CR0001

crt_positive_moduli_prefix_empty

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

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

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. 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))

Constructive proof overview

Generated structural guide

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

The unchanged tactic script uses 2 declared prerequisites and contains 17 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

none

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

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.

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 exact 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

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