BT003Y

beta_modulus_nonzero

Stable checked-use theorem · independently kernel verified

Every Gödel-beta decoding modulus is 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 PA statement

forall c i. ~(S ((S i) * c) = 0)

Structural proof guide

Every Gödel-beta decoding modulus is nonzero.

Direct prerequisites: succ_ne_zero. The authored body proceeds by direct introduction and elimination.

Proof neighborhood

Direct dependencies

BT000C succ_ne_zero

Direct dependents

BT0041 beta_at_exists BT0045 beta_at_of_mod_eq_bound BT0059 beta_exclusive_accumulated_product_step BT005A beta_exclusive_recode_congruence_step BT005D beta_prefix_extend

Formal native tactic body

Dependencies are hypotheses of the historical Alpha-v12 body receipt. The complete historical Alpha-v18 proof bundle independently checks every dependency; current Alpha v25 preserves that checked theorem use without changing Stable membership.

Read the argument

Proof checkpoints

4 script commands · 2 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.

Named ingredients (1)
succ ne zero
01Fix variables and assumptionsL1–2

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

  1. L1
    intro c
  2. L2
    intro i
02Use earlier factsL3–4

Instantiate or apply named facts and discharge the corresponding proof obligations.

  1. L3
    specialize succ_ne_zero ((S i) * c)
  2. L4
    exact succ_ne_zero

Library-wide reading audit

Original exact command ledger · 4 lines
  1. 0001intro c
  2. 0002intro i
  3. 0003specialize succ_ne_zero ((S i) * c)
  4. 0004exact succ_ne_zero