PA0036 · theorem

gcd_exists_relational

Stable checked-use theorem · independently closed

Every pair of naturals has a relational greatest common divisor.

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. ∀ b. ∃ d. IsGCD(d,a,b)

Every purple notation token opens its conservative definition. This is a reading surface; the compiler expands the statement before the unchanged kernel checks it.

Definitions used by this theorem

In the theorem statement

PD0006 IsGCD

1 occurrences

In local proof propositions

PD0001 Le PD0006 IsGCD

2 occurrences

Exact expanded native-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)

Proof neighborhood

Direct theorem prerequisites

PA001A le_refl PA0035 gcd_exists_up_to

Direct theorem dependents

PA0038 euclid_prime_dvd_product PA003M prime_coprime_or_divides

Definition-aware tactic body

Only local propositions introduced by have or suffices are compacted. The untrusted compiler re-expands each one before the original tactic script is replayed; defined notation is never accepted by the kernel. Open the exact replay line beneath every changed command.

Read the argument

Proof checkpoints

11 script commands · 4 reading checkpoints · 2 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)
le reflgcd exists up to
01Fix variables and assumptionsL1–2

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

  1. L1
    intro a
  2. L2
    intro b
02Use earlier factsL3–4

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

  1. L3
    specialize gcd_exists_up_to b
  2. L4
    specialize gcd_exists_up_to b
03Establish hbbL5–6

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply le refl.

  1. L5
  2. L6
    apply le_refl
04Establish hallL7–11

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply gcd exists up to.

  1. L7
    have hall : ∀ z. ∃ d. IsGCD(d,z,b)Definitions: IsGCD(d,z,b)Original native command in the exact edition
  2. L8
    apply gcd_exists_up_to
  3. L9
    exact hbb
  4. L10
    specialize hall a
  5. L11
    exact hall

Library-wide reading audit

Original defined command ledger · 11 lines
  1. 0001intro a
  2. 0002intro b
  3. 0003specialize gcd_exists_up_to b
  4. 0004specialize gcd_exists_up_to b
  5. 0005have hbb : Le(b,b)
    Exact native replay linehave hbb : exists t. t + b = b
  6. 0006apply le_refl
  7. 0007have hall : ∀ z. ∃ d. IsGCD(d,z,b)
    Exact native replay linehave 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