BT002Y

coprime_one_left

Stable checked-use theorem · independently kernel verified

One is coprime to every natural 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.

Exact expanded PA statement

forall a d. (exists x. 1 = d * x) -> (exists y. a = d * y) -> d = 1

Structural proof guide

One is coprime to every natural in the expanded common-divisor relation.

Direct prerequisites: divisor_one. The authored body proceeds by direct introduction and elimination.

Proof neighborhood

Direct dependencies

BT002E divisor_one

Direct dependents

BT005C bounded_beta_exclusive_recode_invariant BT00DH beta_product_pointwise_coprime BT00VE primorial_interval_pairwise_coprime BT00YV coprime_powers BT00YW prime_contribution_prefix_pairwise_coprime

Formal native tactic body

Dependencies are hypotheses of the historical Alpha-v12 body receipt. The complete historical Alpha-v18 proof bundle independently checks every dependency; current Alpha v25 preserves that checked theorem use without changing Stable membership.

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.

Named ingredients (1)
divisor one
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 h1
  4. L4
    intro ha
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 exact command ledger · 7 lines
  1. 0001intro a
  2. 0002intro d
  3. 0003intro h1
  4. 0004intro ha
  5. 0005specialize divisor_one d
  6. 0006apply divisor_one
  7. 0007exact h1