DU0008

dirichlet_kronecker_delta_table_exists

Finite constructive equality decisions and actual beta extensions build the delta table for every N, preserving any zero entry.

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. ∀ w. ∃ E. KroneckerDeltaTable(N,E) ∧ ArithAt(E,0,w)

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

Definition DAG

ArithTable(N,F) · 1ArithAt(F,i,z) · 5KroneckerDeltaTable(N,E) · 3Lt(a,b) · 1

Actual proof prerequisites

arithmetic_signed_table_singleton · checked external prerequisitele_zero · checked external prerequisitedirichlet_kronecker_delta_value_existsdirichlet_kronecker_delta_table_appendarithmetic_signed_table_equal_entry_transport · checked external prerequisitezero_le · checked external prerequisitesucc_le_succ · checked external prerequisite
Original expanded first-order statement 
forall N w. exists E. (((exists dst_positive_code_delta_exists_tabletable dst_positive_scale_delta_exists_tabletable dst_negative_code_delta_exists_tabletable dst_negative_scale_delta_exists_tabletable. (((E) = (((((dst_positive_code_delta_exists_tabletable) + (dst_positive_scale_delta_exists_tabletable)) * S ((dst_positive_code_delta_exists_tabletable) + (dst_positive_scale_delta_exists_tabletable)) + ((dst_positive_scale_delta_exists_tabletable) + (dst_positive_scale_delta_exists_tabletable))) + (((dst_negative_code_delta_exists_tabletable) + (dst_negative_scale_delta_exists_tabletable)) * S ((dst_negative_code_delta_exists_tabletable) + (dst_negative_scale_delta_exists_tabletable)) + ((dst_negative_scale_delta_exists_tabletable) + (dst_negative_scale_delta_exists_tabletable)))) * S ((((dst_positive_code_delta_exists_tabletable) + (dst_positive_scale_delta_exists_tabletable)) * S ((dst_positive_code_delta_exists_tabletable) + (dst_positive_scale_delta_exists_tabletable)) + ((dst_positive_scale_delta_exists_tabletable) + (dst_positive_scale_delta_exists_tabletable))) + (((dst_negative_code_delta_exists_tabletable) + (dst_negative_scale_delta_exists_tabletable)) * S ((dst_negative_code_delta_exists_tabletable) + (dst_negative_scale_delta_exists_tabletable)) + ((dst_negative_scale_delta_exists_tabletable) + (dst_negative_scale_delta_exists_tabletable)))) + ((((dst_negative_code_delta_exists_tabletable) + (dst_negative_scale_delta_exists_tabletable)) * S ((dst_negative_code_delta_exists_tabletable) + (dst_negative_scale_delta_exists_tabletable)) + ((dst_negative_scale_delta_exists_tabletable) + (dst_negative_scale_delta_exists_tabletable))) + (((dst_negative_code_delta_exists_tabletable) + (dst_negative_scale_delta_exists_tabletable)) * S ((dst_negative_code_delta_exists_tabletable) + (dst_negative_scale_delta_exists_tabletable)) + ((dst_negative_scale_delta_exists_tabletable) + (dst_negative_scale_delta_exists_tabletable)))))) /\ (forall dst_index_delta_exists_tabletable. (exists pvs_le_gap_delta_exists_tabletabledomain. pvs_le_gap_delta_exists_tabletabledomain + (dst_index_delta_exists_tabletable) = (N)) -> exists dst_positive_delta_exists_tabletable dst_negative_delta_exists_tabletable dst_value_delta_exists_tabletable. ((((exists ff_h_pvs_delta_exists_tabletableentrypositive. ff_h_pvs_delta_exists_tabletableentrypositive + S (dst_positive_delta_exists_tabletable) = S ((S (dst_index_delta_exists_tabletable)) * dst_positive_scale_delta_exists_tabletable)) /\ exists ff_q_pvs_delta_exists_tabletableentrypositive. dst_positive_code_delta_exists_tabletable = ff_q_pvs_delta_exists_tabletableentrypositive * S ((S (dst_index_delta_exists_tabletable)) * dst_positive_scale_delta_exists_tabletable) + (dst_positive_delta_exists_tabletable))) /\ (((((exists ff_h_pvs_delta_exists_tabletableentrynegative. ff_h_pvs_delta_exists_tabletableentrynegative + S (dst_negative_delta_exists_tabletable) = S ((S (dst_index_delta_exists_tabletable)) * dst_negative_scale_delta_exists_tabletable)) /\ exists ff_q_pvs_delta_exists_tabletableentrynegative. dst_negative_code_delta_exists_tabletable = ff_q_pvs_delta_exists_tabletableentrynegative * S ((S (dst_index_delta_exists_tabletable)) * dst_negative_scale_delta_exists_tabletable) + (dst_negative_delta_exists_tabletable))) /\ (exists ge_balance_positive_delta_exists_tabletableentryvalue ge_balance_negative_delta_exists_tabletableentryvalue. (((((dst_value_delta_exists_tabletable) = 2 * (ge_balance_positive_delta_exists_tabletableentryvalue) /\ (ge_balance_negative_delta_exists_tabletableentryvalue) = 0) \/ exists ge_signed_half_delta_exists_tabletableentryvaluedecode. (((dst_value_delta_exists_tabletable) = 2 * ge_signed_half_delta_exists_tabletableentryvaluedecode + 1 /\ (ge_balance_positive_delta_exists_tabletableentryvalue) = 0) /\ (ge_balance_negative_delta_exists_tabletableentryvalue) = S ge_signed_half_delta_exists_tabletableentryvaluedecode))) /\ ((dst_positive_delta_exists_tabletable) + ge_balance_negative_delta_exists_tabletableentryvalue = (dst_negative_delta_exists_tabletable) + ge_balance_positive_delta_exists_tabletableentryvalue))))))))) /\ (forall du_index_delta_exists_table du_value_delta_exists_table. ~(du_index_delta_exists_table=0) -> (exists pvs_le_gap_delta_exists_tablebound. pvs_le_gap_delta_exists_tablebound + (du_index_delta_exists_table) = (N)) -> (exists dst_positive_code_delta_exists_tableentry dst_positive_scale_delta_exists_tableentry dst_negative_code_delta_exists_tableentry dst_negative_scale_delta_exists_tableentry dst_positive_delta_exists_tableentry dst_negative_delta_exists_tableentry. (((E) = (((((dst_positive_code_delta_exists_tableentry) + (dst_positive_scale_delta_exists_tableentry)) * S ((dst_positive_code_delta_exists_tableentry) + (dst_positive_scale_delta_exists_tableentry)) + ((dst_positive_scale_delta_exists_tableentry) + (dst_positive_scale_delta_exists_tableentry))) + (((dst_negative_code_delta_exists_tableentry) + (dst_negative_scale_delta_exists_tableentry)) * S ((dst_negative_code_delta_exists_tableentry) + (dst_negative_scale_delta_exists_tableentry)) + ((dst_negative_scale_delta_exists_tableentry) + (dst_negative_scale_delta_exists_tableentry)))) * S ((((dst_positive_code_delta_exists_tableentry) + (dst_positive_scale_delta_exists_tableentry)) * S ((dst_positive_code_delta_exists_tableentry) + (dst_positive_scale_delta_exists_tableentry)) + ((dst_positive_scale_delta_exists_tableentry) + (dst_positive_scale_delta_exists_tableentry))) + (((dst_negative_code_delta_exists_tableentry) + (dst_negative_scale_delta_exists_tableentry)) * S ((dst_negative_code_delta_exists_tableentry) + (dst_negative_scale_delta_exists_tableentry)) + ((dst_negative_scale_delta_exists_tableentry) + (dst_negative_scale_delta_exists_tableentry)))) + ((((dst_negative_code_delta_exists_tableentry) + (dst_negative_scale_delta_exists_tableentry)) * S ((dst_negative_code_delta_exists_tableentry) + (dst_negative_scale_delta_exists_tableentry)) + ((dst_negative_scale_delta_exists_tableentry) + (dst_negative_scale_delta_exists_tableentry))) + (((dst_negative_code_delta_exists_tableentry) + (dst_negative_scale_delta_exists_tableentry)) * S ((dst_negative_code_delta_exists_tableentry) + (dst_negative_scale_delta_exists_tableentry)) + ((dst_negative_scale_delta_exists_tableentry) + (dst_negative_scale_delta_exists_tableentry)))))) /\ (((((exists ff_h_pvs_delta_exists_tableentrypositive. ff_h_pvs_delta_exists_tableentrypositive + S (dst_positive_delta_exists_tableentry) = S ((S (du_index_delta_exists_table)) * dst_positive_scale_delta_exists_tableentry)) /\ exists ff_q_pvs_delta_exists_tableentrypositive. dst_positive_code_delta_exists_tableentry = ff_q_pvs_delta_exists_tableentrypositive * S ((S (du_index_delta_exists_table)) * dst_positive_scale_delta_exists_tableentry) + (dst_positive_delta_exists_tableentry))) /\ (((((exists ff_h_pvs_delta_exists_tableentrynegative. ff_h_pvs_delta_exists_tableentrynegative + S (dst_negative_delta_exists_tableentry) = S ((S (du_index_delta_exists_table)) * dst_negative_scale_delta_exists_tableentry)) /\ exists ff_q_pvs_delta_exists_tableentrynegative. dst_negative_code_delta_exists_tableentry = ff_q_pvs_delta_exists_tableentrynegative * S ((S (du_index_delta_exists_table)) * dst_negative_scale_delta_exists_tableentry) + (dst_negative_delta_exists_tableentry))) /\ (exists ge_balance_positive_delta_exists_tableentryvalue ge_balance_negative_delta_exists_tableentryvalue. (((((du_value_delta_exists_table) = 2 * (ge_balance_positive_delta_exists_tableentryvalue) /\ (ge_balance_negative_delta_exists_tableentryvalue) = 0) \/ exists ge_signed_half_delta_exists_tableentryvaluedecode. (((du_value_delta_exists_table) = 2 * ge_signed_half_delta_exists_tableentryvaluedecode + 1 /\ (ge_balance_positive_delta_exists_tableentryvalue) = 0) /\ (ge_balance_negative_delta_exists_tableentryvalue) = S ge_signed_half_delta_exists_tableentryvaluedecode))) /\ ((dst_positive_delta_exists_tableentry) + ge_balance_negative_delta_exists_tableentryvalue = (dst_negative_delta_exists_tableentry) + ge_balance_positive_delta_exists_tableentryvalue))))))))) -> ((((du_index_delta_exists_table)=1 -> (du_value_delta_exists_table)=2) /\ (~((du_index_delta_exists_table)=1) -> (du_value_delta_exists_table)=0)))))) /\ (exists dst_positive_code_delta_exists_zero dst_positive_scale_delta_exists_zero dst_negative_code_delta_exists_zero dst_negative_scale_delta_exists_zero dst_positive_delta_exists_zero dst_negative_delta_exists_zero. (((E) = (((((dst_positive_code_delta_exists_zero) + (dst_positive_scale_delta_exists_zero)) * S ((dst_positive_code_delta_exists_zero) + (dst_positive_scale_delta_exists_zero)) + ((dst_positive_scale_delta_exists_zero) + (dst_positive_scale_delta_exists_zero))) + (((dst_negative_code_delta_exists_zero) + (dst_negative_scale_delta_exists_zero)) * S ((dst_negative_code_delta_exists_zero) + (dst_negative_scale_delta_exists_zero)) + ((dst_negative_scale_delta_exists_zero) + (dst_negative_scale_delta_exists_zero)))) * S ((((dst_positive_code_delta_exists_zero) + (dst_positive_scale_delta_exists_zero)) * S ((dst_positive_code_delta_exists_zero) + (dst_positive_scale_delta_exists_zero)) + ((dst_positive_scale_delta_exists_zero) + (dst_positive_scale_delta_exists_zero))) + (((dst_negative_code_delta_exists_zero) + (dst_negative_scale_delta_exists_zero)) * S ((dst_negative_code_delta_exists_zero) + (dst_negative_scale_delta_exists_zero)) + ((dst_negative_scale_delta_exists_zero) + (dst_negative_scale_delta_exists_zero)))) + ((((dst_negative_code_delta_exists_zero) + (dst_negative_scale_delta_exists_zero)) * S ((dst_negative_code_delta_exists_zero) + (dst_negative_scale_delta_exists_zero)) + ((dst_negative_scale_delta_exists_zero) + (dst_negative_scale_delta_exists_zero))) + (((dst_negative_code_delta_exists_zero) + (dst_negative_scale_delta_exists_zero)) * S ((dst_negative_code_delta_exists_zero) + (dst_negative_scale_delta_exists_zero)) + ((dst_negative_scale_delta_exists_zero) + (dst_negative_scale_delta_exists_zero)))))) /\ (((((exists ff_h_pvs_delta_exists_zeropositive. ff_h_pvs_delta_exists_zeropositive + S (dst_positive_delta_exists_zero) = S ((S (0)) * dst_positive_scale_delta_exists_zero)) /\ exists ff_q_pvs_delta_exists_zeropositive. dst_positive_code_delta_exists_zero = ff_q_pvs_delta_exists_zeropositive * S ((S (0)) * dst_positive_scale_delta_exists_zero) + (dst_positive_delta_exists_zero))) /\ (((((exists ff_h_pvs_delta_exists_zeronegative. ff_h_pvs_delta_exists_zeronegative + S (dst_negative_delta_exists_zero) = S ((S (0)) * dst_negative_scale_delta_exists_zero)) /\ exists ff_q_pvs_delta_exists_zeronegative. dst_negative_code_delta_exists_zero = ff_q_pvs_delta_exists_zeronegative * S ((S (0)) * dst_negative_scale_delta_exists_zero) + (dst_negative_delta_exists_zero))) /\ (exists ge_balance_positive_delta_exists_zerovalue ge_balance_negative_delta_exists_zerovalue. (((((w) = 2 * (ge_balance_positive_delta_exists_zerovalue) /\ (ge_balance_negative_delta_exists_zerovalue) = 0) \/ exists ge_signed_half_delta_exists_zerovaluedecode. (((w) = 2 * ge_signed_half_delta_exists_zerovaluedecode + 1 /\ (ge_balance_positive_delta_exists_zerovalue) = 0) /\ (ge_balance_negative_delta_exists_zerovalue) = S ge_signed_half_delta_exists_zerovaluedecode))) /\ ((dst_positive_delta_exists_zero) + ge_balance_negative_delta_exists_zerovalue = (dst_negative_delta_exists_zero) + ge_balance_positive_delta_exists_zerovalue)))))))))

Complete tactic proof in conservative notation

All 63 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

63 script commands · 21 reading checkpoints · 4 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 (2)
dirichlet kronecker delta value existsdirichlet kronecker delta table append
01Fix variables and assumptionsL1–1

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

  1. L1
    intro N
02Induction on NL2–3

Split the argument into the base and successor obligations. The induction hypothesis is available only in the successor branch.

  1. L2
    induction N
  2. L3
    intro w
03Establish hzL4–6

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply arithmetic signed table singleton.

  1. L4
    have hz : ∃ F. ArithTable(0,F) ∧ ArithAt(F,0,w)Definitions: ArithTable(0,F)ArithAt(F,0,w)Original native command in the exact edition
  2. L5
    specialize arithmetic_signed_table_singleton (w)
  3. L6
    apply arithmetic_signed_table_singleton
04Separate the logical casesL7–8

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

  1. L7
    cases hz
  2. L8
    cases hz_witness
05Construct an explicit witnessL9–9

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

  1. L9
    exists x
06Separate the logical casesL10–11

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

  1. L10
    split
  2. L11
    split
07Use earlier factsL12–12

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

  1. L12
    exact hz_witness_left
08Fix variables and assumptionsL13–17

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

  1. L13
    intro i
  2. L14
    intro z
  3. L15
    intro hi
  4. L16
    intro hb
  5. L17
    intro he
09Separate the logical casesL18–18

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

  1. L18
    exfalso
10Use earlier factsL19–23

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

  1. L19
    apply hi
  2. L20
    specialize le_zero (i)
  3. L21
    apply le_zero
  4. L22
    exact hb
  5. L23
    exact hz_witness_right
11Fix variables and assumptionsL24–24

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

  1. L24
    intro w
12Establish hpL25–27

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

  1. L25
    have hp : ∃ F. KroneckerDeltaTable(N,F) ∧ ArithAt(F,0,w)Definitions: KroneckerDeltaTable(N,F)ArithAt(F,0,w)Original native command in the exact edition
  2. L26
    specialize IH (w)
  3. L27
    apply IH
13Separate the logical casesL28–29

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

  1. L28
    cases hp
  2. L29
    cases hp_witness
14Establish hvL30–32

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply dirichlet kronecker delta value exists.

  1. L30
    have hv : exists z. ((((S N)=1 -> (z)=2) /\ (~((S N)=1) -> (z)=0)))
  2. L31
    specialize dirichlet_kronecker_delta_value_exists (S N)
  3. L32
    apply dirichlet_kronecker_delta_value_exists
15Separate the logical casesL33–33

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

  1. L33
    cases hv
16Establish heL34–40

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply dirichlet kronecker delta table append.

  1. L34
    have he : ∃ G. KroneckerDeltaTable(S N,G) ∧ (∀ y. ∀ z. ∀ n. Lt(y,S N) → ArithAt(x,y,z) → ArithAt(G,y,n) → z = n)Definitions: KroneckerDeltaTable(S N,G)Lt(y,S N)ArithAt(x,y,z)ArithAt(G,y,n)Original native command in the exact edition
  2. L35
    specialize dirichlet_kronecker_delta_table_append (N)
  3. L36
    specialize dirichlet_kronecker_delta_table_append (x)
  4. L37
    specialize dirichlet_kronecker_delta_table_append (x1)
  5. L38
    apply dirichlet_kronecker_delta_table_append
  6. L39
    exact hp_witness_left
  7. L40
    exact hv_witness
17Separate the logical casesL41–43

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

  1. L41
    cases he
  2. L42
    cases he_witness
  3. L43
    cases he_witness_left
18Construct an explicit witnessL44–44

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

  1. L44
    exists x2
19Separate the logical casesL45–45

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

  1. L45
    split
20Use earlier factsL46–55

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

  1. L46
    exact he_witness_left
  2. L47
    specialize arithmetic_signed_table_equal_entry_transport (S N)
  3. L48
    specialize arithmetic_signed_table_equal_entry_transport (x)
  4. L49
    specialize arithmetic_signed_table_equal_entry_transport (x2)
  5. L50
    specialize arithmetic_signed_table_equal_entry_transport (S N)
  6. L51
    specialize arithmetic_signed_table_equal_entry_transport (0)
  7. L52
    specialize arithmetic_signed_table_equal_entry_transport (w)
  8. L53
    apply arithmetic_signed_table_equal_entry_transport
  9. L54
    exact he_witness_left_left
  10. L55
    exact he_witness_right
21Use earlier factsL56–63

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

  1. L56
    specialize zero_le (S N)
  2. L57
    apply zero_le
  3. L58
    specialize succ_le_succ (0)
  4. L59
    specialize succ_le_succ (N)
  5. L60
    apply succ_le_succ
  6. L61
    specialize zero_le (N)
  7. L62
    apply zero_le
  8. L63
    exact hp_witness_right

Library-wide reading audit

Original defined command ledger · 63 lines
  1. 0001intro N
  2. 0002induction N
  3. 0003intro w
  4. 0004have hz : ∃ F. ArithTable(0,F) ∧ ArithAt(F,0,w)
  5. 0005specialize arithmetic_signed_table_singleton (w)
  6. 0006apply arithmetic_signed_table_singleton
  7. 0007cases hz
  8. 0008cases hz_witness
  9. 0009exists x
  10. 0010split
  11. 0011split
  12. 0012exact hz_witness_left
  13. 0013intro i
  14. 0014intro z
  15. 0015intro hi
  16. 0016intro hb
  17. 0017intro he
  18. 0018exfalso
  19. 0019apply hi
  20. 0020specialize le_zero (i)
  21. 0021apply le_zero
  22. 0022exact hb
  23. 0023exact hz_witness_right
  24. 0024intro w
  25. 0025have hp : ∃ F. KroneckerDeltaTable(N,F) ∧ ArithAt(F,0,w)
  26. 0026specialize IH (w)
  27. 0027apply IH
  28. 0028cases hp
  29. 0029cases hp_witness
  30. 0030have hv : exists z. ((((S N)=1 -> (z)=2) /\ (~((S N)=1) -> (z)=0)))
  31. 0031specialize dirichlet_kronecker_delta_value_exists (S N)
  32. 0032apply dirichlet_kronecker_delta_value_exists
  33. 0033cases hv
  34. 0034have he : ∃ G. KroneckerDeltaTable(S N,G) ∧ (∀ y. ∀ z. ∀ n. Lt(y,S N) → ArithAt(x,y,z) → ArithAt(G,y,n) → z = n)
  35. 0035specialize dirichlet_kronecker_delta_table_append (N)
  36. 0036specialize dirichlet_kronecker_delta_table_append (x)
  37. 0037specialize dirichlet_kronecker_delta_table_append (x1)
  38. 0038apply dirichlet_kronecker_delta_table_append
  39. 0039exact hp_witness_left
  40. 0040exact hv_witness
  41. 0041cases he
  42. 0042cases he_witness
  43. 0043cases he_witness_left
  44. 0044exists x2
  45. 0045split
  46. 0046exact he_witness_left
  47. 0047specialize arithmetic_signed_table_equal_entry_transport (S N)
  48. 0048specialize arithmetic_signed_table_equal_entry_transport (x)
  49. 0049specialize arithmetic_signed_table_equal_entry_transport (x2)
  50. 0050specialize arithmetic_signed_table_equal_entry_transport (S N)
  51. 0051specialize arithmetic_signed_table_equal_entry_transport (0)
  52. 0052specialize arithmetic_signed_table_equal_entry_transport (w)
  53. 0053apply arithmetic_signed_table_equal_entry_transport
  54. 0054exact he_witness_left_left
  55. 0055exact he_witness_right
  56. 0056specialize zero_le (S N)
  57. 0057apply zero_le
  58. 0058specialize succ_le_succ (0)
  59. 0059specialize succ_le_succ (N)
  60. 0060apply succ_le_succ
  61. 0061specialize zero_le (N)
  62. 0062apply zero_le
  63. 0063exact hp_witness_right