GF001C

gaussian_unit_decidable

Actual Gaussian units are constructively decidable by computing the actual norm and deciding equality with one.

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. ZPairValid(z) → GUnit(z) ∨ ¬GUnit(z)

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

Definition DAG

ZPairValid(z) · 1GNorm(z,n) · 1GUnit(z) · 2

Actual proof prerequisites

gaussian_norm_exists · checked external prerequisiteeq_decidable · checked external prerequisitegaussian_norm_one_is_unitgaussian_norm_value_transportgaussian_unit_has_norm_onegaussian_norm_functional · checked external prerequisite
Original expanded first-order statement 
forall z. (exists ge_real_positive_unit_decidable_domain ge_real_negative_unit_decidable_domain ge_imaginary_positive_unit_decidable_domain ge_imaginary_negative_unit_decidable_domain. (exists ge_real_code_unit_decidable_domaindecode ge_imaginary_code_unit_decidable_domaindecode. (((z) = ((ge_real_code_unit_decidable_domaindecode) + (ge_imaginary_code_unit_decidable_domaindecode)) * S ((ge_real_code_unit_decidable_domaindecode) + (ge_imaginary_code_unit_decidable_domaindecode)) + ((ge_imaginary_code_unit_decidable_domaindecode) + (ge_imaginary_code_unit_decidable_domaindecode))) /\ (((((ge_real_code_unit_decidable_domaindecode) = 2 * (ge_real_positive_unit_decidable_domain) /\ (ge_real_negative_unit_decidable_domain) = 0) \/ exists ge_signed_half_ge_unit_decidable_domaindecode_real. (((ge_real_code_unit_decidable_domaindecode) = 2 * ge_signed_half_ge_unit_decidable_domaindecode_real + 1 /\ (ge_real_positive_unit_decidable_domain) = 0) /\ (ge_real_negative_unit_decidable_domain) = S ge_signed_half_ge_unit_decidable_domaindecode_real))) /\ ((((ge_imaginary_code_unit_decidable_domaindecode) = 2 * (ge_imaginary_positive_unit_decidable_domain) /\ (ge_imaginary_negative_unit_decidable_domain) = 0) \/ exists ge_signed_half_ge_unit_decidable_domaindecode_imaginary. (((ge_imaginary_code_unit_decidable_domaindecode) = 2 * ge_signed_half_ge_unit_decidable_domaindecode_imaginary + 1 /\ (ge_imaginary_positive_unit_decidable_domain) = 0) /\ (ge_imaginary_negative_unit_decidable_domain) = S ge_signed_half_ge_unit_decidable_domaindecode_imaginary))))))) -> (exists gr_inverse_unit_decidable_yes. (exists ge_first_rp_unit_decidable_yesidentity ge_first_rn_unit_decidable_yesidentity ge_first_ip_unit_decidable_yesidentity ge_first_in_unit_decidable_yesidentity ge_second_rp_unit_decidable_yesidentity ge_second_rn_unit_decidable_yesidentity ge_second_ip_unit_decidable_yesidentity ge_second_in_unit_decidable_yesidentity. ((exists ge_representation_real_code_unit_decidable_yesidentityfirst ge_representation_imaginary_code_unit_decidable_yesidentityfirst. (((z) = ((ge_representation_real_code_unit_decidable_yesidentityfirst) + (ge_representation_imaginary_code_unit_decidable_yesidentityfirst)) * S ((ge_representation_real_code_unit_decidable_yesidentityfirst) + (ge_representation_imaginary_code_unit_decidable_yesidentityfirst)) + ((ge_representation_imaginary_code_unit_decidable_yesidentityfirst) + (ge_representation_imaginary_code_unit_decidable_yesidentityfirst))) /\ ((exists ge_balance_positive_unit_decidable_yesidentityfirstreal ge_balance_negative_unit_decidable_yesidentityfirstreal. (((((ge_representation_real_code_unit_decidable_yesidentityfirst) = 2 * (ge_balance_positive_unit_decidable_yesidentityfirstreal) /\ (ge_balance_negative_unit_decidable_yesidentityfirstreal) = 0) \/ exists ge_signed_half_unit_decidable_yesidentityfirstrealdecode. (((ge_representation_real_code_unit_decidable_yesidentityfirst) = 2 * ge_signed_half_unit_decidable_yesidentityfirstrealdecode + 1 /\ (ge_balance_positive_unit_decidable_yesidentityfirstreal) = 0) /\ (ge_balance_negative_unit_decidable_yesidentityfirstreal) = S ge_signed_half_unit_decidable_yesidentityfirstrealdecode))) /\ ((ge_first_rp_unit_decidable_yesidentity) + ge_balance_negative_unit_decidable_yesidentityfirstreal = (ge_first_rn_unit_decidable_yesidentity) + ge_balance_positive_unit_decidable_yesidentityfirstreal))) /\ (exists ge_balance_positive_unit_decidable_yesidentityfirstimaginary ge_balance_negative_unit_decidable_yesidentityfirstimaginary. (((((ge_representation_imaginary_code_unit_decidable_yesidentityfirst) = 2 * (ge_balance_positive_unit_decidable_yesidentityfirstimaginary) /\ (ge_balance_negative_unit_decidable_yesidentityfirstimaginary) = 0) \/ exists ge_signed_half_unit_decidable_yesidentityfirstimaginarydecode. (((ge_representation_imaginary_code_unit_decidable_yesidentityfirst) = 2 * ge_signed_half_unit_decidable_yesidentityfirstimaginarydecode + 1 /\ (ge_balance_positive_unit_decidable_yesidentityfirstimaginary) = 0) /\ (ge_balance_negative_unit_decidable_yesidentityfirstimaginary) = S ge_signed_half_unit_decidable_yesidentityfirstimaginarydecode))) /\ ((ge_first_ip_unit_decidable_yesidentity) + ge_balance_negative_unit_decidable_yesidentityfirstimaginary = (ge_first_in_unit_decidable_yesidentity) + ge_balance_positive_unit_decidable_yesidentityfirstimaginary)))))) /\ ((exists ge_representation_real_code_unit_decidable_yesidentitysecond ge_representation_imaginary_code_unit_decidable_yesidentitysecond. (((gr_inverse_unit_decidable_yes) = ((ge_representation_real_code_unit_decidable_yesidentitysecond) + (ge_representation_imaginary_code_unit_decidable_yesidentitysecond)) * S ((ge_representation_real_code_unit_decidable_yesidentitysecond) + (ge_representation_imaginary_code_unit_decidable_yesidentitysecond)) + ((ge_representation_imaginary_code_unit_decidable_yesidentitysecond) + (ge_representation_imaginary_code_unit_decidable_yesidentitysecond))) /\ ((exists ge_balance_positive_unit_decidable_yesidentitysecondreal ge_balance_negative_unit_decidable_yesidentitysecondreal. (((((ge_representation_real_code_unit_decidable_yesidentitysecond) = 2 * (ge_balance_positive_unit_decidable_yesidentitysecondreal) /\ (ge_balance_negative_unit_decidable_yesidentitysecondreal) = 0) \/ exists ge_signed_half_unit_decidable_yesidentitysecondrealdecode. (((ge_representation_real_code_unit_decidable_yesidentitysecond) = 2 * ge_signed_half_unit_decidable_yesidentitysecondrealdecode + 1 /\ (ge_balance_positive_unit_decidable_yesidentitysecondreal) = 0) /\ (ge_balance_negative_unit_decidable_yesidentitysecondreal) = S ge_signed_half_unit_decidable_yesidentitysecondrealdecode))) /\ ((ge_second_rp_unit_decidable_yesidentity) + ge_balance_negative_unit_decidable_yesidentitysecondreal = (ge_second_rn_unit_decidable_yesidentity) + ge_balance_positive_unit_decidable_yesidentitysecondreal))) /\ (exists ge_balance_positive_unit_decidable_yesidentitysecondimaginary ge_balance_negative_unit_decidable_yesidentitysecondimaginary. (((((ge_representation_imaginary_code_unit_decidable_yesidentitysecond) = 2 * (ge_balance_positive_unit_decidable_yesidentitysecondimaginary) /\ (ge_balance_negative_unit_decidable_yesidentitysecondimaginary) = 0) \/ exists ge_signed_half_unit_decidable_yesidentitysecondimaginarydecode. (((ge_representation_imaginary_code_unit_decidable_yesidentitysecond) = 2 * ge_signed_half_unit_decidable_yesidentitysecondimaginarydecode + 1 /\ (ge_balance_positive_unit_decidable_yesidentitysecondimaginary) = 0) /\ (ge_balance_negative_unit_decidable_yesidentitysecondimaginary) = S ge_signed_half_unit_decidable_yesidentitysecondimaginarydecode))) /\ ((ge_second_ip_unit_decidable_yesidentity) + ge_balance_negative_unit_decidable_yesidentitysecondimaginary = (ge_second_in_unit_decidable_yesidentity) + ge_balance_positive_unit_decidable_yesidentitysecondimaginary)))))) /\ (exists ge_representation_real_code_unit_decidable_yesidentityoutput ge_representation_imaginary_code_unit_decidable_yesidentityoutput. (((6) = ((ge_representation_real_code_unit_decidable_yesidentityoutput) + (ge_representation_imaginary_code_unit_decidable_yesidentityoutput)) * S ((ge_representation_real_code_unit_decidable_yesidentityoutput) + (ge_representation_imaginary_code_unit_decidable_yesidentityoutput)) + ((ge_representation_imaginary_code_unit_decidable_yesidentityoutput) + (ge_representation_imaginary_code_unit_decidable_yesidentityoutput))) /\ ((exists ge_balance_positive_unit_decidable_yesidentityoutputreal ge_balance_negative_unit_decidable_yesidentityoutputreal. (((((ge_representation_real_code_unit_decidable_yesidentityoutput) = 2 * (ge_balance_positive_unit_decidable_yesidentityoutputreal) /\ (ge_balance_negative_unit_decidable_yesidentityoutputreal) = 0) \/ exists ge_signed_half_unit_decidable_yesidentityoutputrealdecode. (((ge_representation_real_code_unit_decidable_yesidentityoutput) = 2 * ge_signed_half_unit_decidable_yesidentityoutputrealdecode + 1 /\ (ge_balance_positive_unit_decidable_yesidentityoutputreal) = 0) /\ (ge_balance_negative_unit_decidable_yesidentityoutputreal) = S ge_signed_half_unit_decidable_yesidentityoutputrealdecode))) /\ ((((((((ge_first_rp_unit_decidable_yesidentity) * (ge_second_rp_unit_decidable_yesidentity))) + (((ge_first_rn_unit_decidable_yesidentity) * (ge_second_rn_unit_decidable_yesidentity))))) + (((((ge_first_ip_unit_decidable_yesidentity) * (ge_second_in_unit_decidable_yesidentity))) + (((ge_first_in_unit_decidable_yesidentity) * (ge_second_ip_unit_decidable_yesidentity))))))) + ge_balance_negative_unit_decidable_yesidentityoutputreal = (((((((ge_first_rp_unit_decidable_yesidentity) * (ge_second_rn_unit_decidable_yesidentity))) + (((ge_first_rn_unit_decidable_yesidentity) * (ge_second_rp_unit_decidable_yesidentity))))) + (((((ge_first_ip_unit_decidable_yesidentity) * (ge_second_ip_unit_decidable_yesidentity))) + (((ge_first_in_unit_decidable_yesidentity) * (ge_second_in_unit_decidable_yesidentity))))))) + ge_balance_positive_unit_decidable_yesidentityoutputreal))) /\ (exists ge_balance_positive_unit_decidable_yesidentityoutputimaginary ge_balance_negative_unit_decidable_yesidentityoutputimaginary. (((((ge_representation_imaginary_code_unit_decidable_yesidentityoutput) = 2 * (ge_balance_positive_unit_decidable_yesidentityoutputimaginary) /\ (ge_balance_negative_unit_decidable_yesidentityoutputimaginary) = 0) \/ exists ge_signed_half_unit_decidable_yesidentityoutputimaginarydecode. (((ge_representation_imaginary_code_unit_decidable_yesidentityoutput) = 2 * ge_signed_half_unit_decidable_yesidentityoutputimaginarydecode + 1 /\ (ge_balance_positive_unit_decidable_yesidentityoutputimaginary) = 0) /\ (ge_balance_negative_unit_decidable_yesidentityoutputimaginary) = S ge_signed_half_unit_decidable_yesidentityoutputimaginarydecode))) /\ ((((((((ge_first_rp_unit_decidable_yesidentity) * (ge_second_ip_unit_decidable_yesidentity))) + (((ge_first_rn_unit_decidable_yesidentity) * (ge_second_in_unit_decidable_yesidentity))))) + (((((ge_first_ip_unit_decidable_yesidentity) * (ge_second_rp_unit_decidable_yesidentity))) + (((ge_first_in_unit_decidable_yesidentity) * (ge_second_rn_unit_decidable_yesidentity))))))) + ge_balance_negative_unit_decidable_yesidentityoutputimaginary = (((((((ge_first_rp_unit_decidable_yesidentity) * (ge_second_in_unit_decidable_yesidentity))) + (((ge_first_rn_unit_decidable_yesidentity) * (ge_second_ip_unit_decidable_yesidentity))))) + (((((ge_first_ip_unit_decidable_yesidentity) * (ge_second_rn_unit_decidable_yesidentity))) + (((ge_first_in_unit_decidable_yesidentity) * (ge_second_rp_unit_decidable_yesidentity))))))) + ge_balance_positive_unit_decidable_yesidentityoutputimaginary)))))))))) \/ ~(exists gr_inverse_unit_decidable_no. (exists ge_first_rp_unit_decidable_noidentity ge_first_rn_unit_decidable_noidentity ge_first_ip_unit_decidable_noidentity ge_first_in_unit_decidable_noidentity ge_second_rp_unit_decidable_noidentity ge_second_rn_unit_decidable_noidentity ge_second_ip_unit_decidable_noidentity ge_second_in_unit_decidable_noidentity. ((exists ge_representation_real_code_unit_decidable_noidentityfirst ge_representation_imaginary_code_unit_decidable_noidentityfirst. (((z) = ((ge_representation_real_code_unit_decidable_noidentityfirst) + (ge_representation_imaginary_code_unit_decidable_noidentityfirst)) * S ((ge_representation_real_code_unit_decidable_noidentityfirst) + (ge_representation_imaginary_code_unit_decidable_noidentityfirst)) + ((ge_representation_imaginary_code_unit_decidable_noidentityfirst) + (ge_representation_imaginary_code_unit_decidable_noidentityfirst))) /\ ((exists ge_balance_positive_unit_decidable_noidentityfirstreal ge_balance_negative_unit_decidable_noidentityfirstreal. (((((ge_representation_real_code_unit_decidable_noidentityfirst) = 2 * (ge_balance_positive_unit_decidable_noidentityfirstreal) /\ (ge_balance_negative_unit_decidable_noidentityfirstreal) = 0) \/ exists ge_signed_half_unit_decidable_noidentityfirstrealdecode. (((ge_representation_real_code_unit_decidable_noidentityfirst) = 2 * ge_signed_half_unit_decidable_noidentityfirstrealdecode + 1 /\ (ge_balance_positive_unit_decidable_noidentityfirstreal) = 0) /\ (ge_balance_negative_unit_decidable_noidentityfirstreal) = S ge_signed_half_unit_decidable_noidentityfirstrealdecode))) /\ ((ge_first_rp_unit_decidable_noidentity) + ge_balance_negative_unit_decidable_noidentityfirstreal = (ge_first_rn_unit_decidable_noidentity) + ge_balance_positive_unit_decidable_noidentityfirstreal))) /\ (exists ge_balance_positive_unit_decidable_noidentityfirstimaginary ge_balance_negative_unit_decidable_noidentityfirstimaginary. (((((ge_representation_imaginary_code_unit_decidable_noidentityfirst) = 2 * (ge_balance_positive_unit_decidable_noidentityfirstimaginary) /\ (ge_balance_negative_unit_decidable_noidentityfirstimaginary) = 0) \/ exists ge_signed_half_unit_decidable_noidentityfirstimaginarydecode. (((ge_representation_imaginary_code_unit_decidable_noidentityfirst) = 2 * ge_signed_half_unit_decidable_noidentityfirstimaginarydecode + 1 /\ (ge_balance_positive_unit_decidable_noidentityfirstimaginary) = 0) /\ (ge_balance_negative_unit_decidable_noidentityfirstimaginary) = S ge_signed_half_unit_decidable_noidentityfirstimaginarydecode))) /\ ((ge_first_ip_unit_decidable_noidentity) + ge_balance_negative_unit_decidable_noidentityfirstimaginary = (ge_first_in_unit_decidable_noidentity) + ge_balance_positive_unit_decidable_noidentityfirstimaginary)))))) /\ ((exists ge_representation_real_code_unit_decidable_noidentitysecond ge_representation_imaginary_code_unit_decidable_noidentitysecond. (((gr_inverse_unit_decidable_no) = ((ge_representation_real_code_unit_decidable_noidentitysecond) + (ge_representation_imaginary_code_unit_decidable_noidentitysecond)) * S ((ge_representation_real_code_unit_decidable_noidentitysecond) + (ge_representation_imaginary_code_unit_decidable_noidentitysecond)) + ((ge_representation_imaginary_code_unit_decidable_noidentitysecond) + (ge_representation_imaginary_code_unit_decidable_noidentitysecond))) /\ ((exists ge_balance_positive_unit_decidable_noidentitysecondreal ge_balance_negative_unit_decidable_noidentitysecondreal. (((((ge_representation_real_code_unit_decidable_noidentitysecond) = 2 * (ge_balance_positive_unit_decidable_noidentitysecondreal) /\ (ge_balance_negative_unit_decidable_noidentitysecondreal) = 0) \/ exists ge_signed_half_unit_decidable_noidentitysecondrealdecode. (((ge_representation_real_code_unit_decidable_noidentitysecond) = 2 * ge_signed_half_unit_decidable_noidentitysecondrealdecode + 1 /\ (ge_balance_positive_unit_decidable_noidentitysecondreal) = 0) /\ (ge_balance_negative_unit_decidable_noidentitysecondreal) = S ge_signed_half_unit_decidable_noidentitysecondrealdecode))) /\ ((ge_second_rp_unit_decidable_noidentity) + ge_balance_negative_unit_decidable_noidentitysecondreal = (ge_second_rn_unit_decidable_noidentity) + ge_balance_positive_unit_decidable_noidentitysecondreal))) /\ (exists ge_balance_positive_unit_decidable_noidentitysecondimaginary ge_balance_negative_unit_decidable_noidentitysecondimaginary. (((((ge_representation_imaginary_code_unit_decidable_noidentitysecond) = 2 * (ge_balance_positive_unit_decidable_noidentitysecondimaginary) /\ (ge_balance_negative_unit_decidable_noidentitysecondimaginary) = 0) \/ exists ge_signed_half_unit_decidable_noidentitysecondimaginarydecode. (((ge_representation_imaginary_code_unit_decidable_noidentitysecond) = 2 * ge_signed_half_unit_decidable_noidentitysecondimaginarydecode + 1 /\ (ge_balance_positive_unit_decidable_noidentitysecondimaginary) = 0) /\ (ge_balance_negative_unit_decidable_noidentitysecondimaginary) = S ge_signed_half_unit_decidable_noidentitysecondimaginarydecode))) /\ ((ge_second_ip_unit_decidable_noidentity) + ge_balance_negative_unit_decidable_noidentitysecondimaginary = (ge_second_in_unit_decidable_noidentity) + ge_balance_positive_unit_decidable_noidentitysecondimaginary)))))) /\ (exists ge_representation_real_code_unit_decidable_noidentityoutput ge_representation_imaginary_code_unit_decidable_noidentityoutput. (((6) = ((ge_representation_real_code_unit_decidable_noidentityoutput) + (ge_representation_imaginary_code_unit_decidable_noidentityoutput)) * S ((ge_representation_real_code_unit_decidable_noidentityoutput) + (ge_representation_imaginary_code_unit_decidable_noidentityoutput)) + ((ge_representation_imaginary_code_unit_decidable_noidentityoutput) + (ge_representation_imaginary_code_unit_decidable_noidentityoutput))) /\ ((exists ge_balance_positive_unit_decidable_noidentityoutputreal ge_balance_negative_unit_decidable_noidentityoutputreal. (((((ge_representation_real_code_unit_decidable_noidentityoutput) = 2 * (ge_balance_positive_unit_decidable_noidentityoutputreal) /\ (ge_balance_negative_unit_decidable_noidentityoutputreal) = 0) \/ exists ge_signed_half_unit_decidable_noidentityoutputrealdecode. (((ge_representation_real_code_unit_decidable_noidentityoutput) = 2 * ge_signed_half_unit_decidable_noidentityoutputrealdecode + 1 /\ (ge_balance_positive_unit_decidable_noidentityoutputreal) = 0) /\ (ge_balance_negative_unit_decidable_noidentityoutputreal) = S ge_signed_half_unit_decidable_noidentityoutputrealdecode))) /\ ((((((((ge_first_rp_unit_decidable_noidentity) * (ge_second_rp_unit_decidable_noidentity))) + (((ge_first_rn_unit_decidable_noidentity) * (ge_second_rn_unit_decidable_noidentity))))) + (((((ge_first_ip_unit_decidable_noidentity) * (ge_second_in_unit_decidable_noidentity))) + (((ge_first_in_unit_decidable_noidentity) * (ge_second_ip_unit_decidable_noidentity))))))) + ge_balance_negative_unit_decidable_noidentityoutputreal = (((((((ge_first_rp_unit_decidable_noidentity) * (ge_second_rn_unit_decidable_noidentity))) + (((ge_first_rn_unit_decidable_noidentity) * (ge_second_rp_unit_decidable_noidentity))))) + (((((ge_first_ip_unit_decidable_noidentity) * (ge_second_ip_unit_decidable_noidentity))) + (((ge_first_in_unit_decidable_noidentity) * (ge_second_in_unit_decidable_noidentity))))))) + ge_balance_positive_unit_decidable_noidentityoutputreal))) /\ (exists ge_balance_positive_unit_decidable_noidentityoutputimaginary ge_balance_negative_unit_decidable_noidentityoutputimaginary. (((((ge_representation_imaginary_code_unit_decidable_noidentityoutput) = 2 * (ge_balance_positive_unit_decidable_noidentityoutputimaginary) /\ (ge_balance_negative_unit_decidable_noidentityoutputimaginary) = 0) \/ exists ge_signed_half_unit_decidable_noidentityoutputimaginarydecode. (((ge_representation_imaginary_code_unit_decidable_noidentityoutput) = 2 * ge_signed_half_unit_decidable_noidentityoutputimaginarydecode + 1 /\ (ge_balance_positive_unit_decidable_noidentityoutputimaginary) = 0) /\ (ge_balance_negative_unit_decidable_noidentityoutputimaginary) = S ge_signed_half_unit_decidable_noidentityoutputimaginarydecode))) /\ ((((((((ge_first_rp_unit_decidable_noidentity) * (ge_second_ip_unit_decidable_noidentity))) + (((ge_first_rn_unit_decidable_noidentity) * (ge_second_in_unit_decidable_noidentity))))) + (((((ge_first_ip_unit_decidable_noidentity) * (ge_second_rp_unit_decidable_noidentity))) + (((ge_first_in_unit_decidable_noidentity) * (ge_second_rn_unit_decidable_noidentity))))))) + ge_balance_negative_unit_decidable_noidentityoutputimaginary = (((((((ge_first_rp_unit_decidable_noidentity) * (ge_second_in_unit_decidable_noidentity))) + (((ge_first_rn_unit_decidable_noidentity) * (ge_second_ip_unit_decidable_noidentity))))) + (((((ge_first_ip_unit_decidable_noidentity) * (ge_second_rn_unit_decidable_noidentity))) + (((ge_first_in_unit_decidable_noidentity) * (ge_second_rp_unit_decidable_noidentity))))))) + ge_balance_positive_unit_decidable_noidentityoutputimaginary))))))))))

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 · 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.

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 norm one is unitgaussian norm value transportgaussian unit has norm one
01Fix variables and assumptionsL1–2

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

  1. L1
    intro z
  2. L2
    intro hv
02Establish hnL3–6

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

  1. L3
    have hn : ∃ N. GNorm(z,N)Definitions: GNorm(z,N)Original native command in the exact edition
  2. L4
    specialize gaussian_norm_exists (z)
  3. L5
    apply gaussian_norm_exists
  4. L6
    exact hv
03Separate the logical casesL7–7

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

  1. L7
    cases hn
04Establish hcasesL8–11

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

  1. L8
    have hcases : x=1 \/ ~(x=1)
  2. L9
    specialize eq_decidable (x)
  3. L10
    specialize eq_decidable (1)
  4. L11
    apply eq_decidable
05Separate the logical casesL12–13

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

  1. L12
    cases hcases
  2. L13
    left
06Use earlier factsL14–21

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

  1. L14
    specialize gaussian_norm_one_is_unit (z)
  2. L15
    apply gaussian_norm_one_is_unit
  3. L16
    specialize gaussian_norm_value_transport (z)
  4. L17
    specialize gaussian_norm_value_transport (x)
  5. L18
    specialize gaussian_norm_value_transport (1)
  6. L19
    apply gaussian_norm_value_transport
  7. L20
    exact hcases_left
  8. L21
    exact hn_witness
07Separate the logical casesL22–22

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

  1. L22
    right
08Fix variables and assumptionsL23–23

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

  1. L23
    intro hu
09Use earlier factsL24–32

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

  1. L24
    apply hcases_right
  2. L25
    specialize gaussian_norm_functional (z)
  3. L26
    specialize gaussian_norm_functional (x)
  4. L27
    specialize gaussian_norm_functional (1)
  5. L28
    apply gaussian_norm_functional
  6. L29
    exact hn_witness
  7. L30
    specialize gaussian_unit_has_norm_one (z)
  8. L31
    apply gaussian_unit_has_norm_one
  9. L32
    exact hu

Library-wide reading audit

Original defined command ledger · 32 lines
  1. 0001intro z
  2. 0002intro hv
  3. 0003have hn : ∃ N. GNorm(z,N)
  4. 0004specialize gaussian_norm_exists (z)
  5. 0005apply gaussian_norm_exists
  6. 0006exact hv
  7. 0007cases hn
  8. 0008have hcases : x=1 \/ ~(x=1)
  9. 0009specialize eq_decidable (x)
  10. 0010specialize eq_decidable (1)
  11. 0011apply eq_decidable
  12. 0012cases hcases
  13. 0013left
  14. 0014specialize gaussian_norm_one_is_unit (z)
  15. 0015apply gaussian_norm_one_is_unit
  16. 0016specialize gaussian_norm_value_transport (z)
  17. 0017specialize gaussian_norm_value_transport (x)
  18. 0018specialize gaussian_norm_value_transport (1)
  19. 0019apply gaussian_norm_value_transport
  20. 0020exact hcases_left
  21. 0021exact hn_witness
  22. 0022right
  23. 0023intro hu
  24. 0024apply hcases_right
  25. 0025specialize gaussian_norm_functional (z)
  26. 0026specialize gaussian_norm_functional (x)
  27. 0027specialize gaussian_norm_functional (1)
  28. 0028apply gaussian_norm_functional
  29. 0029exact hn_witness
  30. 0030specialize gaussian_unit_has_norm_one (z)
  31. 0031apply gaussian_unit_has_norm_one
  32. 0032exact hu