CD0032

finite_modular_add_modulus

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

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

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_u_mod fms_v_mod. (a+p) + (p) * fms_u_mod = (a) + (p) * fms_v_mod

Constructive proof overview

Generated structural guide

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

The unchanged tactic script uses 1 declared prerequisite and contains 5 exact native proof lines.

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

Proof neighborhood

Direct dependencies

zero_add Stable theorem; checked-use authorized

Direct dependents

CD0033 prime_modular_additive_orbit_hits

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

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.

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