Euclidean domains

This file introduces Euclidean domains and provides the extended Euclidean algorithm. To be precise, a slightly more general version is provided which is sometimes called a transfinite Euclidean domain and differs in the fact that the degree function need not take values in ℕ but can take values in any well-ordered set. Transfinite Euclidean domains were introduced by Motzkin and examples which don't satisfy the classical notion were provided independently by Hiblot and Nagata.

Main definitions

• EuclideanDomain: Defines Euclidean domain with functions quotient and remainder. Instances of Div and Mod are provided, so that one can write a = b * (a / b) + a % b.
• gcd: defines the greatest common divisors of two elements of a Euclidean domain.
• xgcd: given two elements a b : R, xgcd a b defines the pair (x, y) such that x * a + y * b = gcd a b.
• lcm: defines the lowest common multiple of two elements a and b of a Euclidean domain as a * b / (gcd a b)

Main statements

See Algebra.EuclideanDomain.Basic for most of the theorems about Eucliean domains, including Bézout's lemma.

See Algebra.EuclideanDomain.Instances for the facts that ℤ is a Euclidean domain, as is any field.

Notation

≺≺ denotes the well founded relation on the Euclidean domain, e.g. in the example of the polynomial ring over a field, p ≺ q≺ q for polynomials p and q if and only if the degree of p is less than the degree of q.

Implementation details

Instead of working with a valuation, EuclideanDomain is implemented with the existence of a well founded relation r on the integral domain R, which in the example of ℤ would correspond to setting i ≺ j≺ j for integers i and j if the absolute value of i is smaller than the absolute value of j.

References

• [Th. Motzkin, The Euclidean algorithm][MR32592]
• [J.-J. Hiblot, Des anneaux euclidiens dont le plus petit algorithme n'est pas à valeurs finies] [MR399081]
• [M. Nagata, On Euclid algorithm][MR541021]

Tags

Euclidean domain, transfinite Euclidean domain, Bézout's lemma

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class EuclideanDomain (R : Type u) extends CommRing , Nontrivial : Type u

• A division function (denoted /) on R. This satisfies the property b * (a / b) + a % b = a, where % denotes remainder.

  quotient : R → R → R

• Division by zero should always give zero by convention.

  quotient_zero : ∀ (a : R), quotient a 0 = 0

• A remainder function (denoted %) on R. This satisfies the property b * (a / b) + a % b = a, where / denotes quotient.

  remainder : R → R → R

• The property that links the quotient and remainder functions. This allows us to compute GCDs and LCMs.

  quotient_mul_add_remainder_eq : ∀ (a b : R), b * quotient a b + remainder a b = a

• A well-founded relation on R, satisfying r (a % b) b. This ensures that the GCD algorithm always terminates.

  r : R → R → Prop

• The relation r must be well-founded. This ensures that the GCD algorithm always terminates.

  r_wellFounded : WellFounded r

• The relation r satisfies r (a % b) b.

  remainder_lt : (a : R) → {b : R} → b ≠ 0 → r (remainder a b) b

• An additional constraint on r.

  mul_left_not_lt : ∀ (a : R) {b : R}, b ≠ 0 → ¬r (a * b) a

A EuclideanDomain is an non-trivial commutative ring with a division and a remainder, satisfying b * (a / b) + a % b = a. The definition of a Euclidean domain usually includes a valuation function R → ℕ→ ℕ. This definition is slightly generalised to include a well founded relation r with the property that r (a % b) b, instead of a valuation.

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instance EuclideanDomain.instDiv {R : Type u} [inst : EuclideanDomain R] : Div R

Equations

• EuclideanDomain.instDiv = { div := EuclideanDomain.quotient }

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instance EuclideanDomain.instMod {R : Type u} [inst : EuclideanDomain R] : Mod R

Equations

• EuclideanDomain.instMod = { mod := EuclideanDomain.remainder }

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theorem EuclideanDomain.div_add_mod {R : Type u} [inst : EuclideanDomain R] (a : R) (b : R) : b * (a / b) + a % b = a

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theorem EuclideanDomain.mod_add_div {R : Type u} [inst : EuclideanDomain R] (a : R) (b : R) : a % b + b * (a / b) = a

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theorem EuclideanDomain.mod_add_div' {R : Type u} [inst : EuclideanDomain R] (m : R) (k : R) : m % k + m / k * k = m

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theorem EuclideanDomain.div_add_mod' {R : Type u} [inst : EuclideanDomain R] (m : R) (k : R) : m / k * k + m % k = m

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theorem EuclideanDomain.mod_eq_sub_mul_div {R : Type u_1} [inst : EuclideanDomain R] (a : R) (b : R) : a % b = a - b * (a / b)

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theorem EuclideanDomain.mod_lt {R : Type u} [inst : EuclideanDomain R] (a : R) {b : R} : b ≠ 0 → EuclideanDomain.r (a % b) b

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theorem EuclideanDomain.mul_right_not_lt {R : Type u} [inst : EuclideanDomain R] {a : R} (b : R) (h : a ≠ 0) : ¬EuclideanDomain.r (a * b) b

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@[simp]

theorem EuclideanDomain.mod_zero {R : Type u} [inst : EuclideanDomain R] (a : R) : a % 0 = a

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theorem EuclideanDomain.lt_one {R : Type u} [inst : EuclideanDomain R] (a : R) : EuclideanDomain.r a 1 → a = 0

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theorem EuclideanDomain.val_dvd_le {R : Type u} [inst : EuclideanDomain R] (a : R) (b : R) : b ∣ a → a ≠ 0 → ¬EuclideanDomain.r a b

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theorem EuclideanDomain.div_zero {R : Type u} [inst : EuclideanDomain R] (a : R) : a / 0 = 0

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theorem EuclideanDomain.GCD.induction {R : Type u} [inst : EuclideanDomain R] {P : R → R → Prop} (a : R) (b : R) (H0 : (x : R) → P 0 x) (H1 : (a b : R) → a ≠ 0 → P (b % a) a → P a b) : P a b

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def EuclideanDomain.gcd {R : Type u} [inst : EuclideanDomain R] [inst : DecidableEq R] : R → R → R

gcd a b is a (non-unique) element such that gcd a b ∣ a∣ a gcd a b ∣ b∣ b, and for any element c such that c ∣ a∣ a and c ∣ b∣ b, then c ∣ gcd a b∣ gcd a b

Equations

• EuclideanDomain.gcd x = fun b => if a0 : x = 0 then b else let_fun x := (_ : EuclideanDomain.r (b % x) x); EuclideanDomain.gcd (b % x) x

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theorem EuclideanDomain.gcd_zero_left {R : Type u} [inst : EuclideanDomain R] [inst : DecidableEq R] (a : R) : EuclideanDomain.gcd 0 a = a

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def EuclideanDomain.xgcdAux {R : Type u} [inst : EuclideanDomain R] [inst : DecidableEq R] : R → R → R → R → R → R → R × R × R

An implementation of the extended GCD algorithm. At each step we are computing a triple (r, s, t), where r is the next value of the GCD algorithm, to compute the greatest common divisor of the input (say x and y), and s and t are the coefficients in front of x and y to obtain r (i.e. r = s * x + t * y). The function xgcdAux takes in two triples, and from these recursively computes the next triple:

xgcdAux (r, s, t) (r', s', t') = xgcdAux (r' % r, s' - (r' / r) * s, t' - (r' / r) * t) (r, s, t)

Equations

• EuclideanDomain.xgcdAux x = fun s t r' s' t' => if _hr : x = 0 then (r', s', t') else let q := r' / x; EuclideanDomain.xgcdAux (r' % x) (s' - q * s) (t' - q * t) x s t

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theorem EuclideanDomain.xgcd_zero_left {R : Type u} [inst : EuclideanDomain R] [inst : DecidableEq R] {s : R} {t : R} {r' : R} {s' : R} {t' : R} : EuclideanDomain.xgcdAux 0 s t r' s' t' = (r', s', t')

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theorem EuclideanDomain.xgcdAux_rec {R : Type u} [inst : EuclideanDomain R] [inst : DecidableEq R] {r : R} {s : R} {t : R} {r' : R} {s' : R} {t' : R} (h : r ≠ 0) : EuclideanDomain.xgcdAux r s t r' s' t' = EuclideanDomain.xgcdAux (r' % r) (s' - r' / r * s) (t' - r' / r * t) r s t

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def EuclideanDomain.xgcd {R : Type u} [inst : EuclideanDomain R] [inst : DecidableEq R] (x : R) (y : R) : R × R

Use the extended GCD algorithm to generate the a and b values satisfying gcd x y = x * a + y * b.

Equations

• EuclideanDomain.xgcd x y = (EuclideanDomain.xgcdAux x 1 0 y 0 1).snd

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def EuclideanDomain.gcdA {R : Type u} [inst : EuclideanDomain R] [inst : DecidableEq R] (x : R) (y : R) : R

The extended GCD a value in the equation gcd x y = x * a + y * b.

Equations

• EuclideanDomain.gcdA x y = (EuclideanDomain.xgcd x y).fst

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def EuclideanDomain.gcdB {R : Type u} [inst : EuclideanDomain R] [inst : DecidableEq R] (x : R) (y : R) : R

The extended GCD b value in the equation gcd x y = x * a + y * b.

Equations

• EuclideanDomain.gcdB x y = (EuclideanDomain.xgcd x y).snd

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theorem EuclideanDomain.gcdA_zero_left {R : Type u} [inst : EuclideanDomain R] [inst : DecidableEq R] {s : R} : EuclideanDomain.gcdA 0 s = 0

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theorem EuclideanDomain.gcdB_zero_left {R : Type u} [inst : EuclideanDomain R] [inst : DecidableEq R] {s : R} : EuclideanDomain.gcdB 0 s = 1

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theorem EuclideanDomain.xgcd_val {R : Type u} [inst : EuclideanDomain R] [inst : DecidableEq R] (x : R) (y : R) : EuclideanDomain.xgcd x y = (EuclideanDomain.gcdA x y, EuclideanDomain.gcdB x y)

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def EuclideanDomain.lcm {R : Type u} [inst : EuclideanDomain R] [inst : DecidableEq R] (x : R) (y : R) : R

lcm a b is a (non-unique) element such that a ∣ lcm a b∣ lcm a b b ∣ lcm a b∣ lcm a b, and for any element c such that a ∣ c∣ c and b ∣ c∣ c, then lcm a b ∣ c∣ c

Equations

• EuclideanDomain.lcm x y = x * y / EuclideanDomain.gcd x y