BT00SD · Bertrand theorem

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.

Statement with defined notation

∀ q. ∀ j. ∃ bit. bit = 1 ∧ Lt(j,q) ∨ bit = 0 ∧ Lt(q,S j)

Every purple notation token opens its conservative definition. Expanding the displayed statement recovers the exact first-order Peano-arithmetic formula checked by the unchanged kernel.

Definitions used by this theorem

In the theorem statement

PD0002 Lt

2 occurrences

In local proof propositions

none

0 occurrences

Exact expanded native-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))))

Proof neighborhood

Direct theorem prerequisites

BT001G le_or_lt

Direct theorem dependents

BT00SF eisenstein_initial_segment_prefix_exists

Definition-aware tactic body

Only local propositions introduced by have or suffices are compacted. Every changed line has an exact-AST conservative-expansion receipt; the kernel still receives the immutable original tactic script.

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.

Definition notation is shown below. Open the paired exact edition for the original native formulas. Source pairing is not a new equivalence certificate.

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 defined 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