CD0025

finite_modular_additive_complement

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

Alpha v34 checked-use · first admitted v27 · 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. Exact original first-admission records.

Sets are complete characteristic-bit codes with actual finite cardinality witnesses. The proof constructs translations and the sumset; no finite-choice oracle, supplied cardinality conclusion, or unproved polynomial-method premise is used.

Exact theorem in conservative defined notation

∀ p. ∀ a. Lt(a,p) → ∃ x. a + x = p

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

Definition DAG

Lt(a,b) · 1

Actual proof prerequisites

add_comm · checked external prerequisiteadd_succ_left · checked external prerequisite
Original expanded first-order statement 
forall p a. (exists fms_gap_lt. fms_gap_lt + S (a) = (p)) -> exists d. a+d=p

Complete tactic proof in conservative notation

All 8 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

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.

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 assumptionsL1–3

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 defined 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