CR0005

crt_prefix_solution_empty

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

Every natural number solves the empty simultaneous-congruence system.

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 r s b c l x. l = 0 -> (forall gcrt_solution_index_solution_empty gcrt_solution_residue_solution_empty gcrt_solution_modulus_solution_empty. (exists ff_lt_gcrt_solution_empty_bound. ff_lt_gcrt_solution_empty_bound + S gcrt_solution_index_solution_empty = l) -> (((exists ff_h_gcrt_solution_empty_residue. ff_h_gcrt_solution_empty_residue + S (gcrt_solution_residue_solution_empty) = S ((S (gcrt_solution_index_solution_empty)) * s)) /\ exists ff_q_gcrt_solution_empty_residue. r = ff_q_gcrt_solution_empty_residue * S ((S (gcrt_solution_index_solution_empty)) * s) + (gcrt_solution_residue_solution_empty))) -> (((exists ff_h_gcrt_solution_empty_modulus. ff_h_gcrt_solution_empty_modulus + S (gcrt_solution_modulus_solution_empty) = S ((S (gcrt_solution_index_solution_empty)) * c)) /\ exists ff_q_gcrt_solution_empty_modulus. b = ff_q_gcrt_solution_empty_modulus * S ((S (gcrt_solution_index_solution_empty)) * c) + (gcrt_solution_modulus_solution_empty))) -> (exists hgcrt_mod_left_gcrt_solution_empty_congruence hgcrt_mod_right_gcrt_solution_empty_congruence. x + gcrt_solution_modulus_solution_empty * hgcrt_mod_left_gcrt_solution_empty_congruence = gcrt_solution_residue_solution_empty + gcrt_solution_modulus_solution_empty * hgcrt_mod_right_gcrt_solution_empty_congruence))

Constructive proof overview

Generated structural guide

Every natural number solves the empty simultaneous-congruence system.

The unchanged tactic script uses 2 declared prerequisites and contains 22 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

Direct dependents

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

22 script commands · 5 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 assumptionsL1–10

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

  1. L1
    intro r
  2. L2
    intro s
  3. L3
    intro b
  4. L4
    intro c
  5. L5
    intro l
  6. L6
    intro x
  7. L7
    intro hz
  8. L8
    intro i
  9. L9
    intro a
  10. L10
    intro m
02Fix variables and assumptionsL11–13

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

  1. L11
    intro hi
  2. L12
    intro ha
  3. L13
    intro hm
03Separate the logical casesL14–14

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

  1. L14
    exfalso
04Calculate and transport equalitiesL15–15

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

  1. L15
    rewrite hz at hi
05Establish hbadL16–22

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

  1. L16
    have hbad : S i = 0
  2. L17
    specialize le_zero (S i)
  3. L18
    apply le_zero
  4. L19
    exact hi
  5. L20
    specialize succ_ne_zero i
  6. L21
    apply succ_ne_zero
  7. L22
    exact hbad

Library-wide reading audit

Original exact command ledger · 22 lines
  1. 0001intro r
  2. 0002intro s
  3. 0003intro b
  4. 0004intro c
  5. 0005intro l
  6. 0006intro x
  7. 0007intro hz
  8. 0008intro i
  9. 0009intro a
  10. 0010intro m
  11. 0011intro hi
  12. 0012intro ha
  13. 0013intro hm
  14. 0014exfalso
  15. 0015rewrite hz at hi
  16. 0016have hbad : S i = 0
  17. 0017specialize le_zero (S i)
  18. 0018apply le_zero
  19. 0019exact hi
  20. 0020specialize succ_ne_zero i
  21. 0021apply succ_ne_zero
  22. 0022exact hbad

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