CR0003

crt_positive_moduli_prefix_last_nonzero

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

The final decoded modulus of a positive successor list 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 first-order arithmetic statement

forall b c l x. (forall gcrt_positive_index_last_source gcrt_positive_value_last_source. (exists ff_lt_gcrt_last_source_bound. ff_lt_gcrt_last_source_bound + S gcrt_positive_index_last_source = S l) -> (((exists ff_h_gcrt_last_source_entry. ff_h_gcrt_last_source_entry + S (gcrt_positive_value_last_source) = S ((S (gcrt_positive_index_last_source)) * c)) /\ exists ff_q_gcrt_last_source_entry. b = ff_q_gcrt_last_source_entry * S ((S (gcrt_positive_index_last_source)) * c) + (gcrt_positive_value_last_source))) -> ~(gcrt_positive_value_last_source = 0)) -> (((exists ff_h_gcrt_last_value. ff_h_gcrt_last_value + S (x) = S ((S (l)) * c)) /\ exists ff_q_gcrt_last_value. b = ff_q_gcrt_last_value * S ((S (l)) * c) + (x))) -> ~(x = 0)

Constructive proof overview

Generated structural guide

The final decoded modulus of a positive successor list is nonzero.

The unchanged tactic script uses 1 declared prerequisite and contains 14 exact native proof lines.

Alpha v34 checked-use · first admitted v24 · independently kernel and Lean verified; not Stable

Proof neighborhood

Direct dependencies

le_refl Stable theorem; checked-use authorized

Direct dependents

CR000A crt_positive_moduli_prefix_product_nonzero CR0013 crt_pairwise_coprime_prefix_solution_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

14 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.

01Fix variables and assumptionsL1–7

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

  1. L1
    intro b
  2. L2
    intro c
  3. L3
    intro l
  4. L4
    intro x
  5. L5
    intro hpositive
  6. L6
    intro hx
  7. L7
    intro hzero
02Use earlier factsL8–14

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

  1. L8
    specialize hpositive l
  2. L9
    specialize hpositive x
  3. L10
    apply hpositive
  4. L11
    specialize le_refl (S l)
  5. L12
    exact le_refl
  6. L13
    exact hx
  7. L14
    exact hzero

Library-wide reading audit

Original exact command ledger · 14 lines
  1. 0001intro b
  2. 0002intro c
  3. 0003intro l
  4. 0004intro x
  5. 0005intro hpositive
  6. 0006intro hx
  7. 0007intro hzero
  8. 0008specialize hpositive l
  9. 0009specialize hpositive x
  10. 0010apply hpositive
  11. 0011specialize le_refl (S l)
  12. 0012exact le_refl
  13. 0013exact hx
  14. 0014exact hzero

Separate complete second-wave branches: Full G011 proof · Alpha v27.