BX0001

binary_modulus_nontrivial_nonzero

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

Every explicitly guarded modulus m>1 is constructively nonzero.

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 m. (exists ff_modulus_gap_binary_guard. ff_modulus_gap_binary_guard + S 1 = m) -> ~(m = 0)

Constructive proof overview

Generated structural guide

Every explicitly guarded modulus m>1 is constructively nonzero.

The unchanged tactic script uses 2 declared prerequisites and contains 13 exact native proof lines.

Alpha v34 checked-use · first admitted v21 · independently kernel and Lean verified; not Stable

Proof neighborhood

Direct dependencies

add_eq_zero_right Stable theorem; checked-use authorized succ_ne_zero Stable theorem; checked-use authorized

Direct dependents

BX0002 binary_canonical_residue_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

13 script commands · 4 reading checkpoints · 1 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 m
  2. L2
    intro hmodulus
  3. L3
    intro hzero
02Separate the logical casesL4–4

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

  1. L4
    cases hmodulus
03Calculate and transport equalitiesL5–5

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

  1. L5
    rewrite hzero at hmodulus_witness
04Establish hsuccessorL6–13

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply add eq zero right.

  1. L6
    have hsuccessor : S 1 = 0
  2. L7
    specialize add_eq_zero_right x
  3. L8
    specialize add_eq_zero_right (S 1)
  4. L9
    apply add_eq_zero_right
  5. L10
    exact hmodulus_witness
  6. L11
    specialize succ_ne_zero 1
  7. L12
    apply succ_ne_zero
  8. L13
    exact hsuccessor

Library-wide reading audit

Original exact command ledger · 13 lines
  1. 0001intro m
  2. 0002intro hmodulus
  3. 0003intro hzero
  4. 0004cases hmodulus
  5. 0005rewrite hzero at hmodulus_witness
  6. 0006have hsuccessor : S 1 = 0
  7. 0007specialize add_eq_zero_right x
  8. 0008specialize add_eq_zero_right (S 1)
  9. 0009apply add_eq_zero_right
  10. 0010exact hmodulus_witness
  11. 0011specialize succ_ne_zero 1
  12. 0012apply succ_ne_zero
  13. 0013exact hsuccessor