BT0087 · Bertrand theorem

beta_range_entry_eq

Stable checked-use theorem · independently kernel verified

A decoded entry of a Range prefix is its start plus its index.

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. ∀ i. ∀ x. Range(b,c,a,l) → Lt(i,l) → BetaAt(b,c,i,x) → x = a + i

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 PD0013 BetaAt PD0018 Range

3 occurrences

In local proof propositions

PD0013 BetaAt

1 occurrences

Exact expanded native-PA statement
forall b c a l i x. (forall ff_i_entry. (exists ff_lt_entry_bound. ff_lt_entry_bound + S ff_i_entry = l) -> (((exists ff_h_entry_decoded. ff_h_entry_decoded + S (a + ff_i_entry) = S ((S (ff_i_entry)) * c)) /\ exists ff_q_entry_decoded. b = ff_q_entry_decoded * S ((S (ff_i_entry)) * c) + (a + ff_i_entry)))) -> (exists h. h + S i = l) -> (((exists ff_h_range_entry_x. ff_h_range_entry_x + S (x) = S ((S (i)) * c)) /\ exists ff_q_range_entry_x. b = ff_q_range_entry_x * S ((S (i)) * c) + (x))) -> x = a + i

Proof neighborhood

Direct theorem prerequisites

BT0042 beta_at_unique

Direct theorem dependents

BT0088 beta_range_transport_entry BT0092 factorial_succ_decompose

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

21 script commands · 3 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 (1)
beta at unique
01Fix variables and assumptionsL1–9

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 i
  6. L6
    intro x
  7. L7
    intro hrange
  8. L8
    intro hi
  9. L9
    intro hx
02Establish haL10–19

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply hrange.

  1. L10
    have ha : BetaAt(b,c,i,a + i)Definitions: BetaAt(b,c,i,a + i)Original native command in the exact edition
  2. L11
    specialize hrange i
  3. L12
    apply hrange
  4. L13
    exact hi
  5. L14
    specialize beta_at_unique b
  6. L15
    specialize beta_at_unique c
  7. L16
    specialize beta_at_unique i
  8. L17
    specialize beta_at_unique x
  9. L18
    specialize beta_at_unique (a + i)
  10. L19
    apply beta_at_unique
03Use earlier factsL20–21

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

  1. L20
    exact hx
  2. L21
    exact ha

Library-wide reading audit

Original defined command ledger · 21 lines
  1. 0001intro b
  2. 0002intro c
  3. 0003intro a
  4. 0004intro l
  5. 0005intro i
  6. 0006intro x
  7. 0007intro hrange
  8. 0008intro hi
  9. 0009intro hx
  10. 0010have ha : BetaAt(b,c,i,a + i)
    Exact native replay linehave ha : ((exists h. h + S (a + i) = S ((S i) * c)) /\ exists q. b = q * S ((S i) * c) + (a + i))
  11. 0011specialize hrange i
  12. 0012apply hrange
  13. 0013exact hi
  14. 0014specialize beta_at_unique b
  15. 0015specialize beta_at_unique c
  16. 0016specialize beta_at_unique i
  17. 0017specialize beta_at_unique x
  18. 0018specialize beta_at_unique (a + i)
  19. 0019apply beta_at_unique
  20. 0020exact hx
  21. 0021exact ha