BT0084 · Bertrand theorem

beta_range_empty

Stable checked-use theorem · independently kernel verified

Every consecutive beta range of length zero is vacuous.

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

∀ b. ∀ c. ∀ a. ∀ l. l = 0 → Range(b,c,a,l)

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

PD0018 Range

1 occurrences

In local proof propositions

none

0 occurrences

Exact expanded native-PA statement
forall b c a l. l = 0 -> (forall ff_i_empty. (exists ff_lt_empty_bound. ff_lt_empty_bound + S ff_i_empty = l) -> (((exists ff_h_empty_decoded. ff_h_empty_decoded + S (a + ff_i_empty) = S ((S (ff_i_empty)) * c)) /\ exists ff_q_empty_decoded. b = ff_q_empty_decoded * S ((S (ff_i_empty)) * c) + (a + ff_i_empty))))

Proof neighborhood

Direct theorem prerequisites

BT000L add_eq_zero_right BT000C succ_ne_zero

Direct theorem dependents

BT0086 beta_range_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

18 script commands · 4 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.

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 (2)
add eq zero rightsucc ne zero
01Fix variables and assumptionsL1–7

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

  1. L1
    intro b
  2. L2
    intro c
  3. L3
    intro a
  4. L4
    intro l
  5. L5
    intro hl
  6. L6
    intro i
  7. L7
    intro hi
02Calculate and transport equalitiesL8–8

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

  1. L8
    rewrite hl at hi
03Separate the logical casesL9–10

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

  1. L9
    exfalso
  2. L10
    cases hi
04Establish hsiL11–18

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply add eq zero right.

  1. L11
    have hsi : S i = 0
  2. L12
    specialize add_eq_zero_right x
  3. L13
    specialize add_eq_zero_right (S i)
  4. L14
    apply add_eq_zero_right
  5. L15
    exact hi_witness
  6. L16
    specialize succ_ne_zero i
  7. L17
    apply succ_ne_zero
  8. L18
    exact hsi

Library-wide reading audit

Original defined command ledger · 18 lines
  1. 0001intro b
  2. 0002intro c
  3. 0003intro a
  4. 0004intro l
  5. 0005intro hl
  6. 0006intro i
  7. 0007intro hi
  8. 0008rewrite hl at hi
  9. 0009exfalso
  10. 0010cases hi
  11. 0011have hsi : S i = 0
  12. 0012specialize add_eq_zero_right x
  13. 0013specialize add_eq_zero_right (S i)
  14. 0014apply add_eq_zero_right
  15. 0015exact hi_witness
  16. 0016specialize succ_ne_zero i
  17. 0017apply succ_ne_zero
  18. 0018exact hsi