Basics for the Rational Numbers #

Summary #

We define the integral domain structure on ℚ and prove basic lemmas about it. The definition of the field structure on ℚ will be done in Mathlib.Data.Rat.Basic once the Field class has been defined.

Main Definitions #

Rat.divInt n d constructs a rational number q = n / d from n d : ℤ.

Notations #

/. is infix notation for Rat.divInt.

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theorem Rat.pos (a : ℚ) :

0 < a.den

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@[simp]

theorem Rat.ofInt_eq_cast (n : ℤ) :

Rat.ofInt n = ↑n

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@[simp]

theorem Rat.coe_int_num (n : ℤ) :

(↑n).num = n

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@[simp]

theorem Rat.coe_int_den (n : ℤ) :

(↑n).den = 1

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theorem Rat.mkRat_eq (n : ℤ) (d : ℕ) :

mkRat n d = Rat.divInt n ↑d

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@[simp]

theorem Rat.zero_mk (d : ℕ) (h : d ≠ 0) (w : Nat.coprime (Int.natAbs 0) d) :

Rat.mk' 0 d = 0

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@[simp]

theorem Rat.divInt_eq_zero {a : ℤ} {b : ℤ} (b0 : b ≠ 0) :

Rat.divInt a b = 0 ↔ a = 0

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theorem Rat.divInt_ne_zero {a : ℤ} {b : ℤ} (b0 : b ≠ 0) :

Rat.divInt a b ≠ 0 ↔ a ≠ 0

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theorem Rat.normalize_eq_mk' (n : ℤ) (d : ℕ) (h : d ≠ 0) (c : Nat.gcd (Int.natAbs n) d = 1) :

Rat.normalize n d = Rat.mk' n d

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theorem Rat.num_den {a : ℚ} :

Rat.divInt a.num ↑a.den = a

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theorem Rat.num_den' {n : ℤ} {d : ℕ} {h : d ≠ 0} {c : Nat.coprime (Int.natAbs n) d} :

Rat.mk' n d = Rat.divInt n ↑d

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theorem Rat.coe_int_eq_divInt (z : ℤ) :

↑z = Rat.divInt z 1

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def Rat.numDenCasesOn {C : ℚ → Sort u} (a : ℚ) :

((n : ℤ) → (d : ℕ) → 0 < d → Nat.coprime (Int.natAbs n) d → C (Rat.divInt n ↑d)) → C a

Define a (dependent) function or prove ∀ r : ℚ, p r∀ r : ℚ, p r by dealing with rational numbers of the form n /. d with 0 < d and coprime n, d.

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Rat.numDenCasesOn x x = match x, x with | Rat.mk' n d, H => Eq.mpr (_ : C (Rat.mk' n d) = C (Rat.divInt n ↑d)) (H n d (_ : 0 < d) c)

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def Rat.numDenCasesOn' {C : ℚ → Sort u} (a : ℚ) (H : (n : ℤ) → (d : ℕ) → d ≠ 0 → C (Rat.divInt n ↑d)) :

C a

Define a (dependent) function or prove ∀ r : ℚ, p r∀ r : ℚ, p r by dealing with rational numbers of the form n /. d with d ≠ 0≠ 0.

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Rat.numDenCasesOn' a H = Rat.numDenCasesOn a fun n d h x => H n d (_ : d ≠ 0)

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theorem Rat.lift_binop_eq (f : ℚ → ℚ → ℚ) (f₁ : ℤ → ℤ → ℤ → ℤ → ℤ) (f₂ : ℤ → ℤ → ℤ → ℤ → ℤ) (fv : ∀ {n₁ : ℤ} {d₁ : ℕ} {h₁ : d₁ ≠ 0} {c₁ : Nat.coprime (Int.natAbs n₁) d₁} {n₂ : ℤ} {d₂ : ℕ} {h₂ : d₂ ≠ 0} {c₂ : Nat.coprime (Int.natAbs n₂) d₂}, f (Rat.mk' n₁ d₁) (Rat.mk' n₂ d₂) = Rat.divInt (f₁ n₁ (↑d₁) n₂ ↑d₂) (f₂ n₁ (↑d₁) n₂ ↑d₂)) (f0 : ∀ {n₁ d₁ n₂ d₂ : ℤ}, d₁ ≠ 0 → d₂ ≠ 0 → f₂ n₁ d₁ n₂ d₂ ≠ 0) (a : ℤ) (b : ℤ) (c : ℤ) (d : ℤ) (b0 : b ≠ 0) (d0 : d ≠ 0) (H : ∀ {n₁ d₁ n₂ d₂ : ℤ}, a * d₁ = n₁ * b → c * d₂ = n₂ * d → f₁ n₁ d₁ n₂ d₂ * f₂ a b c d = f₁ a b c d * f₂ n₁ d₁ n₂ d₂) :

f (Rat.divInt a b) (Rat.divInt c d) = Rat.divInt (f₁ a b c d) (f₂ a b c d)

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@[simp]

theorem Rat.add_def'' {a : ℤ} {b : ℤ} {c : ℤ} {d : ℤ} (b0 : b ≠ 0) (d0 : d ≠ 0) :

Rat.divInt a b + Rat.divInt c d = Rat.divInt (a * d + c * b) (b * d)

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@[simp]

theorem Rat.neg_def {a : ℤ} {b : ℤ} :

-Rat.divInt a b = Rat.divInt (-a) b

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theorem Rat.divInt_neg_den (n : ℤ) (d : ℤ) :

Rat.divInt n (-d) = Rat.divInt (-n) d

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@[simp]

theorem Rat.sub_def'' {a : ℤ} {b : ℤ} {c : ℤ} {d : ℤ} (b0 : b ≠ 0) (d0 : d ≠ 0) :

Rat.divInt a b - Rat.divInt c d = Rat.divInt (a * d - c * b) (b * d)

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@[simp]

theorem Rat.mul_def' {a : ℤ} {b : ℤ} {c : ℤ} {d : ℤ} (b0 : b ≠ 0) (d0 : d ≠ 0) :

Rat.divInt a b * Rat.divInt c d = Rat.divInt (a * c) (b * d)

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instance Rat.instInvRat :

Inv ℚ

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Rat.instInvRat = { inv := Rat.inv }

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@[simp]

theorem Rat.inv_def' {a : ℤ} {b : ℤ} :

(Rat.divInt a b)⁻¹ = Rat.divInt b a

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theorem Rat.add_zero (a : ℚ) :

a + 0 = a

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theorem Rat.zero_add (a : ℚ) :

0 + a = a

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theorem Rat.add_comm (a : ℚ) (b : ℚ) :

a + b = b + a

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theorem Rat.add_assoc (a : ℚ) (b : ℚ) (c : ℚ) :

a + b + c = a + (b + c)

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theorem Rat.add_left_neg (a : ℚ) :

-a + a = 0

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theorem Rat.divInt_zero_one :

Rat.divInt 0 1 = 0

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theorem Rat.divInt_one_one :

Rat.divInt 1 1 = 1

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@[simp]

theorem Rat.divInt_neg_one_one :

Rat.divInt (-1) 1 = -1

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theorem Rat.mul_assoc (a : ℚ) (b : ℚ) (c : ℚ) :

a * b * c = a * (b * c)

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theorem Rat.add_mul (a : ℚ) (b : ℚ) (c : ℚ) :

(a + b) * c = a * c + b * c

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theorem Rat.mul_add (a : ℚ) (b : ℚ) (c : ℚ) :

a * (b + c) = a * b + a * c

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theorem Rat.zero_ne_one :

0 ≠ 1

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theorem Rat.mul_inv_cancel (a : ℚ) :

a ≠ 0 → a * a⁻¹ = 1

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theorem Rat.inv_mul_cancel (a : ℚ) (h : a ≠ 0) :

a⁻¹ * a = 1

At this point in the import hierarchy we have not defined the Field typeclass. Instead we'll instantiate CommRing and CommGroupWithZero at this point. The Rat.field instance and any field-specific lemmas can be found in Mathlib.Data.Rat.Basic.

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instance Rat.commRing :

CommRing ℚ

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