CD0032

finite_modular_add_modulus

Adding the modulus preserves balanced congruence with explicit zero/one witnesses.

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. ModEq(p,a + p,a)

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

Definition DAG

ModEq(m,a,b) · 1

Actual proof prerequisites

zero_add · checked external prerequisite
Original expanded first-order statement 
forall p a. exists fms_u_mod fms_v_mod. (a+p) + (p) * fms_u_mod = (a) + (p) * fms_v_mod

Complete tactic proof in conservative notation

All 5 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

5 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.

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 assumptionsL1–2

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

  1. L1
    intro p
  2. L2
    intro a
02Construct an explicit witnessL3–4

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

  1. L3
    exists 0
  2. L4
    exists 1
03Calculate and transport equalitiesL5–5

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

  1. L5
    simp [zero_add]

Library-wide reading audit

Original defined command ledger · 5 lines
  1. 0001intro p
  2. 0002intro a
  3. 0003exists 0
  4. 0004exists 1
  5. 0005simp [zero_add]