GF005C

gaussian_divisor_norm_bound

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

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

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 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))

Constructive proof overview

Generated structural guide

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

The unchanged tactic script uses 3 declared prerequisites and contains 41 exact native proof lines.

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

Proof neighborhood

Direct dependencies

GF005B gaussian_divisor_norm_factor GF0016 gaussian_norm_nonzero nonzero_is_succ Stable theorem; checked-use authorized

Direct dependents

none

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

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.

Named ingredients (2)
gaussian divisor norm factorgaussian norm nonzero

Long local formulas use this family’s existing definitions. Each new abbreviation was expanded back to the identical native formula, including its free-variable context. The original edition is preserved below.

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: GNormGMul
  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 exact 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 : exists q Q. ((exists ge_first_rp_norm_bound_product ge_first_rn_norm_bound_product ge_first_ip_norm_bound_product ge_first_in_norm_bound_product ge_second_rp_norm_bound_product ge_second_rn_norm_bound_product ge_second_ip_norm_bound_product ge_second_in_norm_bound_product. ((exists ge_representation_real_code_norm_bound_productfirst ge_representation_imaginary_code_norm_bound_productfirst. (((d) = ((ge_representation_real_code_norm_bound_productfirst) + (ge_representation_imaginary_code_norm_bound_productfirst)) * S ((ge_representation_real_code_norm_bound_productfirst) + (ge_representation_imaginary_code_norm_bound_productfirst)) + ((ge_representation_imaginary_code_norm_bound_productfirst) + (ge_representation_imaginary_code_norm_bound_productfirst))) /\ ((exists ge_balance_positive_norm_bound_productfirstreal ge_balance_negative_norm_bound_productfirstreal. (((((ge_representation_real_code_norm_bound_productfirst) = 2 * (ge_balance_positive_norm_bound_productfirstreal) /\ (ge_balance_negative_norm_bound_productfirstreal) = 0) \/ exists ge_signed_half_norm_bound_productfirstrealdecode. (((ge_representation_real_code_norm_bound_productfirst) = 2 * ge_signed_half_norm_bound_productfirstrealdecode + 1 /\ (ge_balance_positive_norm_bound_productfirstreal) = 0) /\ (ge_balance_negative_norm_bound_productfirstreal) = S ge_signed_half_norm_bound_productfirstrealdecode))) /\ ((ge_first_rp_norm_bound_product) + ge_balance_negative_norm_bound_productfirstreal = (ge_first_rn_norm_bound_product) + ge_balance_positive_norm_bound_productfirstreal))) /\ (exists ge_balance_positive_norm_bound_productfirstimaginary ge_balance_negative_norm_bound_productfirstimaginary. (((((ge_representation_imaginary_code_norm_bound_productfirst) = 2 * (ge_balance_positive_norm_bound_productfirstimaginary) /\ (ge_balance_negative_norm_bound_productfirstimaginary) = 0) \/ exists ge_signed_half_norm_bound_productfirstimaginarydecode. (((ge_representation_imaginary_code_norm_bound_productfirst) = 2 * ge_signed_half_norm_bound_productfirstimaginarydecode + 1 /\ (ge_balance_positive_norm_bound_productfirstimaginary) = 0) /\ (ge_balance_negative_norm_bound_productfirstimaginary) = S ge_signed_half_norm_bound_productfirstimaginarydecode))) /\ ((ge_first_ip_norm_bound_product) + ge_balance_negative_norm_bound_productfirstimaginary = (ge_first_in_norm_bound_product) + ge_balance_positive_norm_bound_productfirstimaginary)))))) /\ ((exists ge_representation_real_code_norm_bound_productsecond ge_representation_imaginary_code_norm_bound_productsecond. (((q) = ((ge_representation_real_code_norm_bound_productsecond) + (ge_representation_imaginary_code_norm_bound_productsecond)) * S ((ge_representation_real_code_norm_bound_productsecond) + (ge_representation_imaginary_code_norm_bound_productsecond)) + ((ge_representation_imaginary_code_norm_bound_productsecond) + (ge_representation_imaginary_code_norm_bound_productsecond))) /\ ((exists ge_balance_positive_norm_bound_productsecondreal ge_balance_negative_norm_bound_productsecondreal. (((((ge_representation_real_code_norm_bound_productsecond) = 2 * (ge_balance_positive_norm_bound_productsecondreal) /\ (ge_balance_negative_norm_bound_productsecondreal) = 0) \/ exists ge_signed_half_norm_bound_productsecondrealdecode. (((ge_representation_real_code_norm_bound_productsecond) = 2 * ge_signed_half_norm_bound_productsecondrealdecode + 1 /\ (ge_balance_positive_norm_bound_productsecondreal) = 0) /\ (ge_balance_negative_norm_bound_productsecondreal) = S ge_signed_half_norm_bound_productsecondrealdecode))) /\ ((ge_second_rp_norm_bound_product) + ge_balance_negative_norm_bound_productsecondreal = (ge_second_rn_norm_bound_product) + ge_balance_positive_norm_bound_productsecondreal))) /\ (exists ge_balance_positive_norm_bound_productsecondimaginary ge_balance_negative_norm_bound_productsecondimaginary. (((((ge_representation_imaginary_code_norm_bound_productsecond) = 2 * (ge_balance_positive_norm_bound_productsecondimaginary) /\ (ge_balance_negative_norm_bound_productsecondimaginary) = 0) \/ exists ge_signed_half_norm_bound_productsecondimaginarydecode. (((ge_representation_imaginary_code_norm_bound_productsecond) = 2 * ge_signed_half_norm_bound_productsecondimaginarydecode + 1 /\ (ge_balance_positive_norm_bound_productsecondimaginary) = 0) /\ (ge_balance_negative_norm_bound_productsecondimaginary) = S ge_signed_half_norm_bound_productsecondimaginarydecode))) /\ ((ge_second_ip_norm_bound_product) + ge_balance_negative_norm_bound_productsecondimaginary = (ge_second_in_norm_bound_product) + ge_balance_positive_norm_bound_productsecondimaginary)))))) /\ (exists ge_representation_real_code_norm_bound_productoutput ge_representation_imaginary_code_norm_bound_productoutput. (((z) = ((ge_representation_real_code_norm_bound_productoutput) + (ge_representation_imaginary_code_norm_bound_productoutput)) * S ((ge_representation_real_code_norm_bound_productoutput) + (ge_representation_imaginary_code_norm_bound_productoutput)) + ((ge_representation_imaginary_code_norm_bound_productoutput) + (ge_representation_imaginary_code_norm_bound_productoutput))) /\ ((exists ge_balance_positive_norm_bound_productoutputreal ge_balance_negative_norm_bound_productoutputreal. (((((ge_representation_real_code_norm_bound_productoutput) = 2 * (ge_balance_positive_norm_bound_productoutputreal) /\ (ge_balance_negative_norm_bound_productoutputreal) = 0) \/ exists ge_signed_half_norm_bound_productoutputrealdecode. (((ge_representation_real_code_norm_bound_productoutput) = 2 * ge_signed_half_norm_bound_productoutputrealdecode + 1 /\ (ge_balance_positive_norm_bound_productoutputreal) = 0) /\ (ge_balance_negative_norm_bound_productoutputreal) = S ge_signed_half_norm_bound_productoutputrealdecode))) /\ ((((((((ge_first_rp_norm_bound_product) * (ge_second_rp_norm_bound_product))) + (((ge_first_rn_norm_bound_product) * (ge_second_rn_norm_bound_product))))) + (((((ge_first_ip_norm_bound_product) * (ge_second_in_norm_bound_product))) + (((ge_first_in_norm_bound_product) * (ge_second_ip_norm_bound_product))))))) + ge_balance_negative_norm_bound_productoutputreal = (((((((ge_first_rp_norm_bound_product) * (ge_second_rn_norm_bound_product))) + (((ge_first_rn_norm_bound_product) * (ge_second_rp_norm_bound_product))))) + (((((ge_first_ip_norm_bound_product) * (ge_second_ip_norm_bound_product))) + (((ge_first_in_norm_bound_product) * (ge_second_in_norm_bound_product))))))) + ge_balance_positive_norm_bound_productoutputreal))) /\ (exists ge_balance_positive_norm_bound_productoutputimaginary ge_balance_negative_norm_bound_productoutputimaginary. (((((ge_representation_imaginary_code_norm_bound_productoutput) = 2 * (ge_balance_positive_norm_bound_productoutputimaginary) /\ (ge_balance_negative_norm_bound_productoutputimaginary) = 0) \/ exists ge_signed_half_norm_bound_productoutputimaginarydecode. (((ge_representation_imaginary_code_norm_bound_productoutput) = 2 * ge_signed_half_norm_bound_productoutputimaginarydecode + 1 /\ (ge_balance_positive_norm_bound_productoutputimaginary) = 0) /\ (ge_balance_negative_norm_bound_productoutputimaginary) = S ge_signed_half_norm_bound_productoutputimaginarydecode))) /\ ((((((((ge_first_rp_norm_bound_product) * (ge_second_ip_norm_bound_product))) + (((ge_first_rn_norm_bound_product) * (ge_second_in_norm_bound_product))))) + (((((ge_first_ip_norm_bound_product) * (ge_second_rp_norm_bound_product))) + (((ge_first_in_norm_bound_product) * (ge_second_rn_norm_bound_product))))))) + ge_balance_negative_norm_bound_productoutputimaginary = (((((((ge_first_rp_norm_bound_product) * (ge_second_in_norm_bound_product))) + (((ge_first_rn_norm_bound_product) * (ge_second_ip_norm_bound_product))))) + (((((ge_first_ip_norm_bound_product) * (ge_second_rn_norm_bound_product))) + (((ge_first_in_norm_bound_product) * (ge_second_rp_norm_bound_product))))))) + ge_balance_positive_norm_bound_productoutputimaginary))))))))) /\ ((exists ge_norm_rp_norm_bound_quotient ge_norm_rn_norm_bound_quotient ge_norm_ip_norm_bound_quotient ge_norm_in_norm_bound_quotient. ((exists ge_representation_real_code_norm_bound_quotientrepresentation ge_representation_imaginary_code_norm_bound_quotientrepresentation. (((q) = ((ge_representation_real_code_norm_bound_quotientrepresentation) + (ge_representation_imaginary_code_norm_bound_quotientrepresentation)) * S ((ge_representation_real_code_norm_bound_quotientrepresentation) + (ge_representation_imaginary_code_norm_bound_quotientrepresentation)) + ((ge_representation_imaginary_code_norm_bound_quotientrepresentation) + (ge_representation_imaginary_code_norm_bound_quotientrepresentation))) /\ ((exists ge_balance_positive_norm_bound_quotientrepresentationreal ge_balance_negative_norm_bound_quotientrepresentationreal. (((((ge_representation_real_code_norm_bound_quotientrepresentation) = 2 * (ge_balance_positive_norm_bound_quotientrepresentationreal) /\ (ge_balance_negative_norm_bound_quotientrepresentationreal) = 0) \/ exists ge_signed_half_norm_bound_quotientrepresentationrealdecode. (((ge_representation_real_code_norm_bound_quotientrepresentation) = 2 * ge_signed_half_norm_bound_quotientrepresentationrealdecode + 1 /\ (ge_balance_positive_norm_bound_quotientrepresentationreal) = 0) /\ (ge_balance_negative_norm_bound_quotientrepresentationreal) = S ge_signed_half_norm_bound_quotientrepresentationrealdecode))) /\ ((ge_norm_rp_norm_bound_quotient) + ge_balance_negative_norm_bound_quotientrepresentationreal = (ge_norm_rn_norm_bound_quotient) + ge_balance_positive_norm_bound_quotientrepresentationreal))) /\ (exists ge_balance_positive_norm_bound_quotientrepresentationimaginary ge_balance_negative_norm_bound_quotientrepresentationimaginary. (((((ge_representation_imaginary_code_norm_bound_quotientrepresentation) = 2 * (ge_balance_positive_norm_bound_quotientrepresentationimaginary) /\ (ge_balance_negative_norm_bound_quotientrepresentationimaginary) = 0) \/ exists ge_signed_half_norm_bound_quotientrepresentationimaginarydecode. (((ge_representation_imaginary_code_norm_bound_quotientrepresentation) = 2 * ge_signed_half_norm_bound_quotientrepresentationimaginarydecode + 1 /\ (ge_balance_positive_norm_bound_quotientrepresentationimaginary) = 0) /\ (ge_balance_negative_norm_bound_quotientrepresentationimaginary) = S ge_signed_half_norm_bound_quotientrepresentationimaginarydecode))) /\ ((ge_norm_ip_norm_bound_quotient) + ge_balance_negative_norm_bound_quotientrepresentationimaginary = (ge_norm_in_norm_bound_quotient) + ge_balance_positive_norm_bound_quotientrepresentationimaginary)))))) /\ (exists ge_real_square_norm_bound_quotientsquare ge_imaginary_square_norm_bound_quotientsquare. ((((((ge_norm_rp_norm_bound_quotient) * (ge_norm_rp_norm_bound_quotient))) + (((ge_norm_rn_norm_bound_quotient) * (ge_norm_rn_norm_bound_quotient)))) = ((ge_real_square_norm_bound_quotientsquare) + (((((ge_norm_rp_norm_bound_quotient) * (ge_norm_rn_norm_bound_quotient))) + (((ge_norm_rn_norm_bound_quotient) * (ge_norm_rp_norm_bound_quotient))))))) /\ ((((((ge_norm_ip_norm_bound_quotient) * (ge_norm_ip_norm_bound_quotient))) + (((ge_norm_in_norm_bound_quotient) * (ge_norm_in_norm_bound_quotient)))) = ((ge_imaginary_square_norm_bound_quotientsquare) + (((((ge_norm_ip_norm_bound_quotient) * (ge_norm_in_norm_bound_quotient))) + (((ge_norm_in_norm_bound_quotient) * (ge_norm_ip_norm_bound_quotient))))))) /\ ((Q) = ge_real_square_norm_bound_quotientsquare + ge_imaginary_square_norm_bound_quotientsquare)))))) /\ (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