BX0001

binary_modulus_nontrivial_nonzero

Every explicitly guarded modulus m>1 is constructively nonzero.

Alpha v34 checked-use · first admitted v21 · 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.

G102 was OPEN when this family was first admitted in Alpha v21. It is now CLOSED in Alpha v23: every arbitrary exponent has actual canonical beta-coded digits, a complete modular execution, and exact counted bound operations≤3*BitLen(e)+2.

Exact theorem in conservative defined notation

∀ m. BinaryModulus(m) → ¬m = 0

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

Definition DAG

BinaryModulus(m) · 1

Actual proof prerequisites

add_eq_zero_right · checked external prerequisitesucc_ne_zero · checked external prerequisite
Original expanded first-order statement 
forall m. (exists ff_modulus_gap_binary_guard. ff_modulus_gap_binary_guard + S 1 = m) -> ~(m = 0)

Complete unchanged native tactic proof

All 13 lines are the exact independently kernel-checked original script.

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.

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 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 defined 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