BT0040 · Bertrand theorem

beta_at_self_of_bound

Stable checked-use theorem · independently kernel verified

A value below a Gödel-beta modulus decodes to itself when used as the code.

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

∀ c. ∀ i. ∀ x. Lt(x,S (S i · c)) → BetaAt(x,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 PD0013 BetaAt

2 occurrences

In local proof propositions

none

0 occurrences

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

Proof neighborhood

Direct theorem prerequisites

BT0004 mul_zero_left BT0000 zero_add

Direct theorem dependents

BT005E beta_prefix_product_trace_exists BT0089 beta_prefix_sum_trace_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

12 script commands · 8 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)
mul zero leftzero add
01Fix variables and assumptionsL1–4

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

  1. L1
    intro c
  2. L2
    intro i
  3. L3
    intro x
  4. L4
    intro hx
02Separate the logical casesL5–5

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

  1. L5
    split
03Use earlier factsL6–6

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

  1. L6
    exact hx
04Construct an explicit witnessL7–7

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

  1. L7
    exists 0
05Use earlier factsL8–8

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

  1. L8
    specialize mul_zero_left (S ((S i) * c))
06Calculate and transport equalitiesL9–9

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

  1. L9
    rewrite mul_zero_left
07Use earlier factsL10–10

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

  1. L10
    specialize zero_add x
08Calculate and transport equalitiesL11–12

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

  1. L11
    rewrite zero_add
  2. L12
    refl

Library-wide reading audit

Original defined command ledger · 12 lines
  1. 0001intro c
  2. 0002intro i
  3. 0003intro x
  4. 0004intro hx
  5. 0005split
  6. 0006exact hx
  7. 0007exists 0
  8. 0008specialize mul_zero_left (S ((S i) * c))
  9. 0009rewrite mul_zero_left
  10. 0010specialize zero_add x
  11. 0011rewrite zero_add
  12. 0012refl