BT0038 · Bertrand theorem

coprime_balanced_bezout

Stable checked-use theorem · independently kernel verified

Coprime inputs admit balanced natural Bezout coefficients with result one.

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. Coprime(a,b) → ∃ x. ∃ y. ∃ z. ∃ n. a · x + b · y = 1 + (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

PD0005 Coprime

1 occurrences

In local proof propositions

PD0006 IsGCD

1 occurrences

Exact expanded native-PA statement
forall a b. (forall d. (exists x. a = d * x) -> (exists y. b = d * y) -> d = 1) -> exists xp yp xn yn. a * xp + b * yp = 1 + (a * xn + b * yn)

Proof neighborhood

Direct theorem prerequisites

BT0035 gcd_balanced_bezout_exists

Direct theorem dependents

BT0039 gauss_coprime_cancel BT004F binary_crt

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

24 script commands · 8 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 (1)
gcd balanced bezout exists
01Fix variables and assumptionsL1–3

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

  1. L1
    intro a
  2. L2
    intro b
  3. L3
    intro hcop
02Establish hgbL4–5

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

  1. L4
    have hgb : ∃ d. IsGCD(d,a,b) ∧ (∃ x. ∃ y. ∃ z. ∃ n. a · x + b · y = d + (a · z + b · n))Definitions: IsGCD(d,a,b)Original native command in the exact edition
  2. L5
    apply gcd_balanced_bezout_exists
03Separate the logical casesL6–9

Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.

  1. L6
    cases hgb
  2. L7
    cases hgb_witness
  3. L8
    cases hgb_witness_left
  4. L9
    cases hgb_witness_left_left
04Establish hdL10–14

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

  1. L10
    have hd : x = 1
  2. L11
    specialize hcop x
  3. L12
    apply hcop
  4. L13
    exact hgb_witness_left_left_left
  5. L14
    exact hgb_witness_left_left_right
05Separate the logical casesL15–18

Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.

  1. L15
    cases hgb_witness_right
  2. L16
    cases hgb_witness_right_witness
  3. L17
    cases hgb_witness_right_witness_witness
  4. L18
    cases hgb_witness_right_witness_witness_witness
06Construct an explicit witnessL19–22

Supply the displayed value, then prove that it has the required property.

  1. L19
    exists x1
  2. L20
    exists x2
  3. L21
    exists x3
  4. L22
    exists x4
07Calculate and transport equalitiesL23–23

Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.

  1. L23
    rewrite hd at hgb_witness_right_witness_witness_witness_witness
08Use earlier factsL24–24

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

  1. L24
    exact hgb_witness_right_witness_witness_witness_witness

Library-wide reading audit

Original defined command ledger · 24 lines
  1. 0001intro a
  2. 0002intro b
  3. 0003intro hcop
  4. 0004have hgb : ∃ d. IsGCD(d,a,b) ∧ (∃ x. ∃ y. ∃ z. ∃ n. a · x + b · y = d + (a · z + b · n))
    Exact native replay linehave hgb : 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))
  5. 0005apply gcd_balanced_bezout_exists
  6. 0006cases hgb
  7. 0007cases hgb_witness
  8. 0008cases hgb_witness_left
  9. 0009cases hgb_witness_left_left
  10. 0010have hd : x = 1
  11. 0011specialize hcop x
  12. 0012apply hcop
  13. 0013exact hgb_witness_left_left_left
  14. 0014exact hgb_witness_left_left_right
  15. 0015cases hgb_witness_right
  16. 0016cases hgb_witness_right_witness
  17. 0017cases hgb_witness_right_witness_witness
  18. 0018cases hgb_witness_right_witness_witness_witness
  19. 0019exists x1
  20. 0020exists x2
  21. 0021exists x3
  22. 0022exists x4
  23. 0023rewrite hd at hgb_witness_right_witness_witness_witness_witness
  24. 0024exact hgb_witness_right_witness_witness_witness_witness