PA0036

gcd_exists_relational

Stable checked-use theorem · independently closed

Every pair of naturals has a relational greatest common divisor.

Exact expanded PA statement

forall a b. exists d. (((exists x. a = d * x) /\ (exists y. b = d * y)) /\ forall c. (exists u. a = c * u) -> (exists v. b = c * v) -> exists w. d = c * w)

Structural proof guide

Generated structural guide

Every pair of naturals has a relational greatest common divisor.

Use the direct prerequisites le_refl, gcd_exists_up_to as previously established PA formulas.

The proof proceeds by intermediate claims (2).

Referenced ingredients

PA001A le_refl PA0035 gcd_exists_up_to

Proof neighborhood

Direct dependencies

PA001A le_refl PA0035 gcd_exists_up_to

Direct dependents

PA0038 euclid_prime_dvd_product PA003M prime_coprime_or_divides

Formal native tactic body

Dependencies are introduced as named hypotheses before line 1. Linked names are exact direct references. This Stable checked-use theorem is independently kernel-checked when replayed.

  1. 0001intro a
  2. 0002intro b
  3. 0003specialize gcd_exists_up_to b
  4. 0004specialize gcd_exists_up_to b
  5. 0005have hbb : exists t. t + b = b
  6. 0006apply le_refl
  7. 0007have hall : forall z. exists d. (((exists x. z = d * x) /\ (exists y. b = d * y)) /\ forall c. (exists u. z = c * u) -> (exists v. b = c * v) -> exists w. d = c * w)
  8. 0008apply gcd_exists_up_to
  9. 0009exact hbb
  10. 0010specialize hall a
  11. 0011exact hall