DU0004

dirichlet_kronecker_delta_value_exists

Constructively decide whether an index is one, obtaining its actual zero-or-one signed value before extending any table.

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.

Signed one is code 2. Positive-only table graphs preserve arbitrary zeroth values, including N=0. The unit and divisor-sum identities are proved, never embedded in definitions. The separate inverse family proves the general unit-at-one criterion; multiplicative-function closure remains open.

Exact theorem in conservative defined notation

∀ n. ∃ z. (n = 1 → z = 2) ∧ (¬n = 1 → z = 0)

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

Definition DAG

none

Actual proof prerequisites

eq_decidable · checked external prerequisite
Original expanded first-order statement 
forall n. exists z. ((((n)=1 -> (z)=2) /\ (~((n)=1) -> (z)=0)))

Complete tactic proof in conservative notation

All 22 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

22 script commands · 17 reading checkpoints · 1 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 assumptionsL1–1

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

  1. L1
    intro n
02Establish hcL2–5

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

  1. L2
    have hc : n=1 \/ ~(n=1)
  2. L3
    specialize eq_decidable (n)
  3. L4
    specialize eq_decidable (1)
  4. L5
    apply eq_decidable
03Separate the logical casesL6–6

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

  1. L6
    cases hc
04Construct an explicit witnessL7–7

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

  1. L7
    exists 2
05Separate the logical casesL8–8

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

  1. L8
    split
06Fix variables and assumptionsL9–9

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

  1. L9
    intro he
07Calculate and transport equalitiesL10–10

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

  1. L10
    refl
08Fix variables and assumptionsL11–11

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

  1. L11
    intro he
09Separate the logical casesL12–12

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

  1. L12
    exfalso
10Use earlier factsL13–14

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

  1. L13
    apply he
  2. L14
    exact hc_left
11Construct an explicit witnessL15–15

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

  1. L15
    exists 0
12Separate the logical casesL16–16

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

  1. L16
    split
13Fix variables and assumptionsL17–17

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

  1. L17
    intro he
14Separate the logical casesL18–18

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

  1. L18
    exfalso
15Use earlier factsL19–20

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

  1. L19
    apply hc_right
  2. L20
    exact he
16Fix variables and assumptionsL21–21

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

  1. L21
    intro he
17Calculate and transport equalitiesL22–22

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

  1. L22
    refl

Library-wide reading audit

Original defined command ledger · 22 lines
  1. 0001intro n
  2. 0002have hc : n=1 \/ ~(n=1)
  3. 0003specialize eq_decidable (n)
  4. 0004specialize eq_decidable (1)
  5. 0005apply eq_decidable
  6. 0006cases hc
  7. 0007exists 2
  8. 0008split
  9. 0009intro he
  10. 0010refl
  11. 0011intro he
  12. 0012exfalso
  13. 0013apply he
  14. 0014exact hc_left
  15. 0015exists 0
  16. 0016split
  17. 0017intro he
  18. 0018exfalso
  19. 0019apply hc_right
  20. 0020exact he
  21. 0021intro he
  22. 0022refl