GF001A

gaussian_norm_one_is_unit

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

Norm one constructs an actual inverse: the canonical conjugate multiplies the value to Gaussian identity code six.

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 z. (exists ge_norm_rp_norm_unit_given ge_norm_rn_norm_unit_given ge_norm_ip_norm_unit_given ge_norm_in_norm_unit_given. ((exists ge_representation_real_code_norm_unit_givenrepresentation ge_representation_imaginary_code_norm_unit_givenrepresentation. (((z) = ((ge_representation_real_code_norm_unit_givenrepresentation) + (ge_representation_imaginary_code_norm_unit_givenrepresentation)) * S ((ge_representation_real_code_norm_unit_givenrepresentation) + (ge_representation_imaginary_code_norm_unit_givenrepresentation)) + ((ge_representation_imaginary_code_norm_unit_givenrepresentation) + (ge_representation_imaginary_code_norm_unit_givenrepresentation))) /\ ((exists ge_balance_positive_norm_unit_givenrepresentationreal ge_balance_negative_norm_unit_givenrepresentationreal. (((((ge_representation_real_code_norm_unit_givenrepresentation) = 2 * (ge_balance_positive_norm_unit_givenrepresentationreal) /\ (ge_balance_negative_norm_unit_givenrepresentationreal) = 0) \/ exists ge_signed_half_norm_unit_givenrepresentationrealdecode. (((ge_representation_real_code_norm_unit_givenrepresentation) = 2 * ge_signed_half_norm_unit_givenrepresentationrealdecode + 1 /\ (ge_balance_positive_norm_unit_givenrepresentationreal) = 0) /\ (ge_balance_negative_norm_unit_givenrepresentationreal) = S ge_signed_half_norm_unit_givenrepresentationrealdecode))) /\ ((ge_norm_rp_norm_unit_given) + ge_balance_negative_norm_unit_givenrepresentationreal = (ge_norm_rn_norm_unit_given) + ge_balance_positive_norm_unit_givenrepresentationreal))) /\ (exists ge_balance_positive_norm_unit_givenrepresentationimaginary ge_balance_negative_norm_unit_givenrepresentationimaginary. (((((ge_representation_imaginary_code_norm_unit_givenrepresentation) = 2 * (ge_balance_positive_norm_unit_givenrepresentationimaginary) /\ (ge_balance_negative_norm_unit_givenrepresentationimaginary) = 0) \/ exists ge_signed_half_norm_unit_givenrepresentationimaginarydecode. (((ge_representation_imaginary_code_norm_unit_givenrepresentation) = 2 * ge_signed_half_norm_unit_givenrepresentationimaginarydecode + 1 /\ (ge_balance_positive_norm_unit_givenrepresentationimaginary) = 0) /\ (ge_balance_negative_norm_unit_givenrepresentationimaginary) = S ge_signed_half_norm_unit_givenrepresentationimaginarydecode))) /\ ((ge_norm_ip_norm_unit_given) + ge_balance_negative_norm_unit_givenrepresentationimaginary = (ge_norm_in_norm_unit_given) + ge_balance_positive_norm_unit_givenrepresentationimaginary)))))) /\ (exists ge_real_square_norm_unit_givensquare ge_imaginary_square_norm_unit_givensquare. ((((((ge_norm_rp_norm_unit_given) * (ge_norm_rp_norm_unit_given))) + (((ge_norm_rn_norm_unit_given) * (ge_norm_rn_norm_unit_given)))) = ((ge_real_square_norm_unit_givensquare) + (((((ge_norm_rp_norm_unit_given) * (ge_norm_rn_norm_unit_given))) + (((ge_norm_rn_norm_unit_given) * (ge_norm_rp_norm_unit_given))))))) /\ ((((((ge_norm_ip_norm_unit_given) * (ge_norm_ip_norm_unit_given))) + (((ge_norm_in_norm_unit_given) * (ge_norm_in_norm_unit_given)))) = ((ge_imaginary_square_norm_unit_givensquare) + (((((ge_norm_ip_norm_unit_given) * (ge_norm_in_norm_unit_given))) + (((ge_norm_in_norm_unit_given) * (ge_norm_ip_norm_unit_given))))))) /\ ((1) = ge_real_square_norm_unit_givensquare + ge_imaginary_square_norm_unit_givensquare)))))) -> (exists gr_inverse_norm_unit_result. (exists ge_first_rp_norm_unit_resultidentity ge_first_rn_norm_unit_resultidentity ge_first_ip_norm_unit_resultidentity ge_first_in_norm_unit_resultidentity ge_second_rp_norm_unit_resultidentity ge_second_rn_norm_unit_resultidentity ge_second_ip_norm_unit_resultidentity ge_second_in_norm_unit_resultidentity. ((exists ge_representation_real_code_norm_unit_resultidentityfirst ge_representation_imaginary_code_norm_unit_resultidentityfirst. (((z) = ((ge_representation_real_code_norm_unit_resultidentityfirst) + (ge_representation_imaginary_code_norm_unit_resultidentityfirst)) * S ((ge_representation_real_code_norm_unit_resultidentityfirst) + (ge_representation_imaginary_code_norm_unit_resultidentityfirst)) + ((ge_representation_imaginary_code_norm_unit_resultidentityfirst) + (ge_representation_imaginary_code_norm_unit_resultidentityfirst))) /\ ((exists ge_balance_positive_norm_unit_resultidentityfirstreal ge_balance_negative_norm_unit_resultidentityfirstreal. (((((ge_representation_real_code_norm_unit_resultidentityfirst) = 2 * (ge_balance_positive_norm_unit_resultidentityfirstreal) /\ (ge_balance_negative_norm_unit_resultidentityfirstreal) = 0) \/ exists ge_signed_half_norm_unit_resultidentityfirstrealdecode. (((ge_representation_real_code_norm_unit_resultidentityfirst) = 2 * ge_signed_half_norm_unit_resultidentityfirstrealdecode + 1 /\ (ge_balance_positive_norm_unit_resultidentityfirstreal) = 0) /\ (ge_balance_negative_norm_unit_resultidentityfirstreal) = S ge_signed_half_norm_unit_resultidentityfirstrealdecode))) /\ ((ge_first_rp_norm_unit_resultidentity) + ge_balance_negative_norm_unit_resultidentityfirstreal = (ge_first_rn_norm_unit_resultidentity) + ge_balance_positive_norm_unit_resultidentityfirstreal))) /\ (exists ge_balance_positive_norm_unit_resultidentityfirstimaginary ge_balance_negative_norm_unit_resultidentityfirstimaginary. (((((ge_representation_imaginary_code_norm_unit_resultidentityfirst) = 2 * (ge_balance_positive_norm_unit_resultidentityfirstimaginary) /\ (ge_balance_negative_norm_unit_resultidentityfirstimaginary) = 0) \/ exists ge_signed_half_norm_unit_resultidentityfirstimaginarydecode. (((ge_representation_imaginary_code_norm_unit_resultidentityfirst) = 2 * ge_signed_half_norm_unit_resultidentityfirstimaginarydecode + 1 /\ (ge_balance_positive_norm_unit_resultidentityfirstimaginary) = 0) /\ (ge_balance_negative_norm_unit_resultidentityfirstimaginary) = S ge_signed_half_norm_unit_resultidentityfirstimaginarydecode))) /\ ((ge_first_ip_norm_unit_resultidentity) + ge_balance_negative_norm_unit_resultidentityfirstimaginary = (ge_first_in_norm_unit_resultidentity) + ge_balance_positive_norm_unit_resultidentityfirstimaginary)))))) /\ ((exists ge_representation_real_code_norm_unit_resultidentitysecond ge_representation_imaginary_code_norm_unit_resultidentitysecond. (((gr_inverse_norm_unit_result) = ((ge_representation_real_code_norm_unit_resultidentitysecond) + (ge_representation_imaginary_code_norm_unit_resultidentitysecond)) * S ((ge_representation_real_code_norm_unit_resultidentitysecond) + (ge_representation_imaginary_code_norm_unit_resultidentitysecond)) + ((ge_representation_imaginary_code_norm_unit_resultidentitysecond) + (ge_representation_imaginary_code_norm_unit_resultidentitysecond))) /\ ((exists ge_balance_positive_norm_unit_resultidentitysecondreal ge_balance_negative_norm_unit_resultidentitysecondreal. (((((ge_representation_real_code_norm_unit_resultidentitysecond) = 2 * (ge_balance_positive_norm_unit_resultidentitysecondreal) /\ (ge_balance_negative_norm_unit_resultidentitysecondreal) = 0) \/ exists ge_signed_half_norm_unit_resultidentitysecondrealdecode. (((ge_representation_real_code_norm_unit_resultidentitysecond) = 2 * ge_signed_half_norm_unit_resultidentitysecondrealdecode + 1 /\ (ge_balance_positive_norm_unit_resultidentitysecondreal) = 0) /\ (ge_balance_negative_norm_unit_resultidentitysecondreal) = S ge_signed_half_norm_unit_resultidentitysecondrealdecode))) /\ ((ge_second_rp_norm_unit_resultidentity) + ge_balance_negative_norm_unit_resultidentitysecondreal = (ge_second_rn_norm_unit_resultidentity) + ge_balance_positive_norm_unit_resultidentitysecondreal))) /\ (exists ge_balance_positive_norm_unit_resultidentitysecondimaginary ge_balance_negative_norm_unit_resultidentitysecondimaginary. (((((ge_representation_imaginary_code_norm_unit_resultidentitysecond) = 2 * (ge_balance_positive_norm_unit_resultidentitysecondimaginary) /\ (ge_balance_negative_norm_unit_resultidentitysecondimaginary) = 0) \/ exists ge_signed_half_norm_unit_resultidentitysecondimaginarydecode. (((ge_representation_imaginary_code_norm_unit_resultidentitysecond) = 2 * ge_signed_half_norm_unit_resultidentitysecondimaginarydecode + 1 /\ (ge_balance_positive_norm_unit_resultidentitysecondimaginary) = 0) /\ (ge_balance_negative_norm_unit_resultidentitysecondimaginary) = S ge_signed_half_norm_unit_resultidentitysecondimaginarydecode))) /\ ((ge_second_ip_norm_unit_resultidentity) + ge_balance_negative_norm_unit_resultidentitysecondimaginary = (ge_second_in_norm_unit_resultidentity) + ge_balance_positive_norm_unit_resultidentitysecondimaginary)))))) /\ (exists ge_representation_real_code_norm_unit_resultidentityoutput ge_representation_imaginary_code_norm_unit_resultidentityoutput. (((6) = ((ge_representation_real_code_norm_unit_resultidentityoutput) + (ge_representation_imaginary_code_norm_unit_resultidentityoutput)) * S ((ge_representation_real_code_norm_unit_resultidentityoutput) + (ge_representation_imaginary_code_norm_unit_resultidentityoutput)) + ((ge_representation_imaginary_code_norm_unit_resultidentityoutput) + (ge_representation_imaginary_code_norm_unit_resultidentityoutput))) /\ ((exists ge_balance_positive_norm_unit_resultidentityoutputreal ge_balance_negative_norm_unit_resultidentityoutputreal. (((((ge_representation_real_code_norm_unit_resultidentityoutput) = 2 * (ge_balance_positive_norm_unit_resultidentityoutputreal) /\ (ge_balance_negative_norm_unit_resultidentityoutputreal) = 0) \/ exists ge_signed_half_norm_unit_resultidentityoutputrealdecode. (((ge_representation_real_code_norm_unit_resultidentityoutput) = 2 * ge_signed_half_norm_unit_resultidentityoutputrealdecode + 1 /\ (ge_balance_positive_norm_unit_resultidentityoutputreal) = 0) /\ (ge_balance_negative_norm_unit_resultidentityoutputreal) = S ge_signed_half_norm_unit_resultidentityoutputrealdecode))) /\ ((((((((ge_first_rp_norm_unit_resultidentity) * (ge_second_rp_norm_unit_resultidentity))) + (((ge_first_rn_norm_unit_resultidentity) * (ge_second_rn_norm_unit_resultidentity))))) + (((((ge_first_ip_norm_unit_resultidentity) * (ge_second_in_norm_unit_resultidentity))) + (((ge_first_in_norm_unit_resultidentity) * (ge_second_ip_norm_unit_resultidentity))))))) + ge_balance_negative_norm_unit_resultidentityoutputreal = (((((((ge_first_rp_norm_unit_resultidentity) * (ge_second_rn_norm_unit_resultidentity))) + (((ge_first_rn_norm_unit_resultidentity) * (ge_second_rp_norm_unit_resultidentity))))) + (((((ge_first_ip_norm_unit_resultidentity) * (ge_second_ip_norm_unit_resultidentity))) + (((ge_first_in_norm_unit_resultidentity) * (ge_second_in_norm_unit_resultidentity))))))) + ge_balance_positive_norm_unit_resultidentityoutputreal))) /\ (exists ge_balance_positive_norm_unit_resultidentityoutputimaginary ge_balance_negative_norm_unit_resultidentityoutputimaginary. (((((ge_representation_imaginary_code_norm_unit_resultidentityoutput) = 2 * (ge_balance_positive_norm_unit_resultidentityoutputimaginary) /\ (ge_balance_negative_norm_unit_resultidentityoutputimaginary) = 0) \/ exists ge_signed_half_norm_unit_resultidentityoutputimaginarydecode. (((ge_representation_imaginary_code_norm_unit_resultidentityoutput) = 2 * ge_signed_half_norm_unit_resultidentityoutputimaginarydecode + 1 /\ (ge_balance_positive_norm_unit_resultidentityoutputimaginary) = 0) /\ (ge_balance_negative_norm_unit_resultidentityoutputimaginary) = S ge_signed_half_norm_unit_resultidentityoutputimaginarydecode))) /\ ((((((((ge_first_rp_norm_unit_resultidentity) * (ge_second_ip_norm_unit_resultidentity))) + (((ge_first_rn_norm_unit_resultidentity) * (ge_second_in_norm_unit_resultidentity))))) + (((((ge_first_ip_norm_unit_resultidentity) * (ge_second_rp_norm_unit_resultidentity))) + (((ge_first_in_norm_unit_resultidentity) * (ge_second_rn_norm_unit_resultidentity))))))) + ge_balance_negative_norm_unit_resultidentityoutputimaginary = (((((((ge_first_rp_norm_unit_resultidentity) * (ge_second_in_norm_unit_resultidentity))) + (((ge_first_rn_norm_unit_resultidentity) * (ge_second_ip_norm_unit_resultidentity))))) + (((((ge_first_ip_norm_unit_resultidentity) * (ge_second_rn_norm_unit_resultidentity))) + (((ge_first_in_norm_unit_resultidentity) * (ge_second_rp_norm_unit_resultidentity))))))) + ge_balance_positive_norm_unit_resultidentityoutputimaginary))))))))))

Constructive proof overview

Generated structural guide

Norm one constructs an actual inverse: the canonical conjugate multiplies the value to Gaussian identity code six.

The unchanged tactic script uses 7 declared prerequisites and contains 62 exact native proof lines.

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

Proof neighborhood

Direct dependencies

gaussian_representation_exists Alpha theorem; checked-use authorized gaussian_conjugate_product_is_norm Alpha theorem; checked-use authorized gaussian_representation_integer_transport Alpha theorem; checked-use authorized gaussian_equal_symmetric Alpha theorem; checked-use authorized GF000E gaussian_one_representation gaussian_multiply_of_representations Alpha theorem; checked-use authorized GF0014 gaussian_multiply_commutative

Direct dependents

GF001B gaussian_unit_iff_norm_one GF001C gaussian_unit_decidable GF003F gaussian_unit_product GF0079 gaussian_search_nonunit_norm_two

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

62 script commands · 10 reading checkpoints · 2 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 one representationgaussian multiply commutative

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 assumptionsL1–2

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

  1. L1
    intro z
  2. L2
    intro h
02Separate the logical casesL3–7

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

  1. L3
    cases h
  2. L4
    cases h_witness
  3. L5
    cases h_witness_witness
  4. L6
    cases h_witness_witness_witness
  5. L7
    cases h_witness_witness_witness_witness
03Establish hinverseL8–13

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

  1. L8
    have hinverse : ∃ u. ZPairRep(u,x,x1,x3,x2)Definitions: ZPairRep
  2. L9
    specialize gaussian_representation_exists (x)
  3. L10
    specialize gaussian_representation_exists (x1)
  4. L11
    specialize gaussian_representation_exists (x3)
  5. L12
    specialize gaussian_representation_exists (x2)
  6. L13
    apply gaussian_representation_exists
04Separate the logical casesL14–14

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

  1. L14
    cases hinverse
05Establish honeL15–24

Establish this local claim before using it. It is not an additional assumption.

  1. L15
    have hone : ZPairRep(6,x · x + x1 · x1 + (x3 · x3 + x2 · x2),x · x1 + x1 · x + (x3 · x2 + x2 · x3),x · x2 + x1 · x3 + (x3 · x + x2 · x1),x · x3 + x1 · x2 + (x3 · x1 + x2 · x))Definitions: ZPairRep
  2. L16
    specialize gaussian_representation_integer_transport (6)
  3. L17
    specialize gaussian_representation_integer_transport (1)
  4. L18
    specialize gaussian_representation_integer_transport (0)
  5. L19
    specialize gaussian_representation_integer_transport (0)
  6. L20
    specialize gaussian_representation_integer_transport (0)
  7. L21
    specialize gaussian_representation_integer_transport (((((((x) * (x))) + (((x1) * (x1))))) + (((((x3) * (x3))) + (((x2) * (x2)))))))
  8. L22
    specialize gaussian_representation_integer_transport (((((((x) * (x1))) + (((x1) * (x))))) + (((((x3) * (x2))) + (((x2) * (x3)))))))
  9. L23
    specialize gaussian_representation_integer_transport (((((((x) * (x2))) + (((x1) * (x3))))) + (((((x3) * (x))) + (((x2) * (x1)))))))
  10. L24
    specialize gaussian_representation_integer_transport (((((((x) * (x3))) + (((x1) * (x2))))) + (((((x3) * (x1))) + (((x2) * (x)))))))
06Use earlier factsL25–34

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

  1. L25
    apply gaussian_representation_integer_transport
  2. L26
    specialize gaussian_equal_symmetric (((((((x) * (x))) + (((x1) * (x1))))) + (((((x3) * (x3))) + (((x2) * (x2)))))))
  3. L27
    specialize gaussian_equal_symmetric (((((((x) * (x1))) + (((x1) * (x))))) + (((((x3) * (x2))) + (((x2) * (x3)))))))
  4. L28
    specialize gaussian_equal_symmetric (((((((x) * (x2))) + (((x1) * (x3))))) + (((((x3) * (x))) + (((x2) * (x1)))))))
  5. L29
    specialize gaussian_equal_symmetric (((((((x) * (x3))) + (((x1) * (x2))))) + (((((x3) * (x1))) + (((x2) * (x)))))))
  6. L30
    specialize gaussian_equal_symmetric (1)
  7. L31
    specialize gaussian_equal_symmetric (0)
  8. L32
    specialize gaussian_equal_symmetric (0)
  9. L33
    specialize gaussian_equal_symmetric (0)
  10. L34
    apply gaussian_equal_symmetric
07Use earlier factsL35–42

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

  1. L35
    specialize gaussian_conjugate_product_is_norm (x)
  2. L36
    specialize gaussian_conjugate_product_is_norm (x1)
  3. L37
    specialize gaussian_conjugate_product_is_norm (x2)
  4. L38
    specialize gaussian_conjugate_product_is_norm (x3)
  5. L39
    specialize gaussian_conjugate_product_is_norm (1)
  6. L40
    apply gaussian_conjugate_product_is_norm
  7. L41
    exact h_witness_witness_witness_witness_right
  8. L42
    exact gaussian_one_representation
08Construct an explicit witnessL43–43

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

  1. L43
    exists (x4)
09Use earlier factsL44–53

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

  1. L44
    specialize gaussian_multiply_commutative (x4)
  2. L45
    specialize gaussian_multiply_commutative (z)
  3. L46
    specialize gaussian_multiply_commutative (6)
  4. L47
    apply gaussian_multiply_commutative
  5. L48
    specialize gaussian_multiply_of_representations (x4)
  6. L49
    specialize gaussian_multiply_of_representations (z)
  7. L50
    specialize gaussian_multiply_of_representations (6)
  8. L51
    specialize gaussian_multiply_of_representations (x)
  9. L52
    specialize gaussian_multiply_of_representations (x1)
  10. L53
    specialize gaussian_multiply_of_representations (x3)
10Use earlier factsL54–62

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

  1. L54
    specialize gaussian_multiply_of_representations (x2)
  2. L55
    specialize gaussian_multiply_of_representations (x)
  3. L56
    specialize gaussian_multiply_of_representations (x1)
  4. L57
    specialize gaussian_multiply_of_representations (x2)
  5. L58
    specialize gaussian_multiply_of_representations (x3)
  6. L59
    apply gaussian_multiply_of_representations
  7. L60
    exact hinverse_witness
  8. L61
    exact h_witness_witness_witness_witness_left
  9. L62
    exact hone

Library-wide reading audit

Original exact command ledger · 62 lines
  1. 0001intro z
  2. 0002intro h
  3. 0003cases h
  4. 0004cases h_witness
  5. 0005cases h_witness_witness
  6. 0006cases h_witness_witness_witness
  7. 0007cases h_witness_witness_witness_witness
  8. 0008have hinverse : exists u. (exists ge_representation_real_code_norm_unit_inverse ge_representation_imaginary_code_norm_unit_inverse. (((u) = ((ge_representation_real_code_norm_unit_inverse) + (ge_representation_imaginary_code_norm_unit_inverse)) * S ((ge_representation_real_code_norm_unit_inverse) + (ge_representation_imaginary_code_norm_unit_inverse)) + ((ge_representation_imaginary_code_norm_unit_inverse) + (ge_representation_imaginary_code_norm_unit_inverse))) /\ ((exists ge_balance_positive_norm_unit_inversereal ge_balance_negative_norm_unit_inversereal. (((((ge_representation_real_code_norm_unit_inverse) = 2 * (ge_balance_positive_norm_unit_inversereal) /\ (ge_balance_negative_norm_unit_inversereal) = 0) \/ exists ge_signed_half_norm_unit_inverserealdecode. (((ge_representation_real_code_norm_unit_inverse) = 2 * ge_signed_half_norm_unit_inverserealdecode + 1 /\ (ge_balance_positive_norm_unit_inversereal) = 0) /\ (ge_balance_negative_norm_unit_inversereal) = S ge_signed_half_norm_unit_inverserealdecode))) /\ ((x) + ge_balance_negative_norm_unit_inversereal = (x1) + ge_balance_positive_norm_unit_inversereal))) /\ (exists ge_balance_positive_norm_unit_inverseimaginary ge_balance_negative_norm_unit_inverseimaginary. (((((ge_representation_imaginary_code_norm_unit_inverse) = 2 * (ge_balance_positive_norm_unit_inverseimaginary) /\ (ge_balance_negative_norm_unit_inverseimaginary) = 0) \/ exists ge_signed_half_norm_unit_inverseimaginarydecode. (((ge_representation_imaginary_code_norm_unit_inverse) = 2 * ge_signed_half_norm_unit_inverseimaginarydecode + 1 /\ (ge_balance_positive_norm_unit_inverseimaginary) = 0) /\ (ge_balance_negative_norm_unit_inverseimaginary) = S ge_signed_half_norm_unit_inverseimaginarydecode))) /\ ((x3) + ge_balance_negative_norm_unit_inverseimaginary = (x2) + ge_balance_positive_norm_unit_inverseimaginary))))))
  9. 0009specialize gaussian_representation_exists (x)
  10. 0010specialize gaussian_representation_exists (x1)
  11. 0011specialize gaussian_representation_exists (x3)
  12. 0012specialize gaussian_representation_exists (x2)
  13. 0013apply gaussian_representation_exists
  14. 0014cases hinverse
  15. 0015have hone : (exists ge_representation_real_code_norm_unit_product ge_representation_imaginary_code_norm_unit_product. (((6) = ((ge_representation_real_code_norm_unit_product) + (ge_representation_imaginary_code_norm_unit_product)) * S ((ge_representation_real_code_norm_unit_product) + (ge_representation_imaginary_code_norm_unit_product)) + ((ge_representation_imaginary_code_norm_unit_product) + (ge_representation_imaginary_code_norm_unit_product))) /\ ((exists ge_balance_positive_norm_unit_productreal ge_balance_negative_norm_unit_productreal. (((((ge_representation_real_code_norm_unit_product) = 2 * (ge_balance_positive_norm_unit_productreal) /\ (ge_balance_negative_norm_unit_productreal) = 0) \/ exists ge_signed_half_norm_unit_productrealdecode. (((ge_representation_real_code_norm_unit_product) = 2 * ge_signed_half_norm_unit_productrealdecode + 1 /\ (ge_balance_positive_norm_unit_productreal) = 0) /\ (ge_balance_negative_norm_unit_productreal) = S ge_signed_half_norm_unit_productrealdecode))) /\ ((((((((x) * (x))) + (((x1) * (x1))))) + (((((x3) * (x3))) + (((x2) * (x2))))))) + ge_balance_negative_norm_unit_productreal = (((((((x) * (x1))) + (((x1) * (x))))) + (((((x3) * (x2))) + (((x2) * (x3))))))) + ge_balance_positive_norm_unit_productreal))) /\ (exists ge_balance_positive_norm_unit_productimaginary ge_balance_negative_norm_unit_productimaginary. (((((ge_representation_imaginary_code_norm_unit_product) = 2 * (ge_balance_positive_norm_unit_productimaginary) /\ (ge_balance_negative_norm_unit_productimaginary) = 0) \/ exists ge_signed_half_norm_unit_productimaginarydecode. (((ge_representation_imaginary_code_norm_unit_product) = 2 * ge_signed_half_norm_unit_productimaginarydecode + 1 /\ (ge_balance_positive_norm_unit_productimaginary) = 0) /\ (ge_balance_negative_norm_unit_productimaginary) = S ge_signed_half_norm_unit_productimaginarydecode))) /\ ((((((((x) * (x2))) + (((x1) * (x3))))) + (((((x3) * (x))) + (((x2) * (x1))))))) + ge_balance_negative_norm_unit_productimaginary = (((((((x) * (x3))) + (((x1) * (x2))))) + (((((x3) * (x1))) + (((x2) * (x))))))) + ge_balance_positive_norm_unit_productimaginary))))))
  16. 0016specialize gaussian_representation_integer_transport (6)
  17. 0017specialize gaussian_representation_integer_transport (1)
  18. 0018specialize gaussian_representation_integer_transport (0)
  19. 0019specialize gaussian_representation_integer_transport (0)
  20. 0020specialize gaussian_representation_integer_transport (0)
  21. 0021specialize gaussian_representation_integer_transport (((((((x) * (x))) + (((x1) * (x1))))) + (((((x3) * (x3))) + (((x2) * (x2)))))))
  22. 0022specialize gaussian_representation_integer_transport (((((((x) * (x1))) + (((x1) * (x))))) + (((((x3) * (x2))) + (((x2) * (x3)))))))
  23. 0023specialize gaussian_representation_integer_transport (((((((x) * (x2))) + (((x1) * (x3))))) + (((((x3) * (x))) + (((x2) * (x1)))))))
  24. 0024specialize gaussian_representation_integer_transport (((((((x) * (x3))) + (((x1) * (x2))))) + (((((x3) * (x1))) + (((x2) * (x)))))))
  25. 0025apply gaussian_representation_integer_transport
  26. 0026specialize gaussian_equal_symmetric (((((((x) * (x))) + (((x1) * (x1))))) + (((((x3) * (x3))) + (((x2) * (x2)))))))
  27. 0027specialize gaussian_equal_symmetric (((((((x) * (x1))) + (((x1) * (x))))) + (((((x3) * (x2))) + (((x2) * (x3)))))))
  28. 0028specialize gaussian_equal_symmetric (((((((x) * (x2))) + (((x1) * (x3))))) + (((((x3) * (x))) + (((x2) * (x1)))))))
  29. 0029specialize gaussian_equal_symmetric (((((((x) * (x3))) + (((x1) * (x2))))) + (((((x3) * (x1))) + (((x2) * (x)))))))
  30. 0030specialize gaussian_equal_symmetric (1)
  31. 0031specialize gaussian_equal_symmetric (0)
  32. 0032specialize gaussian_equal_symmetric (0)
  33. 0033specialize gaussian_equal_symmetric (0)
  34. 0034apply gaussian_equal_symmetric
  35. 0035specialize gaussian_conjugate_product_is_norm (x)
  36. 0036specialize gaussian_conjugate_product_is_norm (x1)
  37. 0037specialize gaussian_conjugate_product_is_norm (x2)
  38. 0038specialize gaussian_conjugate_product_is_norm (x3)
  39. 0039specialize gaussian_conjugate_product_is_norm (1)
  40. 0040apply gaussian_conjugate_product_is_norm
  41. 0041exact h_witness_witness_witness_witness_right
  42. 0042exact gaussian_one_representation
  43. 0043exists (x4)
  44. 0044specialize gaussian_multiply_commutative (x4)
  45. 0045specialize gaussian_multiply_commutative (z)
  46. 0046specialize gaussian_multiply_commutative (6)
  47. 0047apply gaussian_multiply_commutative
  48. 0048specialize gaussian_multiply_of_representations (x4)
  49. 0049specialize gaussian_multiply_of_representations (z)
  50. 0050specialize gaussian_multiply_of_representations (6)
  51. 0051specialize gaussian_multiply_of_representations (x)
  52. 0052specialize gaussian_multiply_of_representations (x1)
  53. 0053specialize gaussian_multiply_of_representations (x3)
  54. 0054specialize gaussian_multiply_of_representations (x2)
  55. 0055specialize gaussian_multiply_of_representations (x)
  56. 0056specialize gaussian_multiply_of_representations (x1)
  57. 0057specialize gaussian_multiply_of_representations (x2)
  58. 0058specialize gaussian_multiply_of_representations (x3)
  59. 0059apply gaussian_multiply_of_representations
  60. 0060exact hinverse_witness
  61. 0061exact h_witness_witness_witness_witness_left
  62. 0062exact hone