CR0001

crt_positive_moduli_prefix_empty

An empty decoded modulus prefix has no zero-valued member.

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. l = 0 → CRTPositiveModuliPrefix(b,c,l)

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

Definition DAG

CRTPositiveModuliPrefix(b,c,l) · 1

Actual proof prerequisites

le_zero · checked external prerequisitesucc_ne_zero · checked external prerequisite
Original expanded first-order statement 
forall b c l. l = 0 -> (forall gcrt_positive_index_empty gcrt_positive_value_empty. (exists ff_lt_gcrt_empty_bound. ff_lt_gcrt_empty_bound + S gcrt_positive_index_empty = l) -> (((exists ff_h_gcrt_empty_entry. ff_h_gcrt_empty_entry + S (gcrt_positive_value_empty) = S ((S (gcrt_positive_index_empty)) * c)) /\ exists ff_q_gcrt_empty_entry. b = ff_q_gcrt_empty_entry * S ((S (gcrt_positive_index_empty)) * c) + (gcrt_positive_value_empty))) -> ~(gcrt_positive_value_empty = 0))

Complete unchanged native tactic proof

All 17 lines are the exact independently kernel-checked original script.

Read the argument

Proof checkpoints

17 script commands · 3 reading checkpoints · 1 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 assumptionsL1–9

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 hz
  5. L5
    intro i
  6. L6
    intro m
  7. L7
    intro hi
  8. L8
    intro hm
  9. L9
    intro hzero
02Calculate and transport equalitiesL10–10

Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.

  1. L10
    rewrite hz at hi
03Establish hbadL11–17

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply le zero.

  1. L11
    have hbad : S i = 0
  2. L12
    specialize le_zero (S i)
  3. L13
    apply le_zero
  4. L14
    exact hi
  5. L15
    specialize succ_ne_zero i
  6. L16
    apply succ_ne_zero
  7. L17
    exact hbad

Library-wide reading audit

Original defined command ledger · 17 lines
  1. 0001intro b
  2. 0002intro c
  3. 0003intro l
  4. 0004intro hz
  5. 0005intro i
  6. 0006intro m
  7. 0007intro hi
  8. 0008intro hm
  9. 0009intro hzero
  10. 0010rewrite hz at hi
  11. 0011have hbad : S i = 0
  12. 0012specialize le_zero (S i)
  13. 0013apply le_zero
  14. 0014exact hi
  15. 0015specialize succ_ne_zero i
  16. 0016apply succ_ne_zero
  17. 0017exact hbad