PA000U · theorem

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.

Statement with defined notation

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

Every purple notation token opens its conservative definition. This is a reading surface; the compiler expands the statement before the unchanged kernel checks it.

Definitions used by this theorem

In the theorem statement

none

0 occurrences

In local proof propositions

none

0 occurrences

Exact expanded native-PA statement
forall c i. ~(S ((S i) * c) = 0)

Proof neighborhood

Direct theorem prerequisites

PA0005 succ_ne_zero

Direct theorem 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

Definition-aware tactic body

Only local propositions introduced by have or suffices are compacted. The untrusted compiler re-expands each one before the original tactic script is replayed; defined notation is never accepted by the kernel. Open the exact replay line beneath every changed command.

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.

Definition notation is shown below. Open the paired exact edition for the original native formulas. Source pairing is not a new equivalence certificate.

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