GF005C

gaussian_divisor_norm_bound

Every actual divisor of a nonzero Gaussian value has norm at most the value norm; the positive quotient-norm gap is constructed explicitly.

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. ∀ D. ∀ N. GDvd(d,z) → GNorm(d,D) → GNorm(z,N) → ¬z = 0 → Le(D,N)

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

Definition DAG

GNorm(z,n) · 3GMul(a,b,c) · 1GDvd(d,z) · 1Le(a,b) · 1

Actual proof prerequisites

gaussian_divisor_norm_factorgaussian_norm_nonzerononzero_is_succ · checked external prerequisite
Original expanded first-order statement 
forall d z D N. (exists gr_quotient_norm_bound_divisor. (exists ge_first_rp_norm_bound_divisorproduct ge_first_rn_norm_bound_divisorproduct ge_first_ip_norm_bound_divisorproduct ge_first_in_norm_bound_divisorproduct ge_second_rp_norm_bound_divisorproduct ge_second_rn_norm_bound_divisorproduct ge_second_ip_norm_bound_divisorproduct ge_second_in_norm_bound_divisorproduct. ((exists ge_representation_real_code_norm_bound_divisorproductfirst ge_representation_imaginary_code_norm_bound_divisorproductfirst. (((d) = ((ge_representation_real_code_norm_bound_divisorproductfirst) + (ge_representation_imaginary_code_norm_bound_divisorproductfirst)) * S ((ge_representation_real_code_norm_bound_divisorproductfirst) + (ge_representation_imaginary_code_norm_bound_divisorproductfirst)) + ((ge_representation_imaginary_code_norm_bound_divisorproductfirst) + (ge_representation_imaginary_code_norm_bound_divisorproductfirst))) /\ ((exists ge_balance_positive_norm_bound_divisorproductfirstreal ge_balance_negative_norm_bound_divisorproductfirstreal. (((((ge_representation_real_code_norm_bound_divisorproductfirst) = 2 * (ge_balance_positive_norm_bound_divisorproductfirstreal) /\ (ge_balance_negative_norm_bound_divisorproductfirstreal) = 0) \/ exists ge_signed_half_norm_bound_divisorproductfirstrealdecode. (((ge_representation_real_code_norm_bound_divisorproductfirst) = 2 * ge_signed_half_norm_bound_divisorproductfirstrealdecode + 1 /\ (ge_balance_positive_norm_bound_divisorproductfirstreal) = 0) /\ (ge_balance_negative_norm_bound_divisorproductfirstreal) = S ge_signed_half_norm_bound_divisorproductfirstrealdecode))) /\ ((ge_first_rp_norm_bound_divisorproduct) + ge_balance_negative_norm_bound_divisorproductfirstreal = (ge_first_rn_norm_bound_divisorproduct) + ge_balance_positive_norm_bound_divisorproductfirstreal))) /\ (exists ge_balance_positive_norm_bound_divisorproductfirstimaginary ge_balance_negative_norm_bound_divisorproductfirstimaginary. (((((ge_representation_imaginary_code_norm_bound_divisorproductfirst) = 2 * (ge_balance_positive_norm_bound_divisorproductfirstimaginary) /\ (ge_balance_negative_norm_bound_divisorproductfirstimaginary) = 0) \/ exists ge_signed_half_norm_bound_divisorproductfirstimaginarydecode. (((ge_representation_imaginary_code_norm_bound_divisorproductfirst) = 2 * ge_signed_half_norm_bound_divisorproductfirstimaginarydecode + 1 /\ (ge_balance_positive_norm_bound_divisorproductfirstimaginary) = 0) /\ (ge_balance_negative_norm_bound_divisorproductfirstimaginary) = S ge_signed_half_norm_bound_divisorproductfirstimaginarydecode))) /\ ((ge_first_ip_norm_bound_divisorproduct) + ge_balance_negative_norm_bound_divisorproductfirstimaginary = (ge_first_in_norm_bound_divisorproduct) + ge_balance_positive_norm_bound_divisorproductfirstimaginary)))))) /\ ((exists ge_representation_real_code_norm_bound_divisorproductsecond ge_representation_imaginary_code_norm_bound_divisorproductsecond. (((gr_quotient_norm_bound_divisor) = ((ge_representation_real_code_norm_bound_divisorproductsecond) + (ge_representation_imaginary_code_norm_bound_divisorproductsecond)) * S ((ge_representation_real_code_norm_bound_divisorproductsecond) + (ge_representation_imaginary_code_norm_bound_divisorproductsecond)) + ((ge_representation_imaginary_code_norm_bound_divisorproductsecond) + (ge_representation_imaginary_code_norm_bound_divisorproductsecond))) /\ ((exists ge_balance_positive_norm_bound_divisorproductsecondreal ge_balance_negative_norm_bound_divisorproductsecondreal. (((((ge_representation_real_code_norm_bound_divisorproductsecond) = 2 * (ge_balance_positive_norm_bound_divisorproductsecondreal) /\ (ge_balance_negative_norm_bound_divisorproductsecondreal) = 0) \/ exists ge_signed_half_norm_bound_divisorproductsecondrealdecode. (((ge_representation_real_code_norm_bound_divisorproductsecond) = 2 * ge_signed_half_norm_bound_divisorproductsecondrealdecode + 1 /\ (ge_balance_positive_norm_bound_divisorproductsecondreal) = 0) /\ (ge_balance_negative_norm_bound_divisorproductsecondreal) = S ge_signed_half_norm_bound_divisorproductsecondrealdecode))) /\ ((ge_second_rp_norm_bound_divisorproduct) + ge_balance_negative_norm_bound_divisorproductsecondreal = (ge_second_rn_norm_bound_divisorproduct) + ge_balance_positive_norm_bound_divisorproductsecondreal))) /\ (exists ge_balance_positive_norm_bound_divisorproductsecondimaginary ge_balance_negative_norm_bound_divisorproductsecondimaginary. (((((ge_representation_imaginary_code_norm_bound_divisorproductsecond) = 2 * (ge_balance_positive_norm_bound_divisorproductsecondimaginary) /\ (ge_balance_negative_norm_bound_divisorproductsecondimaginary) = 0) \/ exists ge_signed_half_norm_bound_divisorproductsecondimaginarydecode. (((ge_representation_imaginary_code_norm_bound_divisorproductsecond) = 2 * ge_signed_half_norm_bound_divisorproductsecondimaginarydecode + 1 /\ (ge_balance_positive_norm_bound_divisorproductsecondimaginary) = 0) /\ (ge_balance_negative_norm_bound_divisorproductsecondimaginary) = S ge_signed_half_norm_bound_divisorproductsecondimaginarydecode))) /\ ((ge_second_ip_norm_bound_divisorproduct) + ge_balance_negative_norm_bound_divisorproductsecondimaginary = (ge_second_in_norm_bound_divisorproduct) + ge_balance_positive_norm_bound_divisorproductsecondimaginary)))))) /\ (exists ge_representation_real_code_norm_bound_divisorproductoutput ge_representation_imaginary_code_norm_bound_divisorproductoutput. (((z) = ((ge_representation_real_code_norm_bound_divisorproductoutput) + (ge_representation_imaginary_code_norm_bound_divisorproductoutput)) * S ((ge_representation_real_code_norm_bound_divisorproductoutput) + (ge_representation_imaginary_code_norm_bound_divisorproductoutput)) + ((ge_representation_imaginary_code_norm_bound_divisorproductoutput) + (ge_representation_imaginary_code_norm_bound_divisorproductoutput))) /\ ((exists ge_balance_positive_norm_bound_divisorproductoutputreal ge_balance_negative_norm_bound_divisorproductoutputreal. (((((ge_representation_real_code_norm_bound_divisorproductoutput) = 2 * (ge_balance_positive_norm_bound_divisorproductoutputreal) /\ (ge_balance_negative_norm_bound_divisorproductoutputreal) = 0) \/ exists ge_signed_half_norm_bound_divisorproductoutputrealdecode. (((ge_representation_real_code_norm_bound_divisorproductoutput) = 2 * ge_signed_half_norm_bound_divisorproductoutputrealdecode + 1 /\ (ge_balance_positive_norm_bound_divisorproductoutputreal) = 0) /\ (ge_balance_negative_norm_bound_divisorproductoutputreal) = S ge_signed_half_norm_bound_divisorproductoutputrealdecode))) /\ ((((((((ge_first_rp_norm_bound_divisorproduct) * (ge_second_rp_norm_bound_divisorproduct))) + (((ge_first_rn_norm_bound_divisorproduct) * (ge_second_rn_norm_bound_divisorproduct))))) + (((((ge_first_ip_norm_bound_divisorproduct) * (ge_second_in_norm_bound_divisorproduct))) + (((ge_first_in_norm_bound_divisorproduct) * (ge_second_ip_norm_bound_divisorproduct))))))) + ge_balance_negative_norm_bound_divisorproductoutputreal = (((((((ge_first_rp_norm_bound_divisorproduct) * (ge_second_rn_norm_bound_divisorproduct))) + (((ge_first_rn_norm_bound_divisorproduct) * (ge_second_rp_norm_bound_divisorproduct))))) + (((((ge_first_ip_norm_bound_divisorproduct) * (ge_second_ip_norm_bound_divisorproduct))) + (((ge_first_in_norm_bound_divisorproduct) * (ge_second_in_norm_bound_divisorproduct))))))) + ge_balance_positive_norm_bound_divisorproductoutputreal))) /\ (exists ge_balance_positive_norm_bound_divisorproductoutputimaginary ge_balance_negative_norm_bound_divisorproductoutputimaginary. (((((ge_representation_imaginary_code_norm_bound_divisorproductoutput) = 2 * (ge_balance_positive_norm_bound_divisorproductoutputimaginary) /\ (ge_balance_negative_norm_bound_divisorproductoutputimaginary) = 0) \/ exists ge_signed_half_norm_bound_divisorproductoutputimaginarydecode. (((ge_representation_imaginary_code_norm_bound_divisorproductoutput) = 2 * ge_signed_half_norm_bound_divisorproductoutputimaginarydecode + 1 /\ (ge_balance_positive_norm_bound_divisorproductoutputimaginary) = 0) /\ (ge_balance_negative_norm_bound_divisorproductoutputimaginary) = S ge_signed_half_norm_bound_divisorproductoutputimaginarydecode))) /\ ((((((((ge_first_rp_norm_bound_divisorproduct) * (ge_second_ip_norm_bound_divisorproduct))) + (((ge_first_rn_norm_bound_divisorproduct) * (ge_second_in_norm_bound_divisorproduct))))) + (((((ge_first_ip_norm_bound_divisorproduct) * (ge_second_rp_norm_bound_divisorproduct))) + (((ge_first_in_norm_bound_divisorproduct) * (ge_second_rn_norm_bound_divisorproduct))))))) + ge_balance_negative_norm_bound_divisorproductoutputimaginary = (((((((ge_first_rp_norm_bound_divisorproduct) * (ge_second_in_norm_bound_divisorproduct))) + (((ge_first_rn_norm_bound_divisorproduct) * (ge_second_ip_norm_bound_divisorproduct))))) + (((((ge_first_ip_norm_bound_divisorproduct) * (ge_second_rn_norm_bound_divisorproduct))) + (((ge_first_in_norm_bound_divisorproduct) * (ge_second_rp_norm_bound_divisorproduct))))))) + ge_balance_positive_norm_bound_divisorproductoutputimaginary)))))))))) -> (exists ge_norm_rp_norm_bound_first ge_norm_rn_norm_bound_first ge_norm_ip_norm_bound_first ge_norm_in_norm_bound_first. ((exists ge_representation_real_code_norm_bound_firstrepresentation ge_representation_imaginary_code_norm_bound_firstrepresentation. (((d) = ((ge_representation_real_code_norm_bound_firstrepresentation) + (ge_representation_imaginary_code_norm_bound_firstrepresentation)) * S ((ge_representation_real_code_norm_bound_firstrepresentation) + (ge_representation_imaginary_code_norm_bound_firstrepresentation)) + ((ge_representation_imaginary_code_norm_bound_firstrepresentation) + (ge_representation_imaginary_code_norm_bound_firstrepresentation))) /\ ((exists ge_balance_positive_norm_bound_firstrepresentationreal ge_balance_negative_norm_bound_firstrepresentationreal. (((((ge_representation_real_code_norm_bound_firstrepresentation) = 2 * (ge_balance_positive_norm_bound_firstrepresentationreal) /\ (ge_balance_negative_norm_bound_firstrepresentationreal) = 0) \/ exists ge_signed_half_norm_bound_firstrepresentationrealdecode. (((ge_representation_real_code_norm_bound_firstrepresentation) = 2 * ge_signed_half_norm_bound_firstrepresentationrealdecode + 1 /\ (ge_balance_positive_norm_bound_firstrepresentationreal) = 0) /\ (ge_balance_negative_norm_bound_firstrepresentationreal) = S ge_signed_half_norm_bound_firstrepresentationrealdecode))) /\ ((ge_norm_rp_norm_bound_first) + ge_balance_negative_norm_bound_firstrepresentationreal = (ge_norm_rn_norm_bound_first) + ge_balance_positive_norm_bound_firstrepresentationreal))) /\ (exists ge_balance_positive_norm_bound_firstrepresentationimaginary ge_balance_negative_norm_bound_firstrepresentationimaginary. (((((ge_representation_imaginary_code_norm_bound_firstrepresentation) = 2 * (ge_balance_positive_norm_bound_firstrepresentationimaginary) /\ (ge_balance_negative_norm_bound_firstrepresentationimaginary) = 0) \/ exists ge_signed_half_norm_bound_firstrepresentationimaginarydecode. (((ge_representation_imaginary_code_norm_bound_firstrepresentation) = 2 * ge_signed_half_norm_bound_firstrepresentationimaginarydecode + 1 /\ (ge_balance_positive_norm_bound_firstrepresentationimaginary) = 0) /\ (ge_balance_negative_norm_bound_firstrepresentationimaginary) = S ge_signed_half_norm_bound_firstrepresentationimaginarydecode))) /\ ((ge_norm_ip_norm_bound_first) + ge_balance_negative_norm_bound_firstrepresentationimaginary = (ge_norm_in_norm_bound_first) + ge_balance_positive_norm_bound_firstrepresentationimaginary)))))) /\ (exists ge_real_square_norm_bound_firstsquare ge_imaginary_square_norm_bound_firstsquare. ((((((ge_norm_rp_norm_bound_first) * (ge_norm_rp_norm_bound_first))) + (((ge_norm_rn_norm_bound_first) * (ge_norm_rn_norm_bound_first)))) = ((ge_real_square_norm_bound_firstsquare) + (((((ge_norm_rp_norm_bound_first) * (ge_norm_rn_norm_bound_first))) + (((ge_norm_rn_norm_bound_first) * (ge_norm_rp_norm_bound_first))))))) /\ ((((((ge_norm_ip_norm_bound_first) * (ge_norm_ip_norm_bound_first))) + (((ge_norm_in_norm_bound_first) * (ge_norm_in_norm_bound_first)))) = ((ge_imaginary_square_norm_bound_firstsquare) + (((((ge_norm_ip_norm_bound_first) * (ge_norm_in_norm_bound_first))) + (((ge_norm_in_norm_bound_first) * (ge_norm_ip_norm_bound_first))))))) /\ ((D) = ge_real_square_norm_bound_firstsquare + ge_imaginary_square_norm_bound_firstsquare)))))) -> (exists ge_norm_rp_norm_bound_total ge_norm_rn_norm_bound_total ge_norm_ip_norm_bound_total ge_norm_in_norm_bound_total. ((exists ge_representation_real_code_norm_bound_totalrepresentation ge_representation_imaginary_code_norm_bound_totalrepresentation. (((z) = ((ge_representation_real_code_norm_bound_totalrepresentation) + (ge_representation_imaginary_code_norm_bound_totalrepresentation)) * S ((ge_representation_real_code_norm_bound_totalrepresentation) + (ge_representation_imaginary_code_norm_bound_totalrepresentation)) + ((ge_representation_imaginary_code_norm_bound_totalrepresentation) + (ge_representation_imaginary_code_norm_bound_totalrepresentation))) /\ ((exists ge_balance_positive_norm_bound_totalrepresentationreal ge_balance_negative_norm_bound_totalrepresentationreal. (((((ge_representation_real_code_norm_bound_totalrepresentation) = 2 * (ge_balance_positive_norm_bound_totalrepresentationreal) /\ (ge_balance_negative_norm_bound_totalrepresentationreal) = 0) \/ exists ge_signed_half_norm_bound_totalrepresentationrealdecode. (((ge_representation_real_code_norm_bound_totalrepresentation) = 2 * ge_signed_half_norm_bound_totalrepresentationrealdecode + 1 /\ (ge_balance_positive_norm_bound_totalrepresentationreal) = 0) /\ (ge_balance_negative_norm_bound_totalrepresentationreal) = S ge_signed_half_norm_bound_totalrepresentationrealdecode))) /\ ((ge_norm_rp_norm_bound_total) + ge_balance_negative_norm_bound_totalrepresentationreal = (ge_norm_rn_norm_bound_total) + ge_balance_positive_norm_bound_totalrepresentationreal))) /\ (exists ge_balance_positive_norm_bound_totalrepresentationimaginary ge_balance_negative_norm_bound_totalrepresentationimaginary. (((((ge_representation_imaginary_code_norm_bound_totalrepresentation) = 2 * (ge_balance_positive_norm_bound_totalrepresentationimaginary) /\ (ge_balance_negative_norm_bound_totalrepresentationimaginary) = 0) \/ exists ge_signed_half_norm_bound_totalrepresentationimaginarydecode. (((ge_representation_imaginary_code_norm_bound_totalrepresentation) = 2 * ge_signed_half_norm_bound_totalrepresentationimaginarydecode + 1 /\ (ge_balance_positive_norm_bound_totalrepresentationimaginary) = 0) /\ (ge_balance_negative_norm_bound_totalrepresentationimaginary) = S ge_signed_half_norm_bound_totalrepresentationimaginarydecode))) /\ ((ge_norm_ip_norm_bound_total) + ge_balance_negative_norm_bound_totalrepresentationimaginary = (ge_norm_in_norm_bound_total) + ge_balance_positive_norm_bound_totalrepresentationimaginary)))))) /\ (exists ge_real_square_norm_bound_totalsquare ge_imaginary_square_norm_bound_totalsquare. ((((((ge_norm_rp_norm_bound_total) * (ge_norm_rp_norm_bound_total))) + (((ge_norm_rn_norm_bound_total) * (ge_norm_rn_norm_bound_total)))) = ((ge_real_square_norm_bound_totalsquare) + (((((ge_norm_rp_norm_bound_total) * (ge_norm_rn_norm_bound_total))) + (((ge_norm_rn_norm_bound_total) * (ge_norm_rp_norm_bound_total))))))) /\ ((((((ge_norm_ip_norm_bound_total) * (ge_norm_ip_norm_bound_total))) + (((ge_norm_in_norm_bound_total) * (ge_norm_in_norm_bound_total)))) = ((ge_imaginary_square_norm_bound_totalsquare) + (((((ge_norm_ip_norm_bound_total) * (ge_norm_in_norm_bound_total))) + (((ge_norm_in_norm_bound_total) * (ge_norm_ip_norm_bound_total))))))) /\ ((N) = ge_real_square_norm_bound_totalsquare + ge_imaginary_square_norm_bound_totalsquare)))))) -> ~(z=0) -> (exists ge_gap_norm_bound. ge_gap_norm_bound + (D) = (N))

Complete tactic proof in conservative notation

All 41 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

41 script commands · 9 reading checkpoints · 3 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 divisor norm factorgaussian norm nonzero
01Fix variables and assumptionsL1–8

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

  1. L1
    intro d
  2. L2
    intro z
  3. L3
    intro D
  4. L4
    intro N
  5. L5
    intro hdiv
  6. L6
    intro hd
  7. L7
    intro hz
  8. L8
    intro hnz
02Establish hfactorL9–17

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply gaussian divisor norm factor.

  1. L9
    have hfactor : ∃ q. ∃ Q. GMul(d,q,z) ∧ (GNorm(q,Q) ∧ N = D · Q)Definitions: GMul(d,q,z)GNorm(q,Q)Original native command in the exact edition
  2. L10
    specialize gaussian_divisor_norm_factor (d)
  3. L11
    specialize gaussian_divisor_norm_factor (z)
  4. L12
    specialize gaussian_divisor_norm_factor (D)
  5. L13
    specialize gaussian_divisor_norm_factor (N)
  6. L14
    apply gaussian_divisor_norm_factor
  7. L15
    exact hdiv
  8. L16
    exact hd
  9. L17
    exact hz
03Separate the logical casesL18–21

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

  1. L18
    cases hfactor
  2. L19
    cases hfactor_witness
  3. L20
    cases hfactor_witness_witness
  4. L21
    cases hfactor_witness_witness_right
04Establish hpositiveL22–31

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply gaussian norm nonzero.

  1. L22
    have hpositive : ~(x1=0)
  2. L23
    intro hzero
  3. L24
    specialize gaussian_norm_nonzero (z)
  4. L25
    specialize gaussian_norm_nonzero (N)
  5. L26
    apply gaussian_norm_nonzero
  6. L27
    exact hz
  7. L28
    exact hnz
  8. L29
    rewrite hzero at hfactor_witness_witness_right_right
  9. L30
    trans D*0
  10. L31
    exact hfactor_witness_witness_right_right
05Calculate and transport equalitiesL32–32

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

  1. L32
    simp
06Establish hsuccL33–36

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply nonzero is succ.

  1. L33
    have hsucc : exists h. x1=S h
  2. L34
    specialize nonzero_is_succ (x1)
  3. L35
    apply nonzero_is_succ
  4. L36
    exact hpositive
07Separate the logical casesL37–37

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

  1. L37
    cases hsucc
08Construct an explicit witnessL38–38

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

  1. L38
    exists (D*x2)
09Calculate and transport equalitiesL39–41

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

  1. L39
    rewrite hsucc_witness at hfactor_witness_witness_right_right
  2. L40
    rewrite hfactor_witness_witness_right_right
  3. L41
    simp

Library-wide reading audit

Original defined command ledger · 41 lines
  1. 0001intro d
  2. 0002intro z
  3. 0003intro D
  4. 0004intro N
  5. 0005intro hdiv
  6. 0006intro hd
  7. 0007intro hz
  8. 0008intro hnz
  9. 0009have hfactor : ∃ q. ∃ Q. GMul(d,q,z) ∧ (GNorm(q,Q) ∧ N = D · Q)
  10. 0010specialize gaussian_divisor_norm_factor (d)
  11. 0011specialize gaussian_divisor_norm_factor (z)
  12. 0012specialize gaussian_divisor_norm_factor (D)
  13. 0013specialize gaussian_divisor_norm_factor (N)
  14. 0014apply gaussian_divisor_norm_factor
  15. 0015exact hdiv
  16. 0016exact hd
  17. 0017exact hz
  18. 0018cases hfactor
  19. 0019cases hfactor_witness
  20. 0020cases hfactor_witness_witness
  21. 0021cases hfactor_witness_witness_right
  22. 0022have hpositive : ~(x1=0)
  23. 0023intro hzero
  24. 0024specialize gaussian_norm_nonzero (z)
  25. 0025specialize gaussian_norm_nonzero (N)
  26. 0026apply gaussian_norm_nonzero
  27. 0027exact hz
  28. 0028exact hnz
  29. 0029rewrite hzero at hfactor_witness_witness_right_right
  30. 0030trans D*0
  31. 0031exact hfactor_witness_witness_right_right
  32. 0032simp
  33. 0033have hsucc : exists h. x1=S h
  34. 0034specialize nonzero_is_succ (x1)
  35. 0035apply nonzero_is_succ
  36. 0036exact hpositive
  37. 0037cases hsucc
  38. 0038exists (D*x2)
  39. 0039rewrite hsucc_witness at hfactor_witness_witness_right_right
  40. 0040rewrite hfactor_witness_witness_right_right
  41. 0041simp