GF0077

gaussian_search_bounded_coordinates_monotone

A larger norm bound preserves the two actual finite signed-coordinate bounds.

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. ∀ D. ∀ N. GNormBoundedCoordinates(z,D) → Le(D,N) → GNormBoundedCoordinates(z,N)

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

Definition DAG

GNormBoundedCoordinates(z,N) · 2Le(a,b) · 2

Actual proof prerequisites

mul_le_mul_left · checked external prerequisitele_trans · checked external prerequisite
Original expanded first-order statement 
forall z D N. (exists gr_search_real_coordinates_old gr_search_imaginary_coordinates_old. (((z)=((gr_search_real_coordinates_old) + (gr_search_imaginary_coordinates_old)) * S ((gr_search_real_coordinates_old) + (gr_search_imaginary_coordinates_old)) + ((gr_search_imaginary_coordinates_old) + (gr_search_imaginary_coordinates_old))) /\ ((exists ge_gap_coordinates_oldreal_bound. ge_gap_coordinates_oldreal_bound + (gr_search_real_coordinates_old) = (2*(D))) /\ (exists ge_gap_coordinates_oldimaginary_bound. ge_gap_coordinates_oldimaginary_bound + (gr_search_imaginary_coordinates_old) = (2*(D)))))) -> (exists ge_gap_coordinates_bound_order. ge_gap_coordinates_bound_order + (D) = (N)) -> (exists gr_search_real_coordinates_new gr_search_imaginary_coordinates_new. (((z)=((gr_search_real_coordinates_new) + (gr_search_imaginary_coordinates_new)) * S ((gr_search_real_coordinates_new) + (gr_search_imaginary_coordinates_new)) + ((gr_search_imaginary_coordinates_new) + (gr_search_imaginary_coordinates_new))) /\ ((exists ge_gap_coordinates_newreal_bound. ge_gap_coordinates_newreal_bound + (gr_search_real_coordinates_new) = (2*(N))) /\ (exists ge_gap_coordinates_newimaginary_bound. ge_gap_coordinates_newimaginary_bound + (gr_search_imaginary_coordinates_new) = (2*(N))))))

Complete tactic proof in conservative notation

All 32 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

32 script commands · 9 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.

01Fix variables and assumptionsL1–5

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

  1. L1
    intro z
  2. L2
    intro D
  3. L3
    intro N
  4. L4
    intro hc
  5. L5
    intro hb
02Separate the logical casesL6–9

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

  1. L6
    cases hc
  2. L7
    cases hc_witness
  3. L8
    cases hc_witness_witness
  4. L9
    cases hc_witness_witness_right
03Establish hscaleL10–15

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply mul le mul left.

  1. L10
    have hscale : Le(2 · D,2 · N)Definitions: Le(2 · D,2 · N)Original native command in the exact edition
  2. L11
    specialize mul_le_mul_left (D)
  3. L12
    specialize mul_le_mul_left (N)
  4. L13
    specialize mul_le_mul_left (2)
  5. L14
    apply mul_le_mul_left
  6. L15
    exact hb
04Construct an explicit witnessL16–17

Supply the displayed value, then prove that it has the required property.

  1. L16
    exists (x)
  2. L17
    exists (x1)
05Separate the logical casesL18–18

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

  1. L18
    split
06Use earlier factsL19–19

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

  1. L19
    exact hc_witness_witness_left
07Separate the logical casesL20–20

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

  1. L20
    split
08Use earlier factsL21–30

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

  1. L21
    specialize le_trans (x)
  2. L22
    specialize le_trans (2*D)
  3. L23
    specialize le_trans (2*N)
  4. L24
    apply le_trans
  5. L25
    exact hc_witness_witness_right_left
  6. L26
    exact hscale
  7. L27
    specialize le_trans (x1)
  8. L28
    specialize le_trans (2*D)
  9. L29
    specialize le_trans (2*N)
  10. L30
    apply le_trans
09Use earlier factsL31–32

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

  1. L31
    exact hc_witness_witness_right_right
  2. L32
    exact hscale

Library-wide reading audit

Original defined command ledger · 32 lines
  1. 0001intro z
  2. 0002intro D
  3. 0003intro N
  4. 0004intro hc
  5. 0005intro hb
  6. 0006cases hc
  7. 0007cases hc_witness
  8. 0008cases hc_witness_witness
  9. 0009cases hc_witness_witness_right
  10. 0010have hscale : Le(2 · D,2 · N)
  11. 0011specialize mul_le_mul_left (D)
  12. 0012specialize mul_le_mul_left (N)
  13. 0013specialize mul_le_mul_left (2)
  14. 0014apply mul_le_mul_left
  15. 0015exact hb
  16. 0016exists (x)
  17. 0017exists (x1)
  18. 0018split
  19. 0019exact hc_witness_witness_left
  20. 0020split
  21. 0021specialize le_trans (x)
  22. 0022specialize le_trans (2*D)
  23. 0023specialize le_trans (2*N)
  24. 0024apply le_trans
  25. 0025exact hc_witness_witness_right_left
  26. 0026exact hscale
  27. 0027specialize le_trans (x1)
  28. 0028specialize le_trans (2*D)
  29. 0029specialize le_trans (2*N)
  30. 0030apply le_trans
  31. 0031exact hc_witness_witness_right_right
  32. 0032exact hscale