ZS0002

signed_zero_window_restrict

Restrict the upper endpoint of an actual zero window without changing any represented values.

Alpha v34 checked-use · first admitted v31 · 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.

The zero window is half-open and its order hypothesis is essential. All folds retain actual beta-coded traces. These are support lemmas for the separately verified full inversion endpoint.

Exact theorem in conservative defined notation

∀ F. ∀ k. ∀ l. ∀ L. Le(l,L) → SignedZeroWindow(F,k,L) → SignedZeroWindow(F,k,l)

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

Definition DAG

SignedZeroWindow(F,k,l) · 2Le(a,b) · 1

Actual proof prerequisites

le_trans · checked external prerequisite
Original expanded first-order statement 
forall F k l L. (exists pvs_le_gap_restrict_bound. pvs_le_gap_restrict_bound + (l) = (L)) -> (forall sfs_index_restrict_source sfs_value_restrict_source. (exists pvs_le_gap_restrict_sourcelower. pvs_le_gap_restrict_sourcelower + (k) = (sfs_index_restrict_source)) -> (exists pvs_gap_restrict_sourceupper. pvs_gap_restrict_sourceupper + S (sfs_index_restrict_source) = (L)) -> (exists dst_positive_code_restrict_sourceentry dst_positive_scale_restrict_sourceentry dst_negative_code_restrict_sourceentry dst_negative_scale_restrict_sourceentry dst_positive_restrict_sourceentry dst_negative_restrict_sourceentry. (((F) = (((((dst_positive_code_restrict_sourceentry) + (dst_positive_scale_restrict_sourceentry)) * S ((dst_positive_code_restrict_sourceentry) + (dst_positive_scale_restrict_sourceentry)) + ((dst_positive_scale_restrict_sourceentry) + (dst_positive_scale_restrict_sourceentry))) + (((dst_negative_code_restrict_sourceentry) + (dst_negative_scale_restrict_sourceentry)) * S ((dst_negative_code_restrict_sourceentry) + (dst_negative_scale_restrict_sourceentry)) + ((dst_negative_scale_restrict_sourceentry) + (dst_negative_scale_restrict_sourceentry)))) * S ((((dst_positive_code_restrict_sourceentry) + (dst_positive_scale_restrict_sourceentry)) * S ((dst_positive_code_restrict_sourceentry) + (dst_positive_scale_restrict_sourceentry)) + ((dst_positive_scale_restrict_sourceentry) + (dst_positive_scale_restrict_sourceentry))) + (((dst_negative_code_restrict_sourceentry) + (dst_negative_scale_restrict_sourceentry)) * S ((dst_negative_code_restrict_sourceentry) + (dst_negative_scale_restrict_sourceentry)) + ((dst_negative_scale_restrict_sourceentry) + (dst_negative_scale_restrict_sourceentry)))) + ((((dst_negative_code_restrict_sourceentry) + (dst_negative_scale_restrict_sourceentry)) * S ((dst_negative_code_restrict_sourceentry) + (dst_negative_scale_restrict_sourceentry)) + ((dst_negative_scale_restrict_sourceentry) + (dst_negative_scale_restrict_sourceentry))) + (((dst_negative_code_restrict_sourceentry) + (dst_negative_scale_restrict_sourceentry)) * S ((dst_negative_code_restrict_sourceentry) + (dst_negative_scale_restrict_sourceentry)) + ((dst_negative_scale_restrict_sourceentry) + (dst_negative_scale_restrict_sourceentry)))))) /\ (((((exists ff_h_pvs_restrict_sourceentrypositive. ff_h_pvs_restrict_sourceentrypositive + S (dst_positive_restrict_sourceentry) = S ((S (sfs_index_restrict_source)) * dst_positive_scale_restrict_sourceentry)) /\ exists ff_q_pvs_restrict_sourceentrypositive. dst_positive_code_restrict_sourceentry = ff_q_pvs_restrict_sourceentrypositive * S ((S (sfs_index_restrict_source)) * dst_positive_scale_restrict_sourceentry) + (dst_positive_restrict_sourceentry))) /\ (((((exists ff_h_pvs_restrict_sourceentrynegative. ff_h_pvs_restrict_sourceentrynegative + S (dst_negative_restrict_sourceentry) = S ((S (sfs_index_restrict_source)) * dst_negative_scale_restrict_sourceentry)) /\ exists ff_q_pvs_restrict_sourceentrynegative. dst_negative_code_restrict_sourceentry = ff_q_pvs_restrict_sourceentrynegative * S ((S (sfs_index_restrict_source)) * dst_negative_scale_restrict_sourceentry) + (dst_negative_restrict_sourceentry))) /\ (exists ge_balance_positive_restrict_sourceentryvalue ge_balance_negative_restrict_sourceentryvalue. (((((sfs_value_restrict_source) = 2 * (ge_balance_positive_restrict_sourceentryvalue) /\ (ge_balance_negative_restrict_sourceentryvalue) = 0) \/ exists ge_signed_half_restrict_sourceentryvaluedecode. (((sfs_value_restrict_source) = 2 * ge_signed_half_restrict_sourceentryvaluedecode + 1 /\ (ge_balance_positive_restrict_sourceentryvalue) = 0) /\ (ge_balance_negative_restrict_sourceentryvalue) = S ge_signed_half_restrict_sourceentryvaluedecode))) /\ ((dst_positive_restrict_sourceentry) + ge_balance_negative_restrict_sourceentryvalue = (dst_negative_restrict_sourceentry) + ge_balance_positive_restrict_sourceentryvalue))))))))) -> sfs_value_restrict_source=0) -> (forall sfs_index_restrict_result sfs_value_restrict_result. (exists pvs_le_gap_restrict_resultlower. pvs_le_gap_restrict_resultlower + (k) = (sfs_index_restrict_result)) -> (exists pvs_gap_restrict_resultupper. pvs_gap_restrict_resultupper + S (sfs_index_restrict_result) = (l)) -> (exists dst_positive_code_restrict_resultentry dst_positive_scale_restrict_resultentry dst_negative_code_restrict_resultentry dst_negative_scale_restrict_resultentry dst_positive_restrict_resultentry dst_negative_restrict_resultentry. (((F) = (((((dst_positive_code_restrict_resultentry) + (dst_positive_scale_restrict_resultentry)) * S ((dst_positive_code_restrict_resultentry) + (dst_positive_scale_restrict_resultentry)) + ((dst_positive_scale_restrict_resultentry) + (dst_positive_scale_restrict_resultentry))) + (((dst_negative_code_restrict_resultentry) + (dst_negative_scale_restrict_resultentry)) * S ((dst_negative_code_restrict_resultentry) + (dst_negative_scale_restrict_resultentry)) + ((dst_negative_scale_restrict_resultentry) + (dst_negative_scale_restrict_resultentry)))) * S ((((dst_positive_code_restrict_resultentry) + (dst_positive_scale_restrict_resultentry)) * S ((dst_positive_code_restrict_resultentry) + (dst_positive_scale_restrict_resultentry)) + ((dst_positive_scale_restrict_resultentry) + (dst_positive_scale_restrict_resultentry))) + (((dst_negative_code_restrict_resultentry) + (dst_negative_scale_restrict_resultentry)) * S ((dst_negative_code_restrict_resultentry) + (dst_negative_scale_restrict_resultentry)) + ((dst_negative_scale_restrict_resultentry) + (dst_negative_scale_restrict_resultentry)))) + ((((dst_negative_code_restrict_resultentry) + (dst_negative_scale_restrict_resultentry)) * S ((dst_negative_code_restrict_resultentry) + (dst_negative_scale_restrict_resultentry)) + ((dst_negative_scale_restrict_resultentry) + (dst_negative_scale_restrict_resultentry))) + (((dst_negative_code_restrict_resultentry) + (dst_negative_scale_restrict_resultentry)) * S ((dst_negative_code_restrict_resultentry) + (dst_negative_scale_restrict_resultentry)) + ((dst_negative_scale_restrict_resultentry) + (dst_negative_scale_restrict_resultentry)))))) /\ (((((exists ff_h_pvs_restrict_resultentrypositive. ff_h_pvs_restrict_resultentrypositive + S (dst_positive_restrict_resultentry) = S ((S (sfs_index_restrict_result)) * dst_positive_scale_restrict_resultentry)) /\ exists ff_q_pvs_restrict_resultentrypositive. dst_positive_code_restrict_resultentry = ff_q_pvs_restrict_resultentrypositive * S ((S (sfs_index_restrict_result)) * dst_positive_scale_restrict_resultentry) + (dst_positive_restrict_resultentry))) /\ (((((exists ff_h_pvs_restrict_resultentrynegative. ff_h_pvs_restrict_resultentrynegative + S (dst_negative_restrict_resultentry) = S ((S (sfs_index_restrict_result)) * dst_negative_scale_restrict_resultentry)) /\ exists ff_q_pvs_restrict_resultentrynegative. dst_negative_code_restrict_resultentry = ff_q_pvs_restrict_resultentrynegative * S ((S (sfs_index_restrict_result)) * dst_negative_scale_restrict_resultentry) + (dst_negative_restrict_resultentry))) /\ (exists ge_balance_positive_restrict_resultentryvalue ge_balance_negative_restrict_resultentryvalue. (((((sfs_value_restrict_result) = 2 * (ge_balance_positive_restrict_resultentryvalue) /\ (ge_balance_negative_restrict_resultentryvalue) = 0) \/ exists ge_signed_half_restrict_resultentryvaluedecode. (((sfs_value_restrict_result) = 2 * ge_signed_half_restrict_resultentryvaluedecode + 1 /\ (ge_balance_positive_restrict_resultentryvalue) = 0) /\ (ge_balance_negative_restrict_resultentryvalue) = S ge_signed_half_restrict_resultentryvaluedecode))) /\ ((dst_positive_restrict_resultentry) + ge_balance_negative_restrict_resultentryvalue = (dst_negative_restrict_resultentry) + ge_balance_positive_restrict_resultentryvalue))))))))) -> sfs_value_restrict_result=0)

Complete tactic proof in conservative notation

All 22 original proof lines are preserved. Only local proposition formulas are abbreviated; every abbreviation has an exact binder-safe expansion check. The linked exact edition contains the unchanged replay script.

Read the argument

Proof checkpoints

22 script commands · 4 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 assumptionsL1–10

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

  1. L1
    intro F
  2. L2
    intro k
  3. L3
    intro l
  4. L4
    intro L
  5. L5
    intro hl
  6. L6
    intro hz
  7. L7
    intro i
  8. L8
    intro z
  9. L9
    intro hki
  10. L10
    intro hil
02Fix variables and assumptionsL11–11

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

  1. L11
    intro hi
03Use earlier factsL12–21

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

  1. L12
    specialize hz (i)
  2. L13
    specialize hz (z)
  3. L14
    apply hz
  4. L15
    exact hki
  5. L16
    specialize le_trans (S i)
  6. L17
    specialize le_trans (l)
  7. L18
    specialize le_trans (L)
  8. L19
    apply le_trans
  9. L20
    exact hil
  10. L21
    exact hl
04Use earlier factsL22–22

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

  1. L22
    exact hi

Library-wide reading audit

Original defined command ledger · 22 lines
  1. 0001intro F
  2. 0002intro k
  3. 0003intro l
  4. 0004intro L
  5. 0005intro hl
  6. 0006intro hz
  7. 0007intro i
  8. 0008intro z
  9. 0009intro hki
  10. 0010intro hil
  11. 0011intro hi
  12. 0012specialize hz (i)
  13. 0013specialize hz (z)
  14. 0014apply hz
  15. 0015exact hki
  16. 0016specialize le_trans (S i)
  17. 0017specialize le_trans (l)
  18. 0018specialize le_trans (L)
  19. 0019apply le_trans
  20. 0020exact hil
  21. 0021exact hl
  22. 0022exact hi