationimaginary) = 0) /\ (ge_balance_negative_nonunit_actual_normrepresentationimaginary) = S ge_signed_half_nonunit_actual_normrepresentationimaginarydecode))) /\ ((ge_norm_ip_nonunit_actual_norm) + ge_balance_negative_nonunit_actual_normrepresentationimaginary = (ge_norm_in_nonunit_actual_norm) + ge_balance_positive_nonunit_actual_normrepresentationimaginary)))))) /\ (exists ge_real_square_nonunit_actual_normsquare ge_imaginary_square_nonunit_actual_normsquare. ((((((ge_norm_rp_nonunit_actual_norm) * (ge_norm_rp_nonunit_actual_norm))) + (((ge_norm_rn_nonunit_actual_norm) * (ge_norm_rn_nonunit_actual_norm)))) = ((ge_real_square_nonunit_actual_normsquare) + (((((ge_norm_rp_nonunit_actual_norm) * (ge_norm_rn_nonunit_actual_norm))) + (((ge_norm_rn_nonunit_actual_norm) * (ge_norm_rp_nonunit_actual_norm))))))) /\ ((((((ge_norm_ip_nonunit_actual_norm) * (ge_norm_ip_nonunit_actual_norm))) + (((ge_norm_in_nonunit_actual_norm) * (ge_norm_in_nonunit_actual_norm)))) = ((ge_imaginary_square_nonunit_actual_normsquare) + (((((ge_norm_ip_nonunit_actual_norm) * (ge_norm_in_nonunit_actual_norm))) + (((ge_norm_in_nonunit_actual_norm) * (ge_norm_ip_nonunit_actual_norm))))))) /\ ((N) = ge_real_square_nonunit_actual_normsquare + ge_imaginary_square_nonunit_actual_normsquare)))))) -> ~(N=0) -> ~(exists gr_inverse_nonunit_given. (exists ge_first_rp_nonunit_givenidentity ge_first_rn_nonunit_givenidentity ge_first_ip_nonunit_givenidentity ge_first_in_nonunit_givenidentity ge_second_rp_nonunit_givenidentity ge_second_rn_nonunit_givenidentity ge_second_ip_nonunit_givenidentity ge_second_in_nonunit_givenidentity. ((exists ge_representation_real_code_nonunit_givenidentityfirst ge_representation_imaginary_code_nonunit_givenidentityfirst. (((z) = ((ge_representation_real_code_nonunit_givenidentityfirst) + (ge_representation_imaginary_code_nonunit_givenidentityfirst)) * S ((ge_representation_real_code_nonunit_givenidentityfirst) + (ge_representation_imaginary_code_nonunit_givenidentityfirst)) + ((ge_representation_imaginary_code_nonunit_givenidentityfirst) + (ge_representation_imaginary_code_nonunit_givenidentityfirst))) /\ ((exists ge_balance_positive_nonunit_givenidentityfirstreal ge_balance_negative_nonunit_givenidentityfirstreal. (((((ge_representation_real_code_nonunit_givenidentityfirst) = 2 * (ge_balance_positive_nonunit_givenidentityfirstreal) /\ (ge_balance_negative_nonunit_givenidentityfirstreal) = 0) \/ exists ge_signed_half_nonunit_givenidentityfirstrealdecode. (((ge_representation_real_code_nonunit_givenidentityfirst) = 2 * ge_signed_half_nonunit_givenidentityfirstrealdecode + 1 /\ (ge_balance_positive_nonunit_givenidentityfirstreal) = 0) /\ (ge_balance_negative_nonunit_givenidentityfirstreal) = S ge_signed_half_nonunit_givenidentityfirstrealdecode))) /\ ((ge_first_rp_nonunit_givenidentity) + ge_balance_negative_nonunit_givenidentityfirstreal = (ge_first_rn_nonunit_givenidentity) + ge_balance_positive_nonunit_givenidentityfirstreal))) /\ (exists ge_balance_positive_nonunit_givenidentityfirstimaginary ge_balance_negative_nonunit_givenidentityfirstimaginary. (((((ge_representation_imaginary_code_nonunit_givenidentityfirst) = 2 * (ge_balance_positive_nonunit_givenidentityfirstimaginary) /\ (ge_balance_negative_nonunit_givenidentityfirstimaginary) = 0) \/ exists ge_signed_half_nonunit_givenidentityfirstimaginarydecode. (((ge_representation_imaginary_code_nonunit_givenidentityfirst) = 2 * ge_signed_half_nonunit_givenidentityfirstimaginarydecode + 1 /\ (ge_balance_positive_nonunit_givenidentityfirstimaginary) = 0) /\ (ge_balance_negative_nonunit_givenidentityfirstimaginary) = S ge_signed_half_nonunit_givenidentityfirstimaginarydecode))) /\ ((ge_first_ip_nonunit_givenidentity) + ge_balance_negative_nonunit_givenidentityfirstimaginary = (ge_first_in_nonunit_givenidentity) + ge_balance_positive_nonunit_givenidentityfirstimaginary)))))) /\ ((exists ge_representation_real_code_nonunit_givenidentitysecond ge_representation_imaginary_code_nonunit_givenidentitysecond. (((gr_inverse_nonunit_given) = ((ge_representation_real_code_nonunit_givenidentitysecond) + (ge_representation_imaginary_code_nonunit_givenidentitysecond)) * S ((ge_representation_real_code_nonunit_givenidentitysecond) + (ge_representation_imaginary_code_nonunit_givenidentitysecond)) + ((ge_representation_imaginary_code_nonunit_givenidentitysecond) + (ge_representation_imaginary_code_nonunit_givenidentitysecond))) /\ ((exists ge_balance_positive_nonunit_givenidentitysecondreal ge_balance_negative_nonunit_givenidentitysecondreal. (((((ge_representation_real_code_nonunit_givenidentitysecond) = 2 * (ge_balance_positive_nonunit_givenidentitysecondreal) /\ (ge_balance_negative_nonunit_givenidentitysecondreal) = 0) \/ exists ge_signed_half_nonunit_givenidentitysecondrealdecode. (((ge_representation_real_code_nonunit_givenidentitysecond) = 2 * ge_signed_half_nonunit_givenidentitysecondrealdecode + 1 /\ (ge_balance_positive_nonunit_givenidentitysecondreal) = 0) /\ (ge_balance_negative_nonunit_givenidentitysecondreal) = S ge_signed_half_nonunit_givenidentitysecondrealdecode))) /\ ((ge_second_rp_nonunit_givenidentity) + ge_balance_negative_nonunit_givenidentitysecondreal = (ge_second_rn_nonunit_givenidentity) + ge_balance_positive_nonunit_givenidentitysecondreal))) /\ (exists ge_balance_positive_nonunit_givenidentitysecondimaginary ge_balance_negative_nonunit_givenidentitysecondimaginary. (((((ge_representation_imaginary_code_nonunit_givenidentitysecond) = 2 * (ge_balance_positive_nonunit_givenidentitysecondimaginary) /\ (ge_balance_negative_nonunit_givenidentitysecondimaginary) = 0) \/ exists ge_signed_half_nonunit_givenidentitysecondimaginarydecode. (((ge_representation_imaginary_code_nonunit_givenidentitysecond) = 2 * ge_signed_half_nonunit_givenidentitysecondimaginarydecode + 1 /\ (ge_balance_positive_nonunit_givenidentitysecondimaginary) = 0) /\ (ge_balance_negative_nonunit_givenidentitysecondimaginary) = S ge_signed_half_nonunit_givenidentitysecondimaginarydecode))) /\ ((ge_second_ip_nonunit_givenidentity) + ge_balance_negative_nonunit_givenidentitysecondimaginary = (ge_second_in_nonunit_givenidentity) + ge_balance_positive_nonunit_givenidentitysecondimaginary)))))) /\ (exists ge_representation_real_code_nonunit_givenidentityoutput ge_representation_imaginary_code_nonunit_givenidentityoutput. (((6) = ((ge_representation_real_code_nonunit_givenidentityoutput) + (ge_representation_imaginary_code_nonunit_givenidentityoutput)) * S ((ge_representation_real_code_nonunit_givenidentityoutput) + (ge_representation_imaginary_code_nonunit_givenidentityoutput)) + ((ge_representation_imaginary_code_nonunit_givenidentityoutput) + (ge_representation_imaginary_code_nonunit_givenidentityoutput))) /\ ((exists ge_balance_positive_nonunit_givenidentityoutputreal ge_balance_negative_nonunit_givenidentityoutputreal. (((((ge_representation_real_code_nonunit_givenidentityoutput) = 2 * (ge_balance_positive_nonunit_givenidentityoutputreal) /\ (ge_balance_negative_nonunit_givenidentityoutputreal) = 0) \/ exists ge_signed_half_nonunit_givenidentityoutputrealdecode. (((ge_representation_real_code_nonunit_givenidentityoutput) = 2 * ge_signed_half_nonunit_givenidentityoutputrealdecode + 1 /\ (ge_balance_positive_nonunit_givenidentityoutputreal) = 0) /\ (ge_balance_negative_nonunit_givenidentityoutputreal) = S ge_signed_half_nonunit_givenidentityoutputrealdecode))) /\ ((((((((ge_first_rp_nonunit_givenidentity) * (ge_second_rp_nonunit_givenidentity))) + (((ge_first_rn_nonunit_givenidentity) * (ge_second_rn_nonunit_givenidentity))))) + (((((ge_first_ip_nonunit_givenidentity) * (ge_second_in_nonunit_givenidentity))) + (((ge_first_in_nonunit_givenidentity) * (ge_second_ip_nonunit_givenidentity))))))) + ge_balance_negative_nonunit_givenidentityoutputreal = (((((((ge_first_rp_nonunit_givenidentity) * (ge_second_rn_nonunit_givenidentity))) + (((ge_first_rn_nonunit_givenidentity) * (ge_second_rp_nonunit_givenidentity))))) + (((((ge_first_ip_nonunit_givenidentity) * (ge_second_ip_nonunit_givenidentity))) + (((ge_first_in_nonunit_givenidentity) * (ge_second_in_nonunit_givenidentity))))))) + ge_balance_positive_nonunit_givenidentityoutputreal))) /\ (exists ge_balance_positive_nonunit_givenidentityoutputimaginary ge_balance_negative_nonunit_givenidentityoutputimaginary. (((((ge_representation_imaginary_code_nonunit_givenidentityoutput) = 2 * (ge_balance_positive_nonunit_givenidentityoutputimaginary) /\ (ge_balance_negative_nonunit_givenidentityoutputimaginary) = 0) \/ exists ge_signed_half_nonunit_givenidentityoutputimaginarydecode. (((ge_representation_imaginary_code_nonunit_givenidentityoutput) = 2 * ge_signed_half_nonunit_givenidentityoutputimaginarydecode + 1 /\ (ge_balance_positive_nonunit_givenidentityoutputimaginary) = 0) /\ (ge_balance_negative_nonunit_givenidentityoutputimaginary) = S ge_signed_half_nonunit_givenidentityoutputimaginarydecode))) /\ ((((((((ge_first_rp_nonunit_givenidentity) * (ge_second_ip_nonunit_givenidentity))) + (((ge_first_rn_nonunit_givenidentity) * (ge_second_in_nonunit_givenidentity))))) + (((((ge_first_ip_nonunit_givenidentity) * (ge_second_rp_nonunit_givenidentity))) + (((ge_first_in_nonunit_givenidentity) * (ge_second_rn_nonunit_givenidentity))))))) + ge_balance_negative_nonunit_givenidentityoutputimaginary = (((((((ge_first_rp_nonunit_givenidentity) * (ge_second_in_nonunit_givenidentity))) + (((ge_first_rn_nonunit_givenidentity) * (ge_second_ip_nonunit_givenidentity))))) + (((((ge_first_ip_nonunit_givenidentity) * (ge_second_rn_nonunit_givenidentity))) + (((ge_first_in_nonunit_givenidentity) * (ge_second_rp_nonunit_givenidentity))))))) + ge_balance_positive_nonunit_givenidentityoutputimaginary)))))))))) -> (exists ge_gap_nonunit_norm_lower. ge_gap_nonunit_norm_lower + (2) = (N))

Complete tactic proof in conservative notation

All 18 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

18 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 (3)
gaussian search two le nonzero not onegaussian norm one is unitgaussian norm value transport
01Fix variables and assumptionsL1–5

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

  1. L1
    intro z
  2. L2
    intro N
  3. L3
    intro hn
  4. L4
    intro hzero
  5. L5
    intro hu
02Use earlier factsL6–8

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

  1. L6
    specialize gaussian_search_two_le_nonzero_not_one (N)
  2. L7
    apply gaussian_search_two_le_nonzero_not_one
  3. L8
    exact hzero
03Fix variables and assumptionsL9–9

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

  1. L9
    intro hone
04Use earlier factsL10–18

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

  1. L10
    apply hu
  2. L11
    specialize gaussian_norm_one_is_unit (z)
  3. L12
    apply gaussian_norm_one_is_unit
  4. L13
    specialize gaussian_norm_value_transport (z)
  5. L14
    specialize gaussian_norm_value_transport (N)
  6. L15
    specialize gaussian_norm_value_transport (1)
  7. L16
    apply gaussian_norm_value_transport
  8. L17
    exact hone
  9. L18
    exact hn

Library-wide reading audit

Original defined command ledger · 18 lines
  1. 0001intro z
  2. 0002intro N
  3. 0003intro hn
  4. 0004intro hzero
  5. 0005intro hu
  6. 0006specialize gaussian_search_two_le_nonzero_not_one (N)
  7. 0007apply gaussian_search_two_le_nonzero_not_one
  8. 0008exact hzero
  9. 0009intro hone
  10. 0010apply hu
  11. 0011specialize gaussian_norm_one_is_unit (z)
  12. 0012apply gaussian_norm_one_is_unit
  13. 0013specialize gaussian_norm_value_transport (z)
  14. 0014specialize gaussian_norm_value_transport (N)
  15. 0015specialize gaussian_norm_value_transport (1)
  16. 0016apply gaussian_norm_value_transport
  17. 0017exact hone
  18. 0018exact hn