DU0002

dirichlet_kronecker_delta_table_one_value

The actual entry at index one is signed one whenever that index lies in the finite table domain.

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. ∀ E. ∀ z. KroneckerDeltaTable(N,E) → Lt(0,N) → ArithAt(E,1,z) → z = 2

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

Definition DAG

ArithAt(F,i,z) · 1KroneckerDeltaTable(N,E) · 1Lt(a,b) · 1

Actual proof prerequisites

none
Original expanded first-order statement 
forall N E z. (((exists dst_positive_code_delta_one_tabletable dst_positive_scale_delta_one_tabletable dst_negative_code_delta_one_tabletable dst_negative_scale_delta_one_tabletable. (((E) = (((((dst_positive_code_delta_one_tabletable) + (dst_positive_scale_delta_one_tabletable)) * S ((dst_positive_code_delta_one_tabletable) + (dst_positive_scale_delta_one_tabletable)) + ((dst_positive_scale_delta_one_tabletable) + (dst_positive_scale_delta_one_tabletable))) + (((dst_negative_code_delta_one_tabletable) + (dst_negative_scale_delta_one_tabletable)) * S ((dst_negative_code_delta_one_tabletable) + (dst_negative_scale_delta_one_tabletable)) + ((dst_negative_scale_delta_one_tabletable) + (dst_negative_scale_delta_one_tabletable)))) * S ((((dst_positive_code_delta_one_tabletable) + (dst_positive_scale_delta_one_tabletable)) * S ((dst_positive_code_delta_one_tabletable) + (dst_positive_scale_delta_one_tabletable)) + ((dst_positive_scale_delta_one_tabletable) + (dst_positive_scale_delta_one_tabletable))) + (((dst_negative_code_delta_one_tabletable) + (dst_negative_scale_delta_one_tabletable)) * S ((dst_negative_code_delta_one_tabletable) + (dst_negative_scale_delta_one_tabletable)) + ((dst_negative_scale_delta_one_tabletable) + (dst_negative_scale_delta_one_tabletable)))) + ((((dst_negative_code_delta_one_tabletable) + (dst_negative_scale_delta_one_tabletable)) * S ((dst_negative_code_delta_one_tabletable) + (dst_negative_scale_delta_one_tabletable)) + ((dst_negative_scale_delta_one_tabletable) + (dst_negative_scale_delta_one_tabletable))) + (((dst_negative_code_delta_one_tabletable) + (dst_negative_scale_delta_one_tabletable)) * S ((dst_negative_code_delta_one_tabletable) + (dst_negative_scale_delta_one_tabletable)) + ((dst_negative_scale_delta_one_tabletable) + (dst_negative_scale_delta_one_tabletable)))))) /\ (forall dst_index_delta_one_tabletable. (exists pvs_le_gap_delta_one_tabletabledomain. pvs_le_gap_delta_one_tabletabledomain + (dst_index_delta_one_tabletable) = (N)) -> exists dst_positive_delta_one_tabletable dst_negative_delta_one_tabletable dst_value_delta_one_tabletable. ((((exists ff_h_pvs_delta_one_tabletableentrypositive. ff_h_pvs_delta_one_tabletableentrypositive + S (dst_positive_delta_one_tabletable) = S ((S (dst_index_delta_one_tabletable)) * dst_positive_scale_delta_one_tabletable)) /\ exists ff_q_pvs_delta_one_tabletableentrypositive. dst_positive_code_delta_one_tabletable = ff_q_pvs_delta_one_tabletableentrypositive * S ((S (dst_index_delta_one_tabletable)) * dst_positive_scale_delta_one_tabletable) + (dst_positive_delta_one_tabletable))) /\ (((((exists ff_h_pvs_delta_one_tabletableentrynegative. ff_h_pvs_delta_one_tabletableentrynegative + S (dst_negative_delta_one_tabletable) = S ((S (dst_index_delta_one_tabletable)) * dst_negative_scale_delta_one_tabletable)) /\ exists ff_q_pvs_delta_one_tabletableentrynegative. dst_negative_code_delta_one_tabletable = ff_q_pvs_delta_one_tabletableentrynegative * S ((S (dst_index_delta_one_tabletable)) * dst_negative_scale_delta_one_tabletable) + (dst_negative_delta_one_tabletable))) /\ (exists ge_balance_positive_delta_one_tabletableentryvalue ge_balance_negative_delta_one_tabletableentryvalue. (((((dst_value_delta_one_tabletable) = 2 * (ge_balance_positive_delta_one_tabletableentryvalue) /\ (ge_balance_negative_delta_one_tabletableentryvalue) = 0) \/ exists ge_signed_half_delta_one_tabletableentryvaluedecode. (((dst_value_delta_one_tabletable) = 2 * ge_signed_half_delta_one_tabletableentryvaluedecode + 1 /\ (ge_balance_positive_delta_one_tabletableentryvalue) = 0) /\ (ge_balance_negative_delta_one_tabletableentryvalue) = S ge_signed_half_delta_one_tabletableentryvaluedecode))) /\ ((dst_positive_delta_one_tabletable) + ge_balance_negative_delta_one_tabletableentryvalue = (dst_negative_delta_one_tabletable) + ge_balance_positive_delta_one_tabletableentryvalue))))))))) /\ (forall du_index_delta_one_table du_value_delta_one_table. ~(du_index_delta_one_table=0) -> (exists pvs_le_gap_delta_one_tablebound. pvs_le_gap_delta_one_tablebound + (du_index_delta_one_table) = (N)) -> (exists dst_positive_code_delta_one_tableentry dst_positive_scale_delta_one_tableentry dst_negative_code_delta_one_tableentry dst_negative_scale_delta_one_tableentry dst_positive_delta_one_tableentry dst_negative_delta_one_tableentry. (((E) = (((((dst_positive_code_delta_one_tableentry) + (dst_positive_scale_delta_one_tableentry)) * S ((dst_positive_code_delta_one_tableentry) + (dst_positive_scale_delta_one_tableentry)) + ((dst_positive_scale_delta_one_tableentry) + (dst_positive_scale_delta_one_tableentry))) + (((dst_negative_code_delta_one_tableentry) + (dst_negative_scale_delta_one_tableentry)) * S ((dst_negative_code_delta_one_tableentry) + (dst_negative_scale_delta_one_tableentry)) + ((dst_negative_scale_delta_one_tableentry) + (dst_negative_scale_delta_one_tableentry)))) * S ((((dst_positive_code_delta_one_tableentry) + (dst_positive_scale_delta_one_tableentry)) * S ((dst_positive_code_delta_one_tableentry) + (dst_positive_scale_delta_one_tableentry)) + ((dst_positive_scale_delta_one_tableentry) + (dst_positive_scale_delta_one_tableentry))) + (((dst_negative_code_delta_one_tableentry) + (dst_negative_scale_delta_one_tableentry)) * S ((dst_negative_code_delta_one_tableentry) + (dst_negative_scale_delta_one_tableentry)) + ((dst_negative_scale_delta_one_tableentry) + (dst_negative_scale_delta_one_tableentry)))) + ((((dst_negative_code_delta_one_tableentry) + (dst_negative_scale_delta_one_tableentry)) * S ((dst_negative_code_delta_one_tableentry) + (dst_negative_scale_delta_one_tableentry)) + ((dst_negative_scale_delta_one_tableentry) + (dst_negative_scale_delta_one_tableentry))) + (((dst_negative_code_delta_one_tableentry) + (dst_negative_scale_delta_one_tableentry)) * S ((dst_negative_code_delta_one_tableentry) + (dst_negative_scale_delta_one_tableentry)) + ((dst_negative_scale_delta_one_tableentry) + (dst_negative_scale_delta_one_tableentry)))))) /\ (((((exists ff_h_pvs_delta_one_tableentrypositive. ff_h_pvs_delta_one_tableentrypositive + S (dst_positive_delta_one_tableentry) = S ((S (du_index_delta_one_table)) * dst_positive_scale_delta_one_tableentry)) /\ exists ff_q_pvs_delta_one_tableentrypositive. dst_positive_code_delta_one_tableentry = ff_q_pvs_delta_one_tableentrypositive * S ((S (du_index_delta_one_table)) * dst_positive_scale_delta_one_tableentry) + (dst_positive_delta_one_tableentry))) /\ (((((exists ff_h_pvs_delta_one_tableentrynegative. ff_h_pvs_delta_one_tableentrynegative + S (dst_negative_delta_one_tableentry) = S ((S (du_index_delta_one_table)) * dst_negative_scale_delta_one_tableentry)) /\ exists ff_q_pvs_delta_one_tableentrynegative. dst_negative_code_delta_one_tableentry = ff_q_pvs_delta_one_tableentrynegative * S ((S (du_index_delta_one_table)) * dst_negative_scale_delta_one_tableentry) + (dst_negative_delta_one_tableentry))) /\ (exists ge_balance_positive_delta_one_tableentryvalue ge_balance_negative_delta_one_tableentryvalue. (((((du_value_delta_one_table) = 2 * (ge_balance_positive_delta_one_tableentryvalue) /\ (ge_balance_negative_delta_one_tableentryvalue) = 0) \/ exists ge_signed_half_delta_one_tableentryvaluedecode. (((du_value_delta_one_table) = 2 * ge_signed_half_delta_one_tableentryvaluedecode + 1 /\ (ge_balance_positive_delta_one_tableentryvalue) = 0) /\ (ge_balance_negative_delta_one_tableentryvalue) = S ge_signed_half_delta_one_tableentryvaluedecode))) /\ ((dst_positive_delta_one_tableentry) + ge_balance_negative_delta_one_tableentryvalue = (dst_negative_delta_one_tableentry) + ge_balance_positive_delta_one_tableentryvalue))))))))) -> ((((du_index_delta_one_table)=1 -> (du_value_delta_one_table)=2) /\ (~((du_index_delta_one_table)=1) -> (du_value_delta_one_table)=0)))))) -> (exists pvs_le_gap_delta_one_bound. pvs_le_gap_delta_one_bound + (1) = (N)) -> (exists dst_positive_code_delta_one_entry dst_positive_scale_delta_one_entry dst_negative_code_delta_one_entry dst_negative_scale_delta_one_entry dst_positive_delta_one_entry dst_negative_delta_one_entry. (((E) = (((((dst_positive_code_delta_one_entry) + (dst_positive_scale_delta_one_entry)) * S ((dst_positive_code_delta_one_entry) + (dst_positive_scale_delta_one_entry)) + ((dst_positive_scale_delta_one_entry) + (dst_positive_scale_delta_one_entry))) + (((dst_negative_code_delta_one_entry) + (dst_negative_scale_delta_one_entry)) * S ((dst_negative_code_delta_one_entry) + (dst_negative_scale_delta_one_entry)) + ((dst_negative_scale_delta_one_entry) + (dst_negative_scale_delta_one_entry)))) * S ((((dst_positive_code_delta_one_entry) + (dst_positive_scale_delta_one_entry)) * S ((dst_positive_code_delta_one_entry) + (dst_positive_scale_delta_one_entry)) + ((dst_positive_scale_delta_one_entry) + (dst_positive_scale_delta_one_entry))) + (((dst_negative_code_delta_one_entry) + (dst_negative_scale_delta_one_entry)) * S ((dst_negative_code_delta_one_entry) + (dst_negative_scale_delta_one_entry)) + ((dst_negative_scale_delta_one_entry) + (dst_negative_scale_delta_one_entry)))) + ((((dst_negative_code_delta_one_entry) + (dst_negative_scale_delta_one_entry)) * S ((dst_negative_code_delta_one_entry) + (dst_negative_scale_delta_one_entry)) + ((dst_negative_scale_delta_one_entry) + (dst_negative_scale_delta_one_entry))) + (((dst_negative_code_delta_one_entry) + (dst_negative_scale_delta_one_entry)) * S ((dst_negative_code_delta_one_entry) + (dst_negative_scale_delta_one_entry)) + ((dst_negative_scale_delta_one_entry) + (dst_negative_scale_delta_one_entry)))))) /\ (((((exists ff_h_pvs_delta_one_entrypositive. ff_h_pvs_delta_one_entrypositive + S (dst_positive_delta_one_entry) = S ((S (1)) * dst_positive_scale_delta_one_entry)) /\ exists ff_q_pvs_delta_one_entrypositive. dst_positive_code_delta_one_entry = ff_q_pvs_delta_one_entrypositive * S ((S (1)) * dst_positive_scale_delta_one_entry) + (dst_positive_delta_one_entry))) /\ (((((exists ff_h_pvs_delta_one_entrynegative. ff_h_pvs_delta_one_entrynegative + S (dst_negative_delta_one_entry) = S ((S (1)) * dst_negative_scale_delta_one_entry)) /\ exists ff_q_pvs_delta_one_entrynegative. dst_negative_code_delta_one_entry = ff_q_pvs_delta_one_entrynegative * S ((S (1)) * dst_negative_scale_delta_one_entry) + (dst_negative_delta_one_entry))) /\ (exists ge_balance_positive_delta_one_entryvalue ge_balance_negative_delta_one_entryvalue. (((((z) = 2 * (ge_balance_positive_delta_one_entryvalue) /\ (ge_balance_negative_delta_one_entryvalue) = 0) \/ exists ge_signed_half_delta_one_entryvaluedecode. (((z) = 2 * ge_signed_half_delta_one_entryvaluedecode + 1 /\ (ge_balance_positive_delta_one_entryvalue) = 0) /\ (ge_balance_negative_delta_one_entryvalue) = S ge_signed_half_delta_one_entryvaluedecode))) /\ ((dst_positive_delta_one_entry) + ge_balance_negative_delta_one_entryvalue = (dst_negative_delta_one_entry) + ge_balance_positive_delta_one_entryvalue))))))))) -> z=2

Complete tactic proof in conservative notation

All 19 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

19 script commands · 6 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 assumptionsL1–6

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

  1. L1
    intro N
  2. L2
    intro E
  3. L3
    intro z
  4. L4
    intro he
  5. L5
    intro hb
  6. L6
    intro hz
02Separate the logical casesL7–7

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

  1. L7
    cases he
03Establish hvL8–16

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

  1. L8
    have hv : (((1)=1 -> (z)=2) /\ (~((1)=1) -> (z)=0))
  2. L9
    specialize he_right (1)
  3. L10
    specialize he_right (z)
  4. L11
    apply he_right
  5. L12
    intro hn
  6. L13
    apply PA1
  7. L14
    exact hn
  8. L15
    exact hb
  9. L16
    exact hz
04Separate the logical casesL17–17

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

  1. L17
    cases hv
05Use earlier factsL18–18

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

  1. L18
    apply hv_left
06Calculate and transport equalitiesL19–19

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

  1. L19
    refl

Library-wide reading audit

Original defined command ledger · 19 lines
  1. 0001intro N
  2. 0002intro E
  3. 0003intro z
  4. 0004intro he
  5. 0005intro hb
  6. 0006intro hz
  7. 0007cases he
  8. 0008have hv : (((1)=1 -> (z)=2) /\ (~((1)=1) -> (z)=0))
  9. 0009specialize he_right (1)
  10. 0010specialize he_right (z)
  11. 0011apply he_right
  12. 0012intro hn
  13. 0013apply PA1
  14. 0014exact hn
  15. 0015exact hb
  16. 0016exact hz
  17. 0017cases hv
  18. 0018apply hv_left
  19. 0019refl