IV0001

dirichlet_unit_at_one_witness

An actual unit-at-one lookup supplies its canonical signed unit code; no inverse property is hidden in the predicate.

Alpha v34 checked-use · first admitted v31 · 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.

For an actual table an inverse exists exactly when N=0 or F(1) is signed +1 or -1. At a positive window this is the unit-at-one criterion; the empty window imposes no condition at one. Every inverse has actual delta and two-sided convolution witnesses. Its zeroth value is arbitrary, so uniqueness is positive-value equality, not equality of codes or of zeroth values. Multiplicative-function closure and full finite signed G009 are admitted in the separate Alpha-v32 multiplicative-convolution family.

Exact theorem in conservative defined notation

∀ F. DirichletUnitAtOne(F) → ∃ x. ArithAt(F,1,x) ∧ SignedUnit(x)

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

Definition DAG

ArithAt(F,i,z) · 1SignedUnit(u) · 1DirichletUnitAtOne(F) · 1

Actual proof prerequisites

none
Original expanded first-order statement 
forall F. ((exists dst_positive_code_unit_witness_sourcepositive dst_positive_scale_unit_witness_sourcepositive dst_negative_code_unit_witness_sourcepositive dst_negative_scale_unit_witness_sourcepositive dst_positive_unit_witness_sourcepositive dst_negative_unit_witness_sourcepositive. (((F) = (((((dst_positive_code_unit_witness_sourcepositive) + (dst_positive_scale_unit_witness_sourcepositive)) * S ((dst_positive_code_unit_witness_sourcepositive) + (dst_positive_scale_unit_witness_sourcepositive)) + ((dst_positive_scale_unit_witness_sourcepositive) + (dst_positive_scale_unit_witness_sourcepositive))) + (((dst_negative_code_unit_witness_sourcepositive) + (dst_negative_scale_unit_witness_sourcepositive)) * S ((dst_negative_code_unit_witness_sourcepositive) + (dst_negative_scale_unit_witness_sourcepositive)) + ((dst_negative_scale_unit_witness_sourcepositive) + (dst_negative_scale_unit_witness_sourcepositive)))) * S ((((dst_positive_code_unit_witness_sourcepositive) + (dst_positive_scale_unit_witness_sourcepositive)) * S ((dst_positive_code_unit_witness_sourcepositive) + (dst_positive_scale_unit_witness_sourcepositive)) + ((dst_positive_scale_unit_witness_sourcepositive) + (dst_positive_scale_unit_witness_sourcepositive))) + (((dst_negative_code_unit_witness_sourcepositive) + (dst_negative_scale_unit_witness_sourcepositive)) * S ((dst_negative_code_unit_witness_sourcepositive) + (dst_negative_scale_unit_witness_sourcepositive)) + ((dst_negative_scale_unit_witness_sourcepositive) + (dst_negative_scale_unit_witness_sourcepositive)))) + ((((dst_negative_code_unit_witness_sourcepositive) + (dst_negative_scale_unit_witness_sourcepositive)) * S ((dst_negative_code_unit_witness_sourcepositive) + (dst_negative_scale_unit_witness_sourcepositive)) + ((dst_negative_scale_unit_witness_sourcepositive) + (dst_negative_scale_unit_witness_sourcepositive))) + (((dst_negative_code_unit_witness_sourcepositive) + (dst_negative_scale_unit_witness_sourcepositive)) * S ((dst_negative_code_unit_witness_sourcepositive) + (dst_negative_scale_unit_witness_sourcepositive)) + ((dst_negative_scale_unit_witness_sourcepositive) + (dst_negative_scale_unit_witness_sourcepositive)))))) /\ (((((exists ff_h_pvs_unit_witness_sourcepositivepositive. ff_h_pvs_unit_witness_sourcepositivepositive + S (dst_positive_unit_witness_sourcepositive) = S ((S (1)) * dst_positive_scale_unit_witness_sourcepositive)) /\ exists ff_q_pvs_unit_witness_sourcepositivepositive. dst_positive_code_unit_witness_sourcepositive = ff_q_pvs_unit_witness_sourcepositivepositive * S ((S (1)) * dst_positive_scale_unit_witness_sourcepositive) + (dst_positive_unit_witness_sourcepositive))) /\ (((((exists ff_h_pvs_unit_witness_sourcepositivenegative. ff_h_pvs_unit_witness_sourcepositivenegative + S (dst_negative_unit_witness_sourcepositive) = S ((S (1)) * dst_negative_scale_unit_witness_sourcepositive)) /\ exists ff_q_pvs_unit_witness_sourcepositivenegative. dst_negative_code_unit_witness_sourcepositive = ff_q_pvs_unit_witness_sourcepositivenegative * S ((S (1)) * dst_negative_scale_unit_witness_sourcepositive) + (dst_negative_unit_witness_sourcepositive))) /\ (exists ge_balance_positive_unit_witness_sourcepositivevalue ge_balance_negative_unit_witness_sourcepositivevalue. (((((2) = 2 * (ge_balance_positive_unit_witness_sourcepositivevalue) /\ (ge_balance_negative_unit_witness_sourcepositivevalue) = 0) \/ exists ge_signed_half_unit_witness_sourcepositivevaluedecode. (((2) = 2 * ge_signed_half_unit_witness_sourcepositivevaluedecode + 1 /\ (ge_balance_positive_unit_witness_sourcepositivevalue) = 0) /\ (ge_balance_negative_unit_witness_sourcepositivevalue) = S ge_signed_half_unit_witness_sourcepositivevaluedecode))) /\ ((dst_positive_unit_witness_sourcepositive) + ge_balance_negative_unit_witness_sourcepositivevalue = (dst_negative_unit_witness_sourcepositive) + ge_balance_positive_unit_witness_sourcepositivevalue))))))))) \/ (exists dst_positive_code_unit_witness_sourcenegative dst_positive_scale_unit_witness_sourcenegative dst_negative_code_unit_witness_sourcenegative dst_negative_scale_unit_witness_sourcenegative dst_positive_unit_witness_sourcenegative dst_negative_unit_witness_sourcenegative. (((F) = (((((dst_positive_code_unit_witness_sourcenegative) + (dst_positive_scale_unit_witness_sourcenegative)) * S ((dst_positive_code_unit_witness_sourcenegative) + (dst_positive_scale_unit_witness_sourcenegative)) + ((dst_positive_scale_unit_witness_sourcenegative) + (dst_positive_scale_unit_witness_sourcenegative))) + (((dst_negative_code_unit_witness_sourcenegative) + (dst_negative_scale_unit_witness_sourcenegative)) * S ((dst_negative_code_unit_witness_sourcenegative) + (dst_negative_scale_unit_witness_sourcenegative)) + ((dst_negative_scale_unit_witness_sourcenegative) + (dst_negative_scale_unit_witness_sourcenegative)))) * S ((((dst_positive_code_unit_witness_sourcenegative) + (dst_positive_scale_unit_witness_sourcenegative)) * S ((dst_positive_code_unit_witness_sourcenegative) + (dst_positive_scale_unit_witness_sourcenegative)) + ((dst_positive_scale_unit_witness_sourcenegative) + (dst_positive_scale_unit_witness_sourcenegative))) + (((dst_negative_code_unit_witness_sourcenegative) + (dst_negative_scale_unit_witness_sourcenegative)) * S ((dst_negative_code_unit_witness_sourcenegative) + (dst_negative_scale_unit_witness_sourcenegative)) + ((dst_negative_scale_unit_witness_sourcenegative) + (dst_negative_scale_unit_witness_sourcenegative)))) + ((((dst_negative_code_unit_witness_sourcenegative) + (dst_negative_scale_unit_witness_sourcenegative)) * S ((dst_negative_code_unit_witness_sourcenegative) + (dst_negative_scale_unit_witness_sourcenegative)) + ((dst_negative_scale_unit_witness_sourcenegative) + (dst_negative_scale_unit_witness_sourcenegative))) + (((dst_negative_code_unit_witness_sourcenegative) + (dst_negative_scale_unit_witness_sourcenegative)) * S ((dst_negative_code_unit_witness_sourcenegative) + (dst_negative_scale_unit_witness_sourcenegative)) + ((dst_negative_scale_unit_witness_sourcenegative) + (dst_negative_scale_unit_witness_sourcenegative)))))) /\ (((((exists ff_h_pvs_unit_witness_sourcenegativepositive. ff_h_pvs_unit_witness_sourcenegativepositive + S (dst_positive_unit_witness_sourcenegative) = S ((S (1)) * dst_positive_scale_unit_witness_sourcenegative)) /\ exists ff_q_pvs_unit_witness_sourcenegativepositive. dst_positive_code_unit_witness_sourcenegative = ff_q_pvs_unit_witness_sourcenegativepositive * S ((S (1)) * dst_positive_scale_unit_witness_sourcenegative) + (dst_positive_unit_witness_sourcenegative))) /\ (((((exists ff_h_pvs_unit_witness_sourcenegativenegative. ff_h_pvs_unit_witness_sourcenegativenegative + S (dst_negative_unit_witness_sourcenegative) = S ((S (1)) * dst_negative_scale_unit_witness_sourcenegative)) /\ exists ff_q_pvs_unit_witness_sourcenegativenegative. dst_negative_code_unit_witness_sourcenegative = ff_q_pvs_unit_witness_sourcenegativenegative * S ((S (1)) * dst_negative_scale_unit_witness_sourcenegative) + (dst_negative_unit_witness_sourcenegative))) /\ (exists ge_balance_positive_unit_witness_sourcenegativevalue ge_balance_negative_unit_witness_sourcenegativevalue. (((((1) = 2 * (ge_balance_positive_unit_witness_sourcenegativevalue) /\ (ge_balance_negative_unit_witness_sourcenegativevalue) = 0) \/ exists ge_signed_half_unit_witness_sourcenegativevaluedecode. (((1) = 2 * ge_signed_half_unit_witness_sourcenegativevaluedecode + 1 /\ (ge_balance_positive_unit_witness_sourcenegativevalue) = 0) /\ (ge_balance_negative_unit_witness_sourcenegativevalue) = S ge_signed_half_unit_witness_sourcenegativevaluedecode))) /\ ((dst_positive_unit_witness_sourcenegative) + ge_balance_negative_unit_witness_sourcenegativevalue = (dst_negative_unit_witness_sourcenegative) + ge_balance_positive_unit_witness_sourcenegativevalue)))))))))) -> exists u. ((exists dst_positive_code_unit_witness_entry dst_positive_scale_unit_witness_entry dst_negative_code_unit_witness_entry dst_negative_scale_unit_witness_entry dst_positive_unit_witness_entry dst_negative_unit_witness_entry. (((F) = (((((dst_positive_code_unit_witness_entry) + (dst_positive_scale_unit_witness_entry)) * S ((dst_positive_code_unit_witness_entry) + (dst_positive_scale_unit_witness_entry)) + ((dst_positive_scale_unit_witness_entry) + (dst_positive_scale_unit_witness_entry))) + (((dst_negative_code_unit_witness_entry) + (dst_negative_scale_unit_witness_entry)) * S ((dst_negative_code_unit_witness_entry) + (dst_negative_scale_unit_witness_entry)) + ((dst_negative_scale_unit_witness_entry) + (dst_negative_scale_unit_witness_entry)))) * S ((((dst_positive_code_unit_witness_entry) + (dst_positive_scale_unit_witness_entry)) * S ((dst_positive_code_unit_witness_entry) + (dst_positive_scale_unit_witness_entry)) + ((dst_positive_scale_unit_witness_entry) + (dst_positive_scale_unit_witness_entry))) + (((dst_negative_code_unit_witness_entry) + (dst_negative_scale_unit_witness_entry)) * S ((dst_negative_code_unit_witness_entry) + (dst_negative_scale_unit_witness_entry)) + ((dst_negative_scale_unit_witness_entry) + (dst_negative_scale_unit_witness_entry)))) + ((((dst_negative_code_unit_witness_entry) + (dst_negative_scale_unit_witness_entry)) * S ((dst_negative_code_unit_witness_entry) + (dst_negative_scale_unit_witness_entry)) + ((dst_negative_scale_unit_witness_entry) + (dst_negative_scale_unit_witness_entry))) + (((dst_negative_code_unit_witness_entry) + (dst_negative_scale_unit_witness_entry)) * S ((dst_negative_code_unit_witness_entry) + (dst_negative_scale_unit_witness_entry)) + ((dst_negative_scale_unit_witness_entry) + (dst_negative_scale_unit_witness_entry)))))) /\ (((((exists ff_h_pvs_unit_witness_entrypositive. ff_h_pvs_unit_witness_entrypositive + S (dst_positive_unit_witness_entry) = S ((S (1)) * dst_positive_scale_unit_witness_entry)) /\ exists ff_q_pvs_unit_witness_entrypositive. dst_positive_code_unit_witness_entry = ff_q_pvs_unit_witness_entrypositive * S ((S (1)) * dst_positive_scale_unit_witness_entry) + (dst_positive_unit_witness_entry))) /\ (((((exists ff_h_pvs_unit_witness_entrynegative. ff_h_pvs_unit_witness_entrynegative + S (dst_negative_unit_witness_entry) = S ((S (1)) * dst_negative_scale_unit_witness_entry)) /\ exists ff_q_pvs_unit_witness_entrynegative. dst_negative_code_unit_witness_entry = ff_q_pvs_unit_witness_entrynegative * S ((S (1)) * dst_negative_scale_unit_witness_entry) + (dst_negative_unit_witness_entry))) /\ (exists ge_balance_positive_unit_witness_entryvalue ge_balance_negative_unit_witness_entryvalue. (((((u) = 2 * (ge_balance_positive_unit_witness_entryvalue) /\ (ge_balance_negative_unit_witness_entryvalue) = 0) \/ exists ge_signed_half_unit_witness_entryvaluedecode. (((u) = 2 * ge_signed_half_unit_witness_entryvaluedecode + 1 /\ (ge_balance_positive_unit_witness_entryvalue) = 0) /\ (ge_balance_negative_unit_witness_entryvalue) = S ge_signed_half_unit_witness_entryvaluedecode))) /\ ((dst_positive_unit_witness_entry) + ge_balance_negative_unit_witness_entryvalue = (dst_negative_unit_witness_entry) + ge_balance_positive_unit_witness_entryvalue))))))))) /\ (((u) = 2 \/ (u) = 1)))

Complete tactic proof in conservative notation

All 13 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

13 script commands · 12 reading checkpoints · 0 local claims

This is a reading aid, not a new proof or a proof-tree certificate. Checkpoint groups are consecutive commands, not inferred branch boundaries. Every step links to the preserved script.

Definition notation is shown below. Open the paired exact edition for the original native formulas. Source pairing is not a new equivalence certificate.

01Fix variables and assumptionsL1–2

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

  1. L1
    intro F
  2. L2
    intro hu
02Separate the logical casesL3–3

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

  1. L3
    cases hu
03Construct an explicit witnessL4–4

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

  1. L4
    exists 2
04Separate the logical casesL5–5

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

  1. L5
    split
05Use earlier factsL6–6

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

  1. L6
    exact hu_left
06Separate the logical casesL7–7

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

  1. L7
    left
07Calculate and transport equalitiesL8–8

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

  1. L8
    refl
08Construct an explicit witnessL9–9

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

  1. L9
    exists 1
09Separate the logical casesL10–10

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

  1. L10
    split
10Use earlier factsL11–11

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

  1. L11
    exact hu_right
11Separate the logical casesL12–12

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

  1. L12
    right
12Calculate and transport equalitiesL13–13

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

  1. L13
    refl

Library-wide reading audit

Original defined command ledger · 13 lines
  1. 0001intro F
  2. 0002intro hu
  3. 0003cases hu
  4. 0004exists 2
  5. 0005split
  6. 0006exact hu_left
  7. 0007left
  8. 0008refl
  9. 0009exists 1
  10. 0010split
  11. 0011exact hu_right
  12. 0012right
  13. 0013refl