CR000E

crt_prefix_lcm_empty

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

The universal-property lcm of the empty decoded modulus list is exactly one.

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. (((forall gcrt_common_index_empty_lcm_own gcrt_common_modulus_empty_lcm_own. (exists ff_lt_gcrt_empty_lcm_own_bound. ff_lt_gcrt_empty_lcm_own_bound + S gcrt_common_index_empty_lcm_own = 0) -> (((exists ff_h_gcrt_empty_lcm_own_entry. ff_h_gcrt_empty_lcm_own_entry + S (gcrt_common_modulus_empty_lcm_own) = S ((S (gcrt_common_index_empty_lcm_own)) * c)) /\ exists ff_q_gcrt_empty_lcm_own_entry. b = ff_q_gcrt_empty_lcm_own_entry * S ((S (gcrt_common_index_empty_lcm_own)) * c) + (gcrt_common_modulus_empty_lcm_own))) -> exists gcrt_common_quotient_empty_lcm_own. 1 = gcrt_common_modulus_empty_lcm_own * gcrt_common_quotient_empty_lcm_own) /\ forall gcrt_lcm_common_empty_lcm. (forall gcrt_common_index_empty_lcm_other gcrt_common_modulus_empty_lcm_other. (exists ff_lt_gcrt_empty_lcm_other_bound. ff_lt_gcrt_empty_lcm_other_bound + S gcrt_common_index_empty_lcm_other = 0) -> (((exists ff_h_gcrt_empty_lcm_other_entry. ff_h_gcrt_empty_lcm_other_entry + S (gcrt_common_modulus_empty_lcm_other) = S ((S (gcrt_common_index_empty_lcm_other)) * c)) /\ exists ff_q_gcrt_empty_lcm_other_entry. b = ff_q_gcrt_empty_lcm_other_entry * S ((S (gcrt_common_index_empty_lcm_other)) * c) + (gcrt_common_modulus_empty_lcm_other))) -> exists gcrt_common_quotient_empty_lcm_other. gcrt_lcm_common_empty_lcm = gcrt_common_modulus_empty_lcm_other * gcrt_common_quotient_empty_lcm_other) -> exists gcrt_lcm_quotient_empty_lcm. gcrt_lcm_common_empty_lcm = 1 * gcrt_lcm_quotient_empty_lcm))

Constructive proof overview

Generated structural guide

The universal-property lcm of the empty decoded modulus list is exactly one.

The unchanged tactic script uses 3 declared prerequisites and contains 19 exact native proof lines.

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

Proof neighborhood

Direct dependencies

le_zero Stable theorem; checked-use authorized succ_ne_zero Stable theorem; checked-use authorized one_multiple Stable theorem; checked-use authorized

Direct dependents

CR0010 crt_prefix_lcm_exists_unique

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

19 script commands · 6 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.

01Fix variables and assumptionsL1–2

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

  1. L1
    intro b
  2. L2
    intro c
02Separate the logical casesL3–3

Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.

  1. L3
    split
03Fix variables and assumptionsL4–7

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

  1. L4
    intro i
  2. L5
    intro m
  3. L6
    intro hi
  4. L7
    intro hm
04Separate the logical casesL8–8

Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.

  1. L8
    exfalso
05Establish hbadL9–18

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

  1. L9
    have hbad : S i = 0
  2. L10
    specialize le_zero (S i)
  3. L11
    apply le_zero
  4. L12
    exact hi
  5. L13
    specialize succ_ne_zero i
  6. L14
    apply succ_ne_zero
  7. L15
    exact hbad
  8. L16
    intro z
  9. L17
    intro hz
  10. L18
    specialize one_multiple z
06Use earlier factsL19–19

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

  1. L19
    exact one_multiple

Library-wide reading audit

Original exact command ledger · 19 lines
  1. 0001intro b
  2. 0002intro c
  3. 0003split
  4. 0004intro i
  5. 0005intro m
  6. 0006intro hi
  7. 0007intro hm
  8. 0008exfalso
  9. 0009have hbad : S i = 0
  10. 0010specialize le_zero (S i)
  11. 0011apply le_zero
  12. 0012exact hi
  13. 0013specialize succ_ne_zero i
  14. 0014apply succ_ne_zero
  15. 0015exact hbad
  16. 0016intro z
  17. 0017intro hz
  18. 0018specialize one_multiple z
  19. 0019exact one_multiple

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