DC0004

dirichlet_convolution_entry_omitted_value

Every actual omitted summand is exactly zero, and a supplied product witness cannot override that branch.

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.

Each retained summand has a witnessed n=d*q and actual signed multiplication. Zero and nondivisors contribute zero. Input and output values at zero are unrestricted; uniqueness is for positive represented values. The separate inverse family proves the unit-at-one criterion. Full G009 multiplicative-function closure is now admitted in the separate Alpha-v32 multiplicative-convolution family.

Exact theorem in conservative defined notation

∀ F. ∀ G. ∀ n. ∀ d. ∀ z. d = 0 ∨ ¬Dvd(d,n) → DirichletEntry(F,G,n,d,z) → z = 0

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

Definition DAG

DirichletEntry(F,G,n,d,z) · 1Dvd(d,n) · 1

Actual proof prerequisites

none
Original expanded first-order statement 
forall F G n d z. (d=0 \/ ~(exists pvs_factor_omitted_guard. (n) = (d) * pvs_factor_omitted_guard)) -> ((((~((d)=0)) /\ (exists dc_quotient_omitted_entry dc_left_omitted_entry dc_right_omitted_entry. (((n)=(d)*dc_quotient_omitted_entry) /\ (((exists dst_positive_code_omitted_entryleft dst_positive_scale_omitted_entryleft dst_negative_code_omitted_entryleft dst_negative_scale_omitted_entryleft dst_positive_omitted_entryleft dst_negative_omitted_entryleft. (((F) = (((((dst_positive_code_omitted_entryleft) + (dst_positive_scale_omitted_entryleft)) * S ((dst_positive_code_omitted_entryleft) + (dst_positive_scale_omitted_entryleft)) + ((dst_positive_scale_omitted_entryleft) + (dst_positive_scale_omitted_entryleft))) + (((dst_negative_code_omitted_entryleft) + (dst_negative_scale_omitted_entryleft)) * S ((dst_negative_code_omitted_entryleft) + (dst_negative_scale_omitted_entryleft)) + ((dst_negative_scale_omitted_entryleft) + (dst_negative_scale_omitted_entryleft)))) * S ((((dst_positive_code_omitted_entryleft) + (dst_positive_scale_omitted_entryleft)) * S ((dst_positive_code_omitted_entryleft) + (dst_positive_scale_omitted_entryleft)) + ((dst_positive_scale_omitted_entryleft) + (dst_positive_scale_omitted_entryleft))) + (((dst_negative_code_omitted_entryleft) + (dst_negative_scale_omitted_entryleft)) * S ((dst_negative_code_omitted_entryleft) + (dst_negative_scale_omitted_entryleft)) + ((dst_negative_scale_omitted_entryleft) + (dst_negative_scale_omitted_entryleft)))) + ((((dst_negative_code_omitted_entryleft) + (dst_negative_scale_omitted_entryleft)) * S ((dst_negative_code_omitted_entryleft) + (dst_negative_scale_omitted_entryleft)) + ((dst_negative_scale_omitted_entryleft) + (dst_negative_scale_omitted_entryleft))) + (((dst_negative_code_omitted_entryleft) + (dst_negative_scale_omitted_entryleft)) * S ((dst_negative_code_omitted_entryleft) + (dst_negative_scale_omitted_entryleft)) + ((dst_negative_scale_omitted_entryleft) + (dst_negative_scale_omitted_entryleft)))))) /\ (((((exists ff_h_pvs_omitted_entryleftpositive. ff_h_pvs_omitted_entryleftpositive + S (dst_positive_omitted_entryleft) = S ((S (d)) * dst_positive_scale_omitted_entryleft)) /\ exists ff_q_pvs_omitted_entryleftpositive. dst_positive_code_omitted_entryleft = ff_q_pvs_omitted_entryleftpositive * S ((S (d)) * dst_positive_scale_omitted_entryleft) + (dst_positive_omitted_entryleft))) /\ (((((exists ff_h_pvs_omitted_entryleftnegative. ff_h_pvs_omitted_entryleftnegative + S (dst_negative_omitted_entryleft) = S ((S (d)) * dst_negative_scale_omitted_entryleft)) /\ exists ff_q_pvs_omitted_entryleftnegative. dst_negative_code_omitted_entryleft = ff_q_pvs_omitted_entryleftnegative * S ((S (d)) * dst_negative_scale_omitted_entryleft) + (dst_negative_omitted_entryleft))) /\ (exists ge_balance_positive_omitted_entryleftvalue ge_balance_negative_omitted_entryleftvalue. (((((dc_left_omitted_entry) = 2 * (ge_balance_positive_omitted_entryleftvalue) /\ (ge_balance_negative_omitted_entryleftvalue) = 0) \/ exists ge_signed_half_omitted_entryleftvaluedecode. (((dc_left_omitted_entry) = 2 * ge_signed_half_omitted_entryleftvaluedecode + 1 /\ (ge_balance_positive_omitted_entryleftvalue) = 0) /\ (ge_balance_negative_omitted_entryleftvalue) = S ge_signed_half_omitted_entryleftvaluedecode))) /\ ((dst_positive_omitted_entryleft) + ge_balance_negative_omitted_entryleftvalue = (dst_negative_omitted_entryleft) + ge_balance_positive_omitted_entryleftvalue))))))))) /\ (((exists dst_positive_code_omitted_entryright dst_positive_scale_omitted_entryright dst_negative_code_omitted_entryright dst_negative_scale_omitted_entryright dst_positive_omitted_entryright dst_negative_omitted_entryright. (((G) = (((((dst_positive_code_omitted_entryright) + (dst_positive_scale_omitted_entryright)) * S ((dst_positive_code_omitted_entryright) + (dst_positive_scale_omitted_entryright)) + ((dst_positive_scale_omitted_entryright) + (dst_positive_scale_omitted_entryright))) + (((dst_negative_code_omitted_entryright) + (dst_negative_scale_omitted_entryright)) * S ((dst_negative_code_omitted_entryright) + (dst_negative_scale_omitted_entryright)) + ((dst_negative_scale_omitted_entryright) + (dst_negative_scale_omitted_entryright)))) * S ((((dst_positive_code_omitted_entryright) + (dst_positive_scale_omitted_entryright)) * S ((dst_positive_code_omitted_entryright) + (dst_positive_scale_omitted_entryright)) + ((dst_positive_scale_omitted_entryright) + (dst_positive_scale_omitted_entryright))) + (((dst_negative_code_omitted_entryright) + (dst_negative_scale_omitted_entryright)) * S ((dst_negative_code_omitted_entryright) + (dst_negative_scale_omitted_entryright)) + ((dst_negative_scale_omitted_entryright) + (dst_negative_scale_omitted_entryright)))) + ((((dst_negative_code_omitted_entryright) + (dst_negative_scale_omitted_entryright)) * S ((dst_negative_code_omitted_entryright) + (dst_negative_scale_omitted_entryright)) + ((dst_negative_scale_omitted_entryright) + (dst_negative_scale_omitted_entryright))) + (((dst_negative_code_omitted_entryright) + (dst_negative_scale_omitted_entryright)) * S ((dst_negative_code_omitted_entryright) + (dst_negative_scale_omitted_entryright)) + ((dst_negative_scale_omitted_entryright) + (dst_negative_scale_omitted_entryright)))))) /\ (((((exists ff_h_pvs_omitted_entryrightpositive. ff_h_pvs_omitted_entryrightpositive + S (dst_positive_omitted_entryright) = S ((S (dc_quotient_omitted_entry)) * dst_positive_scale_omitted_entryright)) /\ exists ff_q_pvs_omitted_entryrightpositive. dst_positive_code_omitted_entryright = ff_q_pvs_omitted_entryrightpositive * S ((S (dc_quotient_omitted_entry)) * dst_positive_scale_omitted_entryright) + (dst_positive_omitted_entryright))) /\ (((((exists ff_h_pvs_omitted_entryrightnegative. ff_h_pvs_omitted_entryrightnegative + S (dst_negative_omitted_entryright) = S ((S (dc_quotient_omitted_entry)) * dst_negative_scale_omitted_entryright)) /\ exists ff_q_pvs_omitted_entryrightnegative. dst_negative_code_omitted_entryright = ff_q_pvs_omitted_entryrightnegative * S ((S (dc_quotient_omitted_entry)) * dst_negative_scale_omitted_entryright) + (dst_negative_omitted_entryright))) /\ (exists ge_balance_positive_omitted_entryrightvalue ge_balance_negative_omitted_entryrightvalue. (((((dc_right_omitted_entry) = 2 * (ge_balance_positive_omitted_entryrightvalue) /\ (ge_balance_negative_omitted_entryrightvalue) = 0) \/ exists ge_signed_half_omitted_entryrightvaluedecode. (((dc_right_omitted_entry) = 2 * ge_signed_half_omitted_entryrightvaluedecode + 1 /\ (ge_balance_positive_omitted_entryrightvalue) = 0) /\ (ge_balance_negative_omitted_entryrightvalue) = S ge_signed_half_omitted_entryrightvaluedecode))) /\ ((dst_positive_omitted_entryright) + ge_balance_negative_omitted_entryrightvalue = (dst_negative_omitted_entryright) + ge_balance_positive_omitted_entryrightvalue))))))))) /\ (exists sto_ap_omitted_entryproduct sto_an_omitted_entryproduct sto_bp_omitted_entryproduct sto_bn_omitted_entryproduct sto_cp_omitted_entryproduct sto_cn_omitted_entryproduct. (((((dc_left_omitted_entry) = 2 * (sto_ap_omitted_entryproduct) /\ (sto_an_omitted_entryproduct) = 0) \/ exists ge_signed_half_omitted_entryproductleft. (((dc_left_omitted_entry) = 2 * ge_signed_half_omitted_entryproductleft + 1 /\ (sto_ap_omitted_entryproduct) = 0) /\ (sto_an_omitted_entryproduct) = S ge_signed_half_omitted_entryproductleft))) /\ ((((((dc_right_omitted_entry) = 2 * (sto_bp_omitted_entryproduct) /\ (sto_bn_omitted_entryproduct) = 0) \/ exists ge_signed_half_omitted_entryproductright. (((dc_right_omitted_entry) = 2 * ge_signed_half_omitted_entryproductright + 1 /\ (sto_bp_omitted_entryproduct) = 0) /\ (sto_bn_omitted_entryproduct) = S ge_signed_half_omitted_entryproductright))) /\ ((((((z) = 2 * (sto_cp_omitted_entryproduct) /\ (sto_cn_omitted_entryproduct) = 0) \/ exists ge_signed_half_omitted_entryproductoutput. (((z) = 2 * ge_signed_half_omitted_entryproductoutput + 1 /\ (sto_cp_omitted_entryproduct) = 0) /\ (sto_cn_omitted_entryproduct) = S ge_signed_half_omitted_entryproductoutput))) /\ ((sto_ap_omitted_entryproduct * sto_bp_omitted_entryproduct + sto_an_omitted_entryproduct * sto_bn_omitted_entryproduct) + sto_cn_omitted_entryproduct = (sto_ap_omitted_entryproduct * sto_bn_omitted_entryproduct + sto_an_omitted_entryproduct * sto_bp_omitted_entryproduct) + sto_cp_omitted_entryproduct))))))))))))))) \/ ((((d)=0 \/ ~(exists pvs_factor_omitted_entrynondivisor. (n) = (d) * pvs_factor_omitted_entrynondivisor)) /\ ((z)=0)))) -> z=0

Complete tactic proof in conservative notation

All 24 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

24 script commands · 7 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 assumptionsL1–7

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

  1. L1
    intro F
  2. L2
    intro G
  3. L3
    intro n
  4. L4
    intro d
  5. L5
    intro z
  6. L6
    intro hc
  7. L7
    intro he
02Separate the logical casesL8–17

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

  1. L8
    cases he
  2. L9
    cases he_left
  3. L10
    cases he_left_right
  4. L11
    cases he_left_right_witness
  5. L12
    cases he_left_right_witness_witness
  6. L13
    cases he_left_right_witness_witness_witness
  7. L14
    cases he_left_right_witness_witness_witness_right
  8. L15
    cases he_left_right_witness_witness_witness_right_right
  9. L16
    exfalso
  10. L17
    cases hc
03Use earlier factsL18–20

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

  1. L18
    apply he_left_left
  2. L19
    exact hc_left
  3. L20
    apply hc_right
04Construct an explicit witnessL21–21

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

  1. L21
    exists x
05Use earlier factsL22–22

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

  1. L22
    exact he_left_right_witness_witness_witness_left
06Separate the logical casesL23–23

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

  1. L23
    cases he_right
07Use earlier factsL24–24

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

  1. L24
    exact he_right_right

Library-wide reading audit

Original defined command ledger · 24 lines
  1. 0001intro F
  2. 0002intro G
  3. 0003intro n
  4. 0004intro d
  5. 0005intro z
  6. 0006intro hc
  7. 0007intro he
  8. 0008cases he
  9. 0009cases he_left
  10. 0010cases he_left_right
  11. 0011cases he_left_right_witness
  12. 0012cases he_left_right_witness_witness
  13. 0013cases he_left_right_witness_witness_witness
  14. 0014cases he_left_right_witness_witness_witness_right
  15. 0015cases he_left_right_witness_witness_witness_right_right
  16. 0016exfalso
  17. 0017cases hc
  18. 0018apply he_left_left
  19. 0019exact hc_left
  20. 0020apply hc_right
  21. 0021exists x
  22. 0022exact he_left_right_witness_witness_witness_left
  23. 0023cases he_right
  24. 0024exact he_right_right