PA000U

beta_modulus_nonzero

Stable checked-use theorem · independently closed

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

Generated structural guide

Every Gödel-beta decoding modulus is nonzero.

Use the direct prerequisites succ_ne_zero as previously established PA formulas.

The proof proceeds by direct introduction and elimination.

Referenced ingredients

PA0005 succ_ne_zero

Proof neighborhood

Direct dependencies

PA0005 succ_ne_zero

Direct dependents

PA001U beta_exclusive_accumulated_product_step PA0029 beta_at_exists PA002G beta_exclusive_recode_congruence_step PA002W beta_at_of_mod_eq_bound PA002X beta_prefix_extend

Formal native tactic body

Dependencies are introduced as named hypotheses before line 1. Linked names are exact direct references. This Stable checked-use theorem is independently kernel-checked when replayed.

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