BT0035 · Bertrand theorem

gcd_balanced_bezout_exists

Stable checked-use theorem · independently kernel verified

Every pair has a relational gcd together with balanced natural Bezout witnesses.

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) ∧ (∃ x. ∃ y. ∃ z. ∃ n. a · x + b · y = d + (a · z + b · n))

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

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) /\ exists xp yp xn yn. a * xp + b * yp = d + (a * xn + b * yn))

Proof neighborhood

Direct theorem prerequisites

BT000E le_refl BT0034 gcd_balanced_bezout_exists_up_to

Direct theorem dependents

BT0038 coprime_balanced_bezout

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

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 balanced bezout 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_balanced_bezout_exists_up_to b
  2. L4
    specialize gcd_balanced_bezout_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 balanced bezout exists up to.

  1. L7
    have hall : ∀ z. ∃ d. IsGCD(d,z,b) ∧ (∃ x. ∃ y. ∃ n. ∃ m. z · x + b · y = d + (z · n + b · m))Definitions: IsGCD(d,z,b)Original native command in the exact edition
  2. L8
    apply gcd_balanced_bezout_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_balanced_bezout_exists_up_to b
  4. 0004specialize gcd_balanced_bezout_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) ∧ (∃ x. ∃ y. ∃ n. ∃ m. z · x + b · y = d + (z · n + b · m))
    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) /\ exists xp yp xn yn. z * xp + b * yp = d + (z * xn + b * yn))
  8. 0008apply gcd_balanced_bezout_exists_up_to
  9. 0009exact hbb
  10. 0010specialize hall a
  11. 0011exact hall