ZS0001

signed_zero_window_empty

A half-open zero window with coinciding endpoints is genuinely empty.

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. SignedZeroWindow(F,k,k)

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

Definition DAG

SignedZeroWindow(F,k,l) · 1

Actual proof prerequisites

lt_not_le · checked external prerequisite
Original expanded first-order statement 
forall F k. (forall sfs_index_empty_window sfs_value_empty_window. (exists pvs_le_gap_empty_windowlower. pvs_le_gap_empty_windowlower + (k) = (sfs_index_empty_window)) -> (exists pvs_gap_empty_windowupper. pvs_gap_empty_windowupper + S (sfs_index_empty_window) = (k)) -> (exists dst_positive_code_empty_windowentry dst_positive_scale_empty_windowentry dst_negative_code_empty_windowentry dst_negative_scale_empty_windowentry dst_positive_empty_windowentry dst_negative_empty_windowentry. (((F) = (((((dst_positive_code_empty_windowentry) + (dst_positive_scale_empty_windowentry)) * S ((dst_positive_code_empty_windowentry) + (dst_positive_scale_empty_windowentry)) + ((dst_positive_scale_empty_windowentry) + (dst_positive_scale_empty_windowentry))) + (((dst_negative_code_empty_windowentry) + (dst_negative_scale_empty_windowentry)) * S ((dst_negative_code_empty_windowentry) + (dst_negative_scale_empty_windowentry)) + ((dst_negative_scale_empty_windowentry) + (dst_negative_scale_empty_windowentry)))) * S ((((dst_positive_code_empty_windowentry) + (dst_positive_scale_empty_windowentry)) * S ((dst_positive_code_empty_windowentry) + (dst_positive_scale_empty_windowentry)) + ((dst_positive_scale_empty_windowentry) + (dst_positive_scale_empty_windowentry))) + (((dst_negative_code_empty_windowentry) + (dst_negative_scale_empty_windowentry)) * S ((dst_negative_code_empty_windowentry) + (dst_negative_scale_empty_windowentry)) + ((dst_negative_scale_empty_windowentry) + (dst_negative_scale_empty_windowentry)))) + ((((dst_negative_code_empty_windowentry) + (dst_negative_scale_empty_windowentry)) * S ((dst_negative_code_empty_windowentry) + (dst_negative_scale_empty_windowentry)) + ((dst_negative_scale_empty_windowentry) + (dst_negative_scale_empty_windowentry))) + (((dst_negative_code_empty_windowentry) + (dst_negative_scale_empty_windowentry)) * S ((dst_negative_code_empty_windowentry) + (dst_negative_scale_empty_windowentry)) + ((dst_negative_scale_empty_windowentry) + (dst_negative_scale_empty_windowentry)))))) /\ (((((exists ff_h_pvs_empty_windowentrypositive. ff_h_pvs_empty_windowentrypositive + S (dst_positive_empty_windowentry) = S ((S (sfs_index_empty_window)) * dst_positive_scale_empty_windowentry)) /\ exists ff_q_pvs_empty_windowentrypositive. dst_positive_code_empty_windowentry = ff_q_pvs_empty_windowentrypositive * S ((S (sfs_index_empty_window)) * dst_positive_scale_empty_windowentry) + (dst_positive_empty_windowentry))) /\ (((((exists ff_h_pvs_empty_windowentrynegative. ff_h_pvs_empty_windowentrynegative + S (dst_negative_empty_windowentry) = S ((S (sfs_index_empty_window)) * dst_negative_scale_empty_windowentry)) /\ exists ff_q_pvs_empty_windowentrynegative. dst_negative_code_empty_windowentry = ff_q_pvs_empty_windowentrynegative * S ((S (sfs_index_empty_window)) * dst_negative_scale_empty_windowentry) + (dst_negative_empty_windowentry))) /\ (exists ge_balance_positive_empty_windowentryvalue ge_balance_negative_empty_windowentryvalue. (((((sfs_value_empty_window) = 2 * (ge_balance_positive_empty_windowentryvalue) /\ (ge_balance_negative_empty_windowentryvalue) = 0) \/ exists ge_signed_half_empty_windowentryvaluedecode. (((sfs_value_empty_window) = 2 * ge_signed_half_empty_windowentryvaluedecode + 1 /\ (ge_balance_positive_empty_windowentryvalue) = 0) /\ (ge_balance_negative_empty_windowentryvalue) = S ge_signed_half_empty_windowentryvaluedecode))) /\ ((dst_positive_empty_windowentry) + ge_balance_negative_empty_windowentryvalue = (dst_negative_empty_windowentry) + ge_balance_positive_empty_windowentryvalue))))))))) -> sfs_value_empty_window=0)

Complete tactic proof in conservative notation

All 13 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

13 script commands · 3 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 assumptionsL1–7

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

  1. L1
    intro F
  2. L2
    intro k
  3. L3
    intro i
  4. L4
    intro z
  5. L5
    intro hki
  6. L6
    intro hik
  7. L7
    intro hz
02Separate the logical casesL8–8

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

  1. L8
    exfalso
03Use earlier factsL9–13

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

  1. L9
    specialize lt_not_le (i)
  2. L10
    specialize lt_not_le (k)
  3. L11
    apply lt_not_le
  4. L12
    exact hik
  5. L13
    exact hki

Library-wide reading audit

Original defined command ledger · 13 lines
  1. 0001intro F
  2. 0002intro k
  3. 0003intro i
  4. 0004intro z
  5. 0005intro hki
  6. 0006intro hik
  7. 0007intro hz
  8. 0008exfalso
  9. 0009specialize lt_not_le (i)
  10. 0010specialize lt_not_le (k)
  11. 0011apply lt_not_le
  12. 0012exact hik
  13. 0013exact hki