DU0007

dirichlet_constant_one_table_exists

For every finite bound, construct a genuine constant-one table retaining any prescribed signed value at zero, including the empty positive 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. ∀ w. ∃ U. ConstantOneTable(N,U) ∧ ArithAt(U,0,w)

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

Definition DAG

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

Actual proof prerequisites

arithmetic_signed_table_singleton · checked external prerequisitele_zero · checked external prerequisitedirichlet_constant_one_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 U. (((exists dst_positive_code_one_exists_tabletable dst_positive_scale_one_exists_tabletable dst_negative_code_one_exists_tabletable dst_negative_scale_one_exists_tabletable. (((U) = (((((dst_positive_code_one_exists_tabletable) + (dst_positive_scale_one_exists_tabletable)) * S ((dst_positive_code_one_exists_tabletable) + (dst_positive_scale_one_exists_tabletable)) + ((dst_positive_scale_one_exists_tabletable) + (dst_positive_scale_one_exists_tabletable))) + (((dst_negative_code_one_exists_tabletable) + (dst_negative_scale_one_exists_tabletable)) * S ((dst_negative_code_one_exists_tabletable) + (dst_negative_scale_one_exists_tabletable)) + ((dst_negative_scale_one_exists_tabletable) + (dst_negative_scale_one_exists_tabletable)))) * S ((((dst_positive_code_one_exists_tabletable) + (dst_positive_scale_one_exists_tabletable)) * S ((dst_positive_code_one_exists_tabletable) + (dst_positive_scale_one_exists_tabletable)) + ((dst_positive_scale_one_exists_tabletable) + (dst_positive_scale_one_exists_tabletable))) + (((dst_negative_code_one_exists_tabletable) + (dst_negative_scale_one_exists_tabletable)) * S ((dst_negative_code_one_exists_tabletable) + (dst_negative_scale_one_exists_tabletable)) + ((dst_negative_scale_one_exists_tabletable) + (dst_negative_scale_one_exists_tabletable)))) + ((((dst_negative_code_one_exists_tabletable) + (dst_negative_scale_one_exists_tabletable)) * S ((dst_negative_code_one_exists_tabletable) + (dst_negative_scale_one_exists_tabletable)) + ((dst_negative_scale_one_exists_tabletable) + (dst_negative_scale_one_exists_tabletable))) + (((dst_negative_code_one_exists_tabletable) + (dst_negative_scale_one_exists_tabletable)) * S ((dst_negative_code_one_exists_tabletable) + (dst_negative_scale_one_exists_tabletable)) + ((dst_negative_scale_one_exists_tabletable) + (dst_negative_scale_one_exists_tabletable)))))) /\ (forall dst_index_one_exists_tabletable. (exists pvs_le_gap_one_exists_tabletabledomain. pvs_le_gap_one_exists_tabletabledomain + (dst_index_one_exists_tabletable) = (N)) -> exists dst_positive_one_exists_tabletable dst_negative_one_exists_tabletable dst_value_one_exists_tabletable. ((((exists ff_h_pvs_one_exists_tabletableentrypositive. ff_h_pvs_one_exists_tabletableentrypositive + S (dst_positive_one_exists_tabletable) = S ((S (dst_index_one_exists_tabletable)) * dst_positive_scale_one_exists_tabletable)) /\ exists ff_q_pvs_one_exists_tabletableentrypositive. dst_positive_code_one_exists_tabletable = ff_q_pvs_one_exists_tabletableentrypositive * S ((S (dst_index_one_exists_tabletable)) * dst_positive_scale_one_exists_tabletable) + (dst_positive_one_exists_tabletable))) /\ (((((exists ff_h_pvs_one_exists_tabletableentrynegative. ff_h_pvs_one_exists_tabletableentrynegative + S (dst_negative_one_exists_tabletable) = S ((S (dst_index_one_exists_tabletable)) * dst_negative_scale_one_exists_tabletable)) /\ exists ff_q_pvs_one_exists_tabletableentrynegative. dst_negative_code_one_exists_tabletable = ff_q_pvs_one_exists_tabletableentrynegative * S ((S (dst_index_one_exists_tabletable)) * dst_negative_scale_one_exists_tabletable) + (dst_negative_one_exists_tabletable))) /\ (exists ge_balance_positive_one_exists_tabletableentryvalue ge_balance_negative_one_exists_tabletableentryvalue. (((((dst_value_one_exists_tabletable) = 2 * (ge_balance_positive_one_exists_tabletableentryvalue) /\ (ge_balance_negative_one_exists_tabletableentryvalue) = 0) \/ exists ge_signed_half_one_exists_tabletableentryvaluedecode. (((dst_value_one_exists_tabletable) = 2 * ge_signed_half_one_exists_tabletableentryvaluedecode + 1 /\ (ge_balance_positive_one_exists_tabletableentryvalue) = 0) /\ (ge_balance_negative_one_exists_tabletableentryvalue) = S ge_signed_half_one_exists_tabletableentryvaluedecode))) /\ ((dst_positive_one_exists_tabletable) + ge_balance_negative_one_exists_tabletableentryvalue = (dst_negative_one_exists_tabletable) + ge_balance_positive_one_exists_tabletableentryvalue))))))))) /\ (forall du_index_one_exists_table du_value_one_exists_table. ~(du_index_one_exists_table=0) -> (exists pvs_le_gap_one_exists_tablebound. pvs_le_gap_one_exists_tablebound + (du_index_one_exists_table) = (N)) -> (exists dst_positive_code_one_exists_tableentry dst_positive_scale_one_exists_tableentry dst_negative_code_one_exists_tableentry dst_negative_scale_one_exists_tableentry dst_positive_one_exists_tableentry dst_negative_one_exists_tableentry. (((U) = (((((dst_positive_code_one_exists_tableentry) + (dst_positive_scale_one_exists_tableentry)) * S ((dst_positive_code_one_exists_tableentry) + (dst_positive_scale_one_exists_tableentry)) + ((dst_positive_scale_one_exists_tableentry) + (dst_positive_scale_one_exists_tableentry))) + (((dst_negative_code_one_exists_tableentry) + (dst_negative_scale_one_exists_tableentry)) * S ((dst_negative_code_one_exists_tableentry) + (dst_negative_scale_one_exists_tableentry)) + ((dst_negative_scale_one_exists_tableentry) + (dst_negative_scale_one_exists_tableentry)))) * S ((((dst_positive_code_one_exists_tableentry) + (dst_positive_scale_one_exists_tableentry)) * S ((dst_positive_code_one_exists_tableentry) + (dst_positive_scale_one_exists_tableentry)) + ((dst_positive_scale_one_exists_tableentry) + (dst_positive_scale_one_exists_tableentry))) + (((dst_negative_code_one_exists_tableentry) + (dst_negative_scale_one_exists_tableentry)) * S ((dst_negative_code_one_exists_tableentry) + (dst_negative_scale_one_exists_tableentry)) + ((dst_negative_scale_one_exists_tableentry) + (dst_negative_scale_one_exists_tableentry)))) + ((((dst_negative_code_one_exists_tableentry) + (dst_negative_scale_one_exists_tableentry)) * S ((dst_negative_code_one_exists_tableentry) + (dst_negative_scale_one_exists_tableentry)) + ((dst_negative_scale_one_exists_tableentry) + (dst_negative_scale_one_exists_tableentry))) + (((dst_negative_code_one_exists_tableentry) + (dst_negative_scale_one_exists_tableentry)) * S ((dst_negative_code_one_exists_tableentry) + (dst_negative_scale_one_exists_tableentry)) + ((dst_negative_scale_one_exists_tableentry) + (dst_negative_scale_one_exists_tableentry)))))) /\ (((((exists ff_h_pvs_one_exists_tableentrypositive. ff_h_pvs_one_exists_tableentrypositive + S (dst_positive_one_exists_tableentry) = S ((S (du_index_one_exists_table)) * dst_positive_scale_one_exists_tableentry)) /\ exists ff_q_pvs_one_exists_tableentrypositive. dst_positive_code_one_exists_tableentry = ff_q_pvs_one_exists_tableentrypositive * S ((S (du_index_one_exists_table)) * dst_positive_scale_one_exists_tableentry) + (dst_positive_one_exists_tableentry))) /\ (((((exists ff_h_pvs_one_exists_tableentrynegative. ff_h_pvs_one_exists_tableentrynegative + S (dst_negative_one_exists_tableentry) = S ((S (du_index_one_exists_table)) * dst_negative_scale_one_exists_tableentry)) /\ exists ff_q_pvs_one_exists_tableentrynegative. dst_negative_code_one_exists_tableentry = ff_q_pvs_one_exists_tableentrynegative * S ((S (du_index_one_exists_table)) * dst_negative_scale_one_exists_tableentry) + (dst_negative_one_exists_tableentry))) /\ (exists ge_balance_positive_one_exists_tableentryvalue ge_balance_negative_one_exists_tableentryvalue. (((((du_value_one_exists_table) = 2 * (ge_balance_positive_one_exists_tableentryvalue) /\ (ge_balance_negative_one_exists_tableentryvalue) = 0) \/ exists ge_signed_half_one_exists_tableentryvaluedecode. (((du_value_one_exists_table) = 2 * ge_signed_half_one_exists_tableentryvaluedecode + 1 /\ (ge_balance_positive_one_exists_tableentryvalue) = 0) /\ (ge_balance_negative_one_exists_tableentryvalue) = S ge_signed_half_one_exists_tableentryvaluedecode))) /\ ((dst_positive_one_exists_tableentry) + ge_balance_negative_one_exists_tableentryvalue = (dst_negative_one_exists_tableentry) + ge_balance_positive_one_exists_tableentryvalue))))))))) -> du_value_one_exists_table=2))) /\ (exists dst_positive_code_one_exists_zero dst_positive_scale_one_exists_zero dst_negative_code_one_exists_zero dst_negative_scale_one_exists_zero dst_positive_one_exists_zero dst_negative_one_exists_zero. (((U) = (((((dst_positive_code_one_exists_zero) + (dst_positive_scale_one_exists_zero)) * S ((dst_positive_code_one_exists_zero) + (dst_positive_scale_one_exists_zero)) + ((dst_positive_scale_one_exists_zero) + (dst_positive_scale_one_exists_zero))) + (((dst_negative_code_one_exists_zero) + (dst_negative_scale_one_exists_zero)) * S ((dst_negative_code_one_exists_zero) + (dst_negative_scale_one_exists_zero)) + ((dst_negative_scale_one_exists_zero) + (dst_negative_scale_one_exists_zero)))) * S ((((dst_positive_code_one_exists_zero) + (dst_positive_scale_one_exists_zero)) * S ((dst_positive_code_one_exists_zero) + (dst_positive_scale_one_exists_zero)) + ((dst_positive_scale_one_exists_zero) + (dst_positive_scale_one_exists_zero))) + (((dst_negative_code_one_exists_zero) + (dst_negative_scale_one_exists_zero)) * S ((dst_negative_code_one_exists_zero) + (dst_negative_scale_one_exists_zero)) + ((dst_negative_scale_one_exists_zero) + (dst_negative_scale_one_exists_zero)))) + ((((dst_negative_code_one_exists_zero) + (dst_negative_scale_one_exists_zero)) * S ((dst_negative_code_one_exists_zero) + (dst_negative_scale_one_exists_zero)) + ((dst_negative_scale_one_exists_zero) + (dst_negative_scale_one_exists_zero))) + (((dst_negative_code_one_exists_zero) + (dst_negative_scale_one_exists_zero)) * S ((dst_negative_code_one_exists_zero) + (dst_negative_scale_one_exists_zero)) + ((dst_negative_scale_one_exists_zero) + (dst_negative_scale_one_exists_zero)))))) /\ (((((exists ff_h_pvs_one_exists_zeropositive. ff_h_pvs_one_exists_zeropositive + S (dst_positive_one_exists_zero) = S ((S (0)) * dst_positive_scale_one_exists_zero)) /\ exists ff_q_pvs_one_exists_zeropositive. dst_positive_code_one_exists_zero = ff_q_pvs_one_exists_zeropositive * S ((S (0)) * dst_positive_scale_one_exists_zero) + (dst_positive_one_exists_zero))) /\ (((((exists ff_h_pvs_one_exists_zeronegative. ff_h_pvs_one_exists_zeronegative + S (dst_negative_one_exists_zero) = S ((S (0)) * dst_negative_scale_one_exists_zero)) /\ exists ff_q_pvs_one_exists_zeronegative. dst_negative_code_one_exists_zero = ff_q_pvs_one_exists_zeronegative * S ((S (0)) * dst_negative_scale_one_exists_zero) + (dst_negative_one_exists_zero))) /\ (exists ge_balance_positive_one_exists_zerovalue ge_balance_negative_one_exists_zerovalue. (((((w) = 2 * (ge_balance_positive_one_exists_zerovalue) /\ (ge_balance_negative_one_exists_zerovalue) = 0) \/ exists ge_signed_half_one_exists_zerovaluedecode. (((w) = 2 * ge_signed_half_one_exists_zerovaluedecode + 1 /\ (ge_balance_positive_one_exists_zerovalue) = 0) /\ (ge_balance_negative_one_exists_zerovalue) = S ge_signed_half_one_exists_zerovaluedecode))) /\ ((dst_positive_one_exists_zero) + ge_balance_negative_one_exists_zerovalue = (dst_negative_one_exists_zero) + ge_balance_positive_one_exists_zerovalue)))))))))

Complete tactic proof in conservative notation

All 57 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

57 script commands · 19 reading checkpoints · 3 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 (1)
dirichlet constant one 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. ConstantOneTable(N,F) ∧ ArithAt(F,0,w)Definitions: ConstantOneTable(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 heL30–34

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

  1. L30
    have he : ∃ G. ConstantOneTable(S N,G) ∧ (∀ y. ∀ z. ∀ n. Lt(y,S N) → ArithAt(x,y,z) → ArithAt(G,y,n) → z = n)Definitions: ConstantOneTable(S N,G)Lt(y,S N)ArithAt(x,y,z)ArithAt(G,y,n)Original native command in the exact edition
  2. L31
    specialize dirichlet_constant_one_table_append (N)
  3. L32
    specialize dirichlet_constant_one_table_append (x)
  4. L33
    apply dirichlet_constant_one_table_append
  5. L34
    exact hp_witness_left
15Separate the logical casesL35–37

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

  1. L35
    cases he
  2. L36
    cases he_witness
  3. L37
    cases he_witness_left
16Construct an explicit witnessL38–38

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

  1. L38
    exists x1
17Separate the logical casesL39–39

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

  1. L39
    split
18Use earlier factsL40–49

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

  1. L40
    exact he_witness_left
  2. L41
    specialize arithmetic_signed_table_equal_entry_transport (S N)
  3. L42
    specialize arithmetic_signed_table_equal_entry_transport (x)
  4. L43
    specialize arithmetic_signed_table_equal_entry_transport (x1)
  5. L44
    specialize arithmetic_signed_table_equal_entry_transport (S N)
  6. L45
    specialize arithmetic_signed_table_equal_entry_transport (0)
  7. L46
    specialize arithmetic_signed_table_equal_entry_transport (w)
  8. L47
    apply arithmetic_signed_table_equal_entry_transport
  9. L48
    exact he_witness_left_left
  10. L49
    exact he_witness_right
19Use earlier factsL50–57

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

  1. L50
    specialize zero_le (S N)
  2. L51
    apply zero_le
  3. L52
    specialize succ_le_succ (0)
  4. L53
    specialize succ_le_succ (N)
  5. L54
    apply succ_le_succ
  6. L55
    specialize zero_le (N)
  7. L56
    apply zero_le
  8. L57
    exact hp_witness_right

Library-wide reading audit

Original defined command ledger · 57 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. ConstantOneTable(N,F) ∧ ArithAt(F,0,w)
  26. 0026specialize IH (w)
  27. 0027apply IH
  28. 0028cases hp
  29. 0029cases hp_witness
  30. 0030have he : ∃ G. ConstantOneTable(S N,G) ∧ (∀ y. ∀ z. ∀ n. Lt(y,S N) → ArithAt(x,y,z) → ArithAt(G,y,n) → z = n)
  31. 0031specialize dirichlet_constant_one_table_append (N)
  32. 0032specialize dirichlet_constant_one_table_append (x)
  33. 0033apply dirichlet_constant_one_table_append
  34. 0034exact hp_witness_left
  35. 0035cases he
  36. 0036cases he_witness
  37. 0037cases he_witness_left
  38. 0038exists x1
  39. 0039split
  40. 0040exact he_witness_left
  41. 0041specialize arithmetic_signed_table_equal_entry_transport (S N)
  42. 0042specialize arithmetic_signed_table_equal_entry_transport (x)
  43. 0043specialize arithmetic_signed_table_equal_entry_transport (x1)
  44. 0044specialize arithmetic_signed_table_equal_entry_transport (S N)
  45. 0045specialize arithmetic_signed_table_equal_entry_transport (0)
  46. 0046specialize arithmetic_signed_table_equal_entry_transport (w)
  47. 0047apply arithmetic_signed_table_equal_entry_transport
  48. 0048exact he_witness_left_left
  49. 0049exact he_witness_right
  50. 0050specialize zero_le (S N)
  51. 0051apply zero_le
  52. 0052specialize succ_le_succ (0)
  53. 0053specialize succ_le_succ (N)
  54. 0054apply succ_le_succ
  55. 0055specialize zero_le (N)
  56. 0056apply zero_le
  57. 0057exact hp_witness_right