CR0002

crt_positive_moduli_prefix_drop_last

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

Positivity of a successor-length modulus list restricts to its predecessor prefix.

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. (forall gcrt_positive_index_drop_source gcrt_positive_value_drop_source. (exists ff_lt_gcrt_drop_source_bound. ff_lt_gcrt_drop_source_bound + S gcrt_positive_index_drop_source = S l) -> (((exists ff_h_gcrt_drop_source_entry. ff_h_gcrt_drop_source_entry + S (gcrt_positive_value_drop_source) = S ((S (gcrt_positive_index_drop_source)) * c)) /\ exists ff_q_gcrt_drop_source_entry. b = ff_q_gcrt_drop_source_entry * S ((S (gcrt_positive_index_drop_source)) * c) + (gcrt_positive_value_drop_source))) -> ~(gcrt_positive_value_drop_source = 0)) -> (forall gcrt_positive_index_drop_target gcrt_positive_value_drop_target. (exists ff_lt_gcrt_drop_target_bound. ff_lt_gcrt_drop_target_bound + S gcrt_positive_index_drop_target = l) -> (((exists ff_h_gcrt_drop_target_entry. ff_h_gcrt_drop_target_entry + S (gcrt_positive_value_drop_target) = S ((S (gcrt_positive_index_drop_target)) * c)) /\ exists ff_q_gcrt_drop_target_entry. b = ff_q_gcrt_drop_target_entry * S ((S (gcrt_positive_index_drop_target)) * c) + (gcrt_positive_value_drop_target))) -> ~(gcrt_positive_value_drop_target = 0))

Constructive proof overview

Generated structural guide

Positivity of a successor-length modulus list restricts to its predecessor prefix.

The unchanged tactic script uses 1 declared prerequisite and contains 18 exact native proof lines.

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

Proof neighborhood

Direct dependencies

le_succ 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

18 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 assumptionsL1–9

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 hpositive
  5. L5
    intro i
  6. L6
    intro m
  7. L7
    intro hi
  8. L8
    intro hm
  9. L9
    intro hzero
02Use earlier factsL10–18

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

  1. L10
    specialize hpositive i
  2. L11
    specialize hpositive m
  3. L12
    apply hpositive
  4. L13
    specialize le_succ (S i)
  5. L14
    specialize le_succ l
  6. L15
    apply le_succ
  7. L16
    exact hi
  8. L17
    exact hm
  9. L18
    exact hzero

Library-wide reading audit

Original exact command ledger · 18 lines
  1. 0001intro b
  2. 0002intro c
  3. 0003intro l
  4. 0004intro hpositive
  5. 0005intro i
  6. 0006intro m
  7. 0007intro hi
  8. 0008intro hm
  9. 0009intro hzero
  10. 0010specialize hpositive i
  11. 0011specialize hpositive m
  12. 0012apply hpositive
  13. 0013specialize le_succ (S i)
  14. 0014specialize le_succ l
  15. 0015apply le_succ
  16. 0016exact hi
  17. 0017exact hm
  18. 0018exact hzero

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