BT00SD

eisenstein_initial_segment_indicator_choice

Alpha v34 checked-use theorem · independently kernel and Lean verified; not Stable

Every position has a constructive exact threshold-indicator bit.

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 PA statement

forall q j. exists bit. (((bit = 1 /\ (exists eis_le_gap_initial_segment_point_inside. eis_le_gap_initial_segment_point_inside + (S j) = q)) \/ (bit = 0 /\ (exists eis_lt_gap_initial_segment_point_outside. eis_lt_gap_initial_segment_point_outside + S (q) = S j))))

Structural proof guide

Every position has a constructive exact threshold-indicator bit.

Direct prerequisites: le_or_lt. The authored body proceeds by case analysis (1).

Proof neighborhood

Direct dependencies

BT001G le_or_lt

Direct dependents

BT00SF eisenstein_initial_segment_prefix_exists

Formal native tactic body

Dependencies are hypotheses of the historical Alpha-v12 body receipt. The complete historical Alpha-v18 proof bundle independently checks every dependency; current Alpha v25 preserves that checked theorem use without changing Stable membership.

Read the argument

Proof checkpoints

15 script commands · 11 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.

Named ingredients (1)
le or lt
01Fix variables and assumptionsL1–2

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

  1. L1
    intro q
  2. L2
    intro j
02Use earlier factsL3–4

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

  1. L3
    specialize le_or_lt (S j)
  2. L4
    specialize le_or_lt q
03Separate the logical casesL5–5

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

  1. L5
    cases le_or_lt
04Construct an explicit witnessL6–6

Supply the displayed value, then prove that it has the required property.

  1. L6
    exists 1
05Separate the logical casesL7–8

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

  1. L7
    left
  2. L8
    split
06Calculate and transport equalitiesL9–9

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

  1. L9
    refl
07Use earlier factsL10–10

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

  1. L10
    exact le_or_lt_left
08Construct an explicit witnessL11–11

Supply the displayed value, then prove that it has the required property.

  1. L11
    exists 0
09Separate the logical casesL12–13

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

  1. L12
    right
  2. L13
    split
10Calculate and transport equalitiesL14–14

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

  1. L14
    refl
11Use earlier factsL15–15

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

  1. L15
    exact le_or_lt_right

Library-wide reading audit

Original exact command ledger · 15 lines
  1. 0001intro q
  2. 0002intro j
  3. 0003specialize le_or_lt (S j)
  4. 0004specialize le_or_lt q
  5. 0005cases le_or_lt
  6. 0006exists 1
  7. 0007left
  8. 0008split
  9. 0009refl
  10. 0010exact le_or_lt_left
  11. 0011exists 0
  12. 0012right
  13. 0013split
  14. 0014refl
  15. 0015exact le_or_lt_right