MX004E

dirichlet_convolution_entry_nonzero_support

Alpha v34 independently verified · alpha_closed; checked-use authorized; not Stable

A genuinely nonzero convolution summand supplies a positive divisor and its actual quotient; omitted indices cannot enter the support.

Current library: Alpha v34, 4,223 checked-use theorems; Stable remains 432. Historical first admissions, original proof editions, and non-admitted aliases are preserved.

Exact expanded first-order arithmetic statement

forall F G n d z. ((((~((d)=0)) /\ (exists dc_quotient_support_entry dc_left_support_entry dc_right_support_entry. (((n)=(d)*dc_quotient_support_entry) /\ (((exists dst_positive_code_support_entryleft dst_positive_scale_support_entryleft dst_negative_code_support_entryleft dst_negative_scale_support_entryleft dst_positive_support_entryleft dst_negative_support_entryleft. (((F) = (((((dst_positive_code_support_entryleft) + (dst_positive_scale_support_entryleft)) * S ((dst_positive_code_support_entryleft) + (dst_positive_scale_support_entryleft)) + ((dst_positive_scale_support_entryleft) + (dst_positive_scale_support_entryleft))) + (((dst_negative_code_support_entryleft) + (dst_negative_scale_support_entryleft)) * S ((dst_negative_code_support_entryleft) + (dst_negative_scale_support_entryleft)) + ((dst_negative_scale_support_entryleft) + (dst_negative_scale_support_entryleft)))) * S ((((dst_positive_code_support_entryleft) + (dst_positive_scale_support_entryleft)) * S ((dst_positive_code_support_entryleft) + (dst_positive_scale_support_entryleft)) + ((dst_positive_scale_support_entryleft) + (dst_positive_scale_support_entryleft))) + (((dst_negative_code_support_entryleft) + (dst_negative_scale_support_entryleft)) * S ((dst_negative_code_support_entryleft) + (dst_negative_scale_support_entryleft)) + ((dst_negative_scale_support_entryleft) + (dst_negative_scale_support_entryleft)))) + ((((dst_negative_code_support_entryleft) + (dst_negative_scale_support_entryleft)) * S ((dst_negative_code_support_entryleft) + (dst_negative_scale_support_entryleft)) + ((dst_negative_scale_support_entryleft) + (dst_negative_scale_support_entryleft))) + (((dst_negative_code_support_entryleft) + (dst_negative_scale_support_entryleft)) * S ((dst_negative_code_support_entryleft) + (dst_negative_scale_support_entryleft)) + ((dst_negative_scale_support_entryleft) + (dst_negative_scale_support_entryleft)))))) /\ (((((exists ff_h_pvs_support_entryleftpositive. ff_h_pvs_support_entryleftpositive + S (dst_positive_support_entryleft) = S ((S (d)) * dst_positive_scale_support_entryleft)) /\ exists ff_q_pvs_support_entryleftpositive. dst_positive_code_support_entryleft = ff_q_pvs_support_entryleftpositive * S ((S (d)) * dst_positive_scale_support_entryleft) + (dst_positive_support_entryleft))) /\ (((((exists ff_h_pvs_support_entryleftnegative. ff_h_pvs_support_entryleftnegative + S (dst_negative_support_entryleft) = S ((S (d)) * dst_negative_scale_support_entryleft)) /\ exists ff_q_pvs_support_entryleftnegative. dst_negative_code_support_entryleft = ff_q_pvs_support_entryleftnegative * S ((S (d)) * dst_negative_scale_support_entryleft) + (dst_negative_support_entryleft))) /\ (exists ge_balance_positive_support_entryleftvalue ge_balance_negative_support_entryleftvalue. (((((dc_left_support_entry) = 2 * (ge_balance_positive_support_entryleftvalue) /\ (ge_balance_negative_support_entryleftvalue) = 0) \/ exists ge_signed_half_support_entryleftvaluedecode. (((dc_left_support_entry) = 2 * ge_signed_half_support_entryleftvaluedecode + 1 /\ (ge_balance_positive_support_entryleftvalue) = 0) /\ (ge_balance_negative_support_entryleftvalue) = S ge_signed_half_support_entryleftvaluedecode))) /\ ((dst_positive_support_entryleft) + ge_balance_negative_support_entryleftvalue = (dst_negative_support_entryleft) + ge_balance_positive_support_entryleftvalue))))))))) /\ (((exists dst_positive_code_support_entryright dst_positive_scale_support_entryright dst_negative_code_support_entryright dst_negative_scale_support_entryright dst_positive_support_entryright dst_negative_support_entryright. (((G) = (((((dst_positive_code_support_entryright) + (dst_positive_scale_support_entryright)) * S ((dst_positive_code_support_entryright) + (dst_positive_scale_support_entryright)) + ((dst_positive_scale_support_entryright) + (dst_positive_scale_support_entryright))) + (((dst_negative_code_support_entryright) + (dst_negative_scale_support_entryright)) * S ((dst_negative_code_support_entryright) + (dst_negative_scale_support_entryright)) + ((dst_negative_scale_support_entryright) + (dst_negative_scale_support_entryright)))) * S ((((dst_positive_code_support_entryright) + (dst_positive_scale_support_entryright)) * S ((dst_positive_code_support_entryright) + (dst_positive_scale_support_entryright)) + ((dst_positive_scale_support_entryright) + (dst_positive_scale_support_entryright))) + (((dst_negative_code_support_entryright) + (dst_negative_scale_support_entryright)) * S ((dst_negative_code_support_entryright) + (dst_negative_scale_support_entryright)) + ((dst_negative_scale_support_entryright) + (dst_negative_scale_support_entryright)))) + ((((dst_negative_code_support_entryright) + (dst_negative_scale_support_entryright)) * S ((dst_negative_code_support_entryright) + (dst_negative_scale_support_entryright)) + ((dst_negative_scale_support_entryright) + (dst_negative_scale_support_entryright))) + (((dst_negative_code_support_entryright) + (dst_negative_scale_support_entryright)) * S ((dst_negative_code_support_entryright) + (dst_negative_scale_support_entryright)) + ((dst_negative_scale_support_entryright) + (dst_negative_scale_support_entryright)))))) /\ (((((exists ff_h_pvs_support_entryrightpositive. ff_h_pvs_support_entryrightpositive + S (dst_positive_support_entryright) = S ((S (dc_quotient_support_entry)) * dst_positive_scale_support_entryright)) /\ exists ff_q_pvs_support_entryrightpositive. dst_positive_code_support_entryright = ff_q_pvs_support_entryrightpositive * S ((S (dc_quotient_support_entry)) * dst_positive_scale_support_entryright) + (dst_positive_support_entryright))) /\ (((((exists ff_h_pvs_support_entryrightnegative. ff_h_pvs_support_entryrightnegative + S (dst_negative_support_entryright) = S ((S (dc_quotient_support_entry)) * dst_negative_scale_support_entryright)) /\ exists ff_q_pvs_support_entryrightnegative. dst_negative_code_support_entryright = ff_q_pvs_support_entryrightnegative * S ((S (dc_quotient_support_entry)) * dst_negative_scale_support_entryright) + (dst_negative_support_entryright))) /\ (exists ge_balance_positive_support_entryrightvalue ge_balance_negative_support_entryrightvalue. (((((dc_right_support_entry) = 2 * (ge_balance_positive_support_entryrightvalue) /\ (ge_balance_negative_support_entryrightvalue) = 0) \/ exists ge_signed_half_support_entryrightvaluedecode. (((dc_right_support_entry) = 2 * ge_signed_half_support_entryrightvaluedecode + 1 /\ (ge_balance_positive_support_entryrightvalue) = 0) /\ (ge_balance_negative_support_entryrightvalue) = S ge_signed_half_support_entryrightvaluedecode))) /\ ((dst_positive_support_entryright) + ge_balance_negative_support_entryrightvalue = (dst_negative_support_entryright) + ge_balance_positive_support_entryrightvalue))))))))) /\ (exists sto_ap_support_entryproduct sto_an_support_entryproduct sto_bp_support_entryproduct sto_bn_support_entryproduct sto_cp_support_entryproduct sto_cn_support_entryproduct. (((((dc_left_support_entry) = 2 * (sto_ap_support_entryproduct) /\ (sto_an_support_entryproduct) = 0) \/ exists ge_signed_half_support_entryproductleft. (((dc_left_support_entry) = 2 * ge_signed_half_support_entryproductleft + 1 /\ (sto_ap_support_entryproduct) = 0) /\ (sto_an_support_entryproduct) = S ge_signed_half_support_entryproductleft))) /\ ((((((dc_right_support_entry) = 2 * (sto_bp_support_entryproduct) /\ (sto_bn_support_entryproduct) = 0) \/ exists ge_signed_half_support_entryproductright. (((dc_right_support_entry) = 2 * ge_signed_half_support_entryproductright + 1 /\ (sto_bp_support_entryproduct) = 0) /\ (sto_bn_support_entryproduct) = S ge_signed_half_support_entryproductright))) /\ ((((((z) = 2 * (sto_cp_support_entryproduct) /\ (sto_cn_support_entryproduct) = 0) \/ exists ge_signed_half_support_entryproductoutput. (((z) = 2 * ge_signed_half_support_entryproductoutput + 1 /\ (sto_cp_support_entryproduct) = 0) /\ (sto_cn_support_entryproduct) = S ge_signed_half_support_entryproductoutput))) /\ ((sto_ap_support_entryproduct * sto_bp_support_entryproduct + sto_an_support_entryproduct * sto_bn_support_entryproduct) + sto_cn_support_entryproduct = (sto_ap_support_entryproduct * sto_bn_support_entryproduct + sto_an_support_entryproduct * sto_bp_support_entryproduct) + sto_cp_support_entryproduct))))))))))))))) \/ ((((d)=0 \/ ~(exists pvs_factor_support_entrynondivisor. (n) = (d) * pvs_factor_support_entrynondivisor)) /\ ((z)=0)))) -> ~(z=0) -> ((~(d=0)) /\ (exists pvs_factor_support_divisor. (n) = (d) * pvs_factor_support_divisor))

Constructive proof overview

Generated structural guide

A genuinely nonzero convolution summand supplies a positive divisor and its actual quotient; omitted indices cannot enter the support.

The unchanged tactic script uses 0 declared prerequisites and contains 23 exact native proof lines.

Alpha v34 checked-use · first admitted v32 · independently kernel and Lean verified; not Stable

Proof neighborhood

Direct dependencies

none

Direct dependents

MX0051 dirichlet_coprime_grid_nonzero_coordinates MX0054 dirichlet_coprime_grid_support_covering

Formal native tactic body

Dependencies are introduced as named hypotheses before line 1. Local theorem links identify exact declared prerequisites. This exact body belongs to a complete independently kernel-checked constructive proof bundle and has Alpha checked-use authority; it does not imply Stable membership.

Read the argument

Proof checkpoints

23 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.

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 he
  7. L7
    intro hz
02Separate the logical casesL8–16

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
    split
03Use earlier factsL17–17

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

  1. L17
    exact he_left_left
04Construct an explicit witnessL18–18

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

  1. L18
    exists x
05Use earlier factsL19–19

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

  1. L19
    exact he_left_right_witness_witness_witness_left
06Separate the logical casesL20–21

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

  1. L20
    cases he_right
  2. L21
    exfalso
07Use earlier factsL22–23

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

  1. L22
    apply hz
  2. L23
    exact he_right_right

Library-wide reading audit

Original exact command ledger · 23 lines
  1. 0001intro F
  2. 0002intro G
  3. 0003intro n
  4. 0004intro d
  5. 0005intro z
  6. 0006intro he
  7. 0007intro hz
  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. 0016split
  17. 0017exact he_left_left
  18. 0018exists x
  19. 0019exact he_left_right_witness_witness_witness_left
  20. 0020cases he_right
  21. 0021exfalso
  22. 0022apply hz
  23. 0023exact he_right_right