DC0001

dirichlet_convolution_entry_zero

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

The zeroth summand is canonical zero without looking at either input value at zero.

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. ((((~((0)=0)) /\ (exists dc_quotient_zero_result dc_left_zero_result dc_right_zero_result. (((n)=(0)*dc_quotient_zero_result) /\ (((exists dst_positive_code_zero_resultleft dst_positive_scale_zero_resultleft dst_negative_code_zero_resultleft dst_negative_scale_zero_resultleft dst_positive_zero_resultleft dst_negative_zero_resultleft. (((F) = (((((dst_positive_code_zero_resultleft) + (dst_positive_scale_zero_resultleft)) * S ((dst_positive_code_zero_resultleft) + (dst_positive_scale_zero_resultleft)) + ((dst_positive_scale_zero_resultleft) + (dst_positive_scale_zero_resultleft))) + (((dst_negative_code_zero_resultleft) + (dst_negative_scale_zero_resultleft)) * S ((dst_negative_code_zero_resultleft) + (dst_negative_scale_zero_resultleft)) + ((dst_negative_scale_zero_resultleft) + (dst_negative_scale_zero_resultleft)))) * S ((((dst_positive_code_zero_resultleft) + (dst_positive_scale_zero_resultleft)) * S ((dst_positive_code_zero_resultleft) + (dst_positive_scale_zero_resultleft)) + ((dst_positive_scale_zero_resultleft) + (dst_positive_scale_zero_resultleft))) + (((dst_negative_code_zero_resultleft) + (dst_negative_scale_zero_resultleft)) * S ((dst_negative_code_zero_resultleft) + (dst_negative_scale_zero_resultleft)) + ((dst_negative_scale_zero_resultleft) + (dst_negative_scale_zero_resultleft)))) + ((((dst_negative_code_zero_resultleft) + (dst_negative_scale_zero_resultleft)) * S ((dst_negative_code_zero_resultleft) + (dst_negative_scale_zero_resultleft)) + ((dst_negative_scale_zero_resultleft) + (dst_negative_scale_zero_resultleft))) + (((dst_negative_code_zero_resultleft) + (dst_negative_scale_zero_resultleft)) * S ((dst_negative_code_zero_resultleft) + (dst_negative_scale_zero_resultleft)) + ((dst_negative_scale_zero_resultleft) + (dst_negative_scale_zero_resultleft)))))) /\ (((((exists ff_h_pvs_zero_resultleftpositive. ff_h_pvs_zero_resultleftpositive + S (dst_positive_zero_resultleft) = S ((S (0)) * dst_positive_scale_zero_resultleft)) /\ exists ff_q_pvs_zero_resultleftpositive. dst_positive_code_zero_resultleft = ff_q_pvs_zero_resultleftpositive * S ((S (0)) * dst_positive_scale_zero_resultleft) + (dst_positive_zero_resultleft))) /\ (((((exists ff_h_pvs_zero_resultleftnegative. ff_h_pvs_zero_resultleftnegative + S (dst_negative_zero_resultleft) = S ((S (0)) * dst_negative_scale_zero_resultleft)) /\ exists ff_q_pvs_zero_resultleftnegative. dst_negative_code_zero_resultleft = ff_q_pvs_zero_resultleftnegative * S ((S (0)) * dst_negative_scale_zero_resultleft) + (dst_negative_zero_resultleft))) /\ (exists ge_balance_positive_zero_resultleftvalue ge_balance_negative_zero_resultleftvalue. (((((dc_left_zero_result) = 2 * (ge_balance_positive_zero_resultleftvalue) /\ (ge_balance_negative_zero_resultleftvalue) = 0) \/ exists ge_signed_half_zero_resultleftvaluedecode. (((dc_left_zero_result) = 2 * ge_signed_half_zero_resultleftvaluedecode + 1 /\ (ge_balance_positive_zero_resultleftvalue) = 0) /\ (ge_balance_negative_zero_resultleftvalue) = S ge_signed_half_zero_resultleftvaluedecode))) /\ ((dst_positive_zero_resultleft) + ge_balance_negative_zero_resultleftvalue = (dst_negative_zero_resultleft) + ge_balance_positive_zero_resultleftvalue))))))))) /\ (((exists dst_positive_code_zero_resultright dst_positive_scale_zero_resultright dst_negative_code_zero_resultright dst_negative_scale_zero_resultright dst_positive_zero_resultright dst_negative_zero_resultright. (((G) = (((((dst_positive_code_zero_resultright) + (dst_positive_scale_zero_resultright)) * S ((dst_positive_code_zero_resultright) + (dst_positive_scale_zero_resultright)) + ((dst_positive_scale_zero_resultright) + (dst_positive_scale_zero_resultright))) + (((dst_negative_code_zero_resultright) + (dst_negative_scale_zero_resultright)) * S ((dst_negative_code_zero_resultright) + (dst_negative_scale_zero_resultright)) + ((dst_negative_scale_zero_resultright) + (dst_negative_scale_zero_resultright)))) * S ((((dst_positive_code_zero_resultright) + (dst_positive_scale_zero_resultright)) * S ((dst_positive_code_zero_resultright) + (dst_positive_scale_zero_resultright)) + ((dst_positive_scale_zero_resultright) + (dst_positive_scale_zero_resultright))) + (((dst_negative_code_zero_resultright) + (dst_negative_scale_zero_resultright)) * S ((dst_negative_code_zero_resultright) + (dst_negative_scale_zero_resultright)) + ((dst_negative_scale_zero_resultright) + (dst_negative_scale_zero_resultright)))) + ((((dst_negative_code_zero_resultright) + (dst_negative_scale_zero_resultright)) * S ((dst_negative_code_zero_resultright) + (dst_negative_scale_zero_resultright)) + ((dst_negative_scale_zero_resultright) + (dst_negative_scale_zero_resultright))) + (((dst_negative_code_zero_resultright) + (dst_negative_scale_zero_resultright)) * S ((dst_negative_code_zero_resultright) + (dst_negative_scale_zero_resultright)) + ((dst_negative_scale_zero_resultright) + (dst_negative_scale_zero_resultright)))))) /\ (((((exists ff_h_pvs_zero_resultrightpositive. ff_h_pvs_zero_resultrightpositive + S (dst_positive_zero_resultright) = S ((S (dc_quotient_zero_result)) * dst_positive_scale_zero_resultright)) /\ exists ff_q_pvs_zero_resultrightpositive. dst_positive_code_zero_resultright = ff_q_pvs_zero_resultrightpositive * S ((S (dc_quotient_zero_result)) * dst_positive_scale_zero_resultright) + (dst_positive_zero_resultright))) /\ (((((exists ff_h_pvs_zero_resultrightnegative. ff_h_pvs_zero_resultrightnegative + S (dst_negative_zero_resultright) = S ((S (dc_quotient_zero_result)) * dst_negative_scale_zero_resultright)) /\ exists ff_q_pvs_zero_resultrightnegative. dst_negative_code_zero_resultright = ff_q_pvs_zero_resultrightnegative * S ((S (dc_quotient_zero_result)) * dst_negative_scale_zero_resultright) + (dst_negative_zero_resultright))) /\ (exists ge_balance_positive_zero_resultrightvalue ge_balance_negative_zero_resultrightvalue. (((((dc_right_zero_result) = 2 * (ge_balance_positive_zero_resultrightvalue) /\ (ge_balance_negative_zero_resultrightvalue) = 0) \/ exists ge_signed_half_zero_resultrightvaluedecode. (((dc_right_zero_result) = 2 * ge_signed_half_zero_resultrightvaluedecode + 1 /\ (ge_balance_positive_zero_resultrightvalue) = 0) /\ (ge_balance_negative_zero_resultrightvalue) = S ge_signed_half_zero_resultrightvaluedecode))) /\ ((dst_positive_zero_resultright) + ge_balance_negative_zero_resultrightvalue = (dst_negative_zero_resultright) + ge_balance_positive_zero_resultrightvalue))))))))) /\ (exists sto_ap_zero_resultproduct sto_an_zero_resultproduct sto_bp_zero_resultproduct sto_bn_zero_resultproduct sto_cp_zero_resultproduct sto_cn_zero_resultproduct. (((((dc_left_zero_result) = 2 * (sto_ap_zero_resultproduct) /\ (sto_an_zero_resultproduct) = 0) \/ exists ge_signed_half_zero_resultproductleft. (((dc_left_zero_result) = 2 * ge_signed_half_zero_resultproductleft + 1 /\ (sto_ap_zero_resultproduct) = 0) /\ (sto_an_zero_resultproduct) = S ge_signed_half_zero_resultproductleft))) /\ ((((((dc_right_zero_result) = 2 * (sto_bp_zero_resultproduct) /\ (sto_bn_zero_resultproduct) = 0) \/ exists ge_signed_half_zero_resultproductright. (((dc_right_zero_result) = 2 * ge_signed_half_zero_resultproductright + 1 /\ (sto_bp_zero_resultproduct) = 0) /\ (sto_bn_zero_resultproduct) = S ge_signed_half_zero_resultproductright))) /\ ((((((0) = 2 * (sto_cp_zero_resultproduct) /\ (sto_cn_zero_resultproduct) = 0) \/ exists ge_signed_half_zero_resultproductoutput. (((0) = 2 * ge_signed_half_zero_resultproductoutput + 1 /\ (sto_cp_zero_resultproduct) = 0) /\ (sto_cn_zero_resultproduct) = S ge_signed_half_zero_resultproductoutput))) /\ ((sto_ap_zero_resultproduct * sto_bp_zero_resultproduct + sto_an_zero_resultproduct * sto_bn_zero_resultproduct) + sto_cn_zero_resultproduct = (sto_ap_zero_resultproduct * sto_bn_zero_resultproduct + sto_an_zero_resultproduct * sto_bp_zero_resultproduct) + sto_cp_zero_resultproduct))))))))))))))) \/ ((((0)=0 \/ ~(exists pvs_factor_zero_resultnondivisor. (n) = (0) * pvs_factor_zero_resultnondivisor)) /\ ((0)=0))))

Constructive proof overview

Generated structural guide

The zeroth summand is canonical zero without looking at either input value at zero.

The unchanged tactic script uses 0 declared prerequisites and contains 8 exact native proof lines.

Alpha v34 checked-use · first admitted v31 · independently kernel and Lean verified; not Stable

Proof neighborhood

Direct dependencies

none

Direct dependents

DC0007 dirichlet_convolution_entry_exists

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

8 script commands · 3 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 assumptionsL1–3

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

  1. L1
    intro F
  2. L2
    intro G
  3. L3
    intro n
02Separate the logical casesL4–6

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

  1. L4
    right
  2. L5
    split
  3. L6
    left
03Calculate and transport equalitiesL7–8

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

  1. L7
    refl
  2. L8
    refl

Library-wide reading audit

Original exact command ledger · 8 lines
  1. 0001intro F
  2. 0002intro G
  3. 0003intro n
  4. 0004right
  5. 0005split
  6. 0006left
  7. 0007refl
  8. 0008refl