BT0045 · Bertrand theorem

beta_at_of_mod_eq_bound

Stable checked-use theorem · independently kernel verified

A bounded value congruent to a code is its expanded Gödel-beta value.

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. ∀ i. ∀ x. Lt(x,S (S i · c)) → ModEq(S (S i · c),b,x) → BetaAt(b,c,i,x)

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 PD0008 ModEq PD0013 BetaAt

3 occurrences

In local proof propositions

none

0 occurrences

Exact expanded native-PA statement
forall b c i x. (exists h. h + S x = S ((S i) * c)) -> (exists u v. b + S ((S i) * c) * u = x + S ((S i) * c) * v) -> ((exists h. h + S x = S ((S i) * c)) /\ exists q. b = q * S ((S i) * c) + x)

Proof neighborhood

Direct theorem prerequisites

BT003Y beta_modulus_nonzero BT003X mod_eq_to_remainder_decomposition

Direct theorem dependents

BT005D beta_prefix_extend

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

17 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.

Named ingredients (2)
beta modulus nonzeromod eq to remainder decomposition
01Fix variables and assumptionsL1–6

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

  1. L1
    intro b
  2. L2
    intro c
  3. L3
    intro i
  4. L4
    intro x
  5. L5
    intro hx
  6. L6
    intro hmod
02Separate the logical casesL7–7

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

  1. L7
    split
03Use earlier factsL8–17

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

  1. L8
    exact hx
  2. L9
    specialize mod_eq_to_remainder_decomposition (S ((S i) * c))
  3. L10
    specialize mod_eq_to_remainder_decomposition b
  4. L11
    specialize mod_eq_to_remainder_decomposition x
  5. L12
    apply mod_eq_to_remainder_decomposition
  6. L13
    specialize beta_modulus_nonzero c
  7. L14
    specialize beta_modulus_nonzero i
  8. L15
    exact beta_modulus_nonzero
  9. L16
    exact hx
  10. L17
    exact hmod

Library-wide reading audit

Original defined command ledger · 17 lines
  1. 0001intro b
  2. 0002intro c
  3. 0003intro i
  4. 0004intro x
  5. 0005intro hx
  6. 0006intro hmod
  7. 0007split
  8. 0008exact hx
  9. 0009specialize mod_eq_to_remainder_decomposition (S ((S i) * c))
  10. 0010specialize mod_eq_to_remainder_decomposition b
  11. 0011specialize mod_eq_to_remainder_decomposition x
  12. 0012apply mod_eq_to_remainder_decomposition
  13. 0013specialize beta_modulus_nonzero c
  14. 0014specialize beta_modulus_nonzero i
  15. 0015exact beta_modulus_nonzero
  16. 0016exact hx
  17. 0017exact hmod