BT002X · Bertrand theorem

coprime_one_right

Stable checked-use theorem · independently kernel verified

Every natural is coprime to one in the expanded common-divisor relation.

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

∀ a. Coprime(a,1)

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

1 occurrences

In local proof propositions

none

0 occurrences

Exact expanded native-PA statement
forall a d. (exists x. a = d * x) -> (exists y. 1 = d * y) -> d = 1

Proof neighborhood

Direct theorem prerequisites

Direct theorem dependents

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

7 script commands · 2 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)
01Fix variables and assumptionsL1–4

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

  1. L1
    intro a
  2. L2
    intro d
  3. L3
    intro ha
  4. L4
    intro h1
02Use earlier factsL5–7

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

  1. L5
    specialize divisor_one d
  2. L6
    apply divisor_one
  3. L7
    exact h1

Library-wide reading audit

Original defined command ledger · 7 lines
  1. 0001intro a
  2. 0002intro d
  3. 0003intro ha
  4. 0004intro h1
  5. 0005specialize divisor_one d
  6. 0006apply divisor_one
  7. 0007exact h1