CR0003

crt_positive_moduli_prefix_last_nonzero

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

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

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.

Historical partial components only: this chapter proves canonical solutions for finite positive pairwise-coprime systems and exact LCM solution classes. G011 is now closed in the separate Alpha-v27 generalized-crt branch for arbitrary pairwise-compatible systems, including noncoprime moduli. Full G011 proof · Alpha v27

Exact theorem in conservative defined notation

∀ b. ∀ c. ∀ l. ∀ x. CRTPositiveModuliPrefix(b,c,S l) → Beta(b,c,l,x) → ¬x = 0

Every linked abbreviation expands hygienically to the identical original native formula.

Definition DAG

Beta(b,c,i,x) · 1CRTPositiveModuliPrefix(b,c,l) · 1

Actual proof prerequisites

le_refl · checked external prerequisite
Original expanded first-order 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)

Complete unchanged native tactic proof

All 14 lines are the exact independently kernel-checked original script.

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.

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

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 defined 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