CD0025

finite_modular_additive_complement

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

A canonical residue has a genuine natural additive complement, including the zero boundary.

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 original first-admission records.

Exact expanded first-order arithmetic statement

forall p a. (exists fms_gap_lt. fms_gap_lt + S (a) = (p)) -> exists d. a+d=p

Constructive proof overview

Generated structural guide

A canonical residue has a genuine natural additive complement, including the zero boundary.

The unchanged tactic script uses 2 declared prerequisites and contains 8 exact native proof lines.

Alpha v34 checked-use · first admitted v27 · independently kernel and Lean verified; not Stable

Proof neighborhood

Direct dependencies

add_comm Stable theorem; checked-use authorized add_succ_left Stable theorem; checked-use authorized

Direct dependents

CD002F finite_modular_sumset_prefix_exists CD0033 prime_modular_additive_orbit_hits CD0038 finite_modular_dyson_transform_exists CD0046 prime_cauchy_davenport_cover_bound

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 · 5 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 p
  2. L2
    intro a
  3. L3
    intro ha
02Separate the logical casesL4–4

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

  1. L4
    cases ha
03Construct an explicit witnessL5–5

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

  1. L5
    exists S x
04Calculate and transport equalitiesL6–7

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

  1. L6
    trans x+S a
  2. L7
    simp [add_comm, add_succ_left]
05Use earlier factsL8–8

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

  1. L8
    exact ha_witness

Library-wide reading audit

Original exact command ledger · 8 lines
  1. 0001intro p
  2. 0002intro a
  3. 0003intro ha
  4. 0004cases ha
  5. 0005exists S x
  6. 0006trans x+S a
  7. 0007simp [add_comm, add_succ_left]
  8. 0008exact ha_witness