ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- WeierstrassCurve.Jacobian.map_smul
protected lemma map_smul (u : R) : f ∘ (u • P) = f u • (f ∘ P) 
Arguments: {R : Type u} [CommRing R] {S : Type u_1} [CommRing S] (f : R →+* S) (P : Fin 3 → R) (u : R)
Goal: ⇑f ∘ (u • P) = f u • ⇑f ∘ P
-- WeierstrassCurve.Jacobian.map_addZ
@[simp] lemma map_addZ : addZ (f ∘ P) (f ∘ Q) = f (addZ P Q) 
Arguments: {R : Type u} [CommRing R] {S : Type u_1} [CommRing S] (f : R →+* S) (P Q : Fin 3 → R)
Goal: WeierstrassCurve.Jacobian.addZ (⇑f ∘ P) (⇑f ∘ Q) = f (WeierstrassCurve.Jacobian.addZ P Q)
-- WeierstrassCurve.Jacobian.map_addX
@[simp] lemma map_addX : addX (W'.map f) (f ∘ P) (f ∘ Q) = f (W'.addX P Q) 
Arguments: {R : Type u} [CommRing R] {W' : WeierstrassCurve.Jacobian R} {S : Type u_1} [CommRing S] (f : R →+* S) (P Q : Fin 3 → R)
Goal: WeierstrassCurve.Jacobian.addX (WeierstrassCurve.map W' f) (⇑f ∘ P) (⇑f ∘ Q) = f (W'.addX P Q)
-- WeierstrassCurve.Jacobian.map_negAddY
@[simp] lemma map_negAddY : negAddY (W'.map f) (f ∘ P) (f ∘ Q) = f (W'.negAddY P Q) 
Arguments: {R : Type u} [CommRing R] {W' : WeierstrassCurve.Jacobian R} {S : Type u_1} [CommRing S] (f : R →+* S) (P Q : Fin 3 → R)
Goal: WeierstrassCurve.Jacobian.negAddY (WeierstrassCurve.map W' f) (⇑f ∘ P) (⇑f ∘ Q) = f (W'.negAddY P Q)
-- WeierstrassCurve.Jacobian.map_negY
@[simp] lemma map_negY : negY (W'.map f) (f ∘ P) = f (W'.negY P) 
Arguments: {R : Type u} [CommRing R] {W' : WeierstrassCurve.Jacobian R} {S : Type u_1} [CommRing S] (f : R →+* S) (P : Fin 3 → R)
Goal: WeierstrassCurve.Jacobian.negY (WeierstrassCurve.map W' f) (⇑f ∘ P) = f (W'.negY P)
-- WeierstrassCurve.Jacobian.map_neg
@[simp] protected lemma map_neg : neg (W'.map f) (f ∘ P) = f ∘ W'.neg P 
Arguments: {R : Type u} [CommRing R] {W' : WeierstrassCurve.Jacobian R} {S : Type u_1} [CommRing S] (f : R →+* S) (P : Fin 3 → R)
Goal: WeierstrassCurve.Jacobian.neg (WeierstrassCurve.map W' f) (⇑f ∘ P) = ⇑f ∘ W'.neg P
-- WeierstrassCurve.Jacobian.map_addY
@[simp] lemma map_addY : addY (W'.map f) (f ∘ P) (f ∘ Q) = f (W'.addY P Q) 
Arguments: {R : Type u} [CommRing R] {W' : WeierstrassCurve.Jacobian R} {S : Type u_1} [CommRing S] (f : R →+* S) (P Q : Fin 3 → R)
Goal: WeierstrassCurve.Jacobian.addY (WeierstrassCurve.map W' f) (⇑f ∘ P) (⇑f ∘ Q) = f (W'.addY P Q)
-- WeierstrassCurve.Jacobian.map_addXYZ
@[simp] lemma map_addXYZ : addXYZ (W'.map f) (f ∘ P) (f ∘ Q) = f ∘ addXYZ W' P Q 
Arguments: {R : Type u} [CommRing R] {W' : WeierstrassCurve.Jacobian R} {S : Type u_1} [CommRing S] (f : R →+* S) (P Q : Fin 3 → R)
Goal: WeierstrassCurve.Jacobian.addXYZ (WeierstrassCurve.map W' f) (⇑f ∘ P) (⇑f ∘ Q) = ⇑f ∘ W'.addXYZ P Q
-- WeierstrassCurve.Jacobian.map_polynomial
lemma map_polynomial : (W'.map f).toJacobian.polynomial = MvPolynomial.map f W'.polynomial 
Arguments: {R : Type u} [CommRing R] {W' : WeierstrassCurve.Jacobian R} {S : Type u_1} [CommRing S] (f : R →+* S)
Goal: (WeierstrassCurve.map W' f).toJacobian.polynomial = (MvPolynomial.map f) W'.polynomial
-- WeierstrassCurve.Jacobian.map_polynomialX
lemma map_polynomialX : (W'.map f).toJacobian.polynomialX = MvPolynomial.map f W'.polynomialX 
Arguments: {R : Type u} [CommRing R] {W' : WeierstrassCurve.Jacobian R} {S : Type u_1} [CommRing S] (f : R →+* S)
Goal: (WeierstrassCurve.map W' f).toJacobian.polynomialX = (MvPolynomial.map f) W'.polynomialX
-- WeierstrassCurve.Jacobian.map_polynomialY
lemma map_polynomialY : (W'.map f).toJacobian.polynomialY = MvPolynomial.map f W'.polynomialY 
Arguments: {R : Type u} [CommRing R] {W' : WeierstrassCurve.Jacobian R} {S : Type u_1} [CommRing S] (f : R →+* S)
Goal: (WeierstrassCurve.map W' f).toJacobian.polynomialY = (MvPolynomial.map f) W'.polynomialY
-- WeierstrassCurve.Jacobian.map_polynomialZ
lemma map_polynomialZ : (W'.map f).toJacobian.polynomialZ = MvPolynomial.map f W'.polynomialZ 
Arguments: {R : Type u} [CommRing R] {W' : WeierstrassCurve.Jacobian R} {S : Type u_1} [CommRing S] (f : R →+* S)
Goal: (WeierstrassCurve.map W' f).toJacobian.polynomialZ = (MvPolynomial.map f) W'.polynomialZ
-- WeierstrassCurve.Jacobian.map_dblZ
@[simp] lemma map_dblZ : dblZ (W'.map f) (f ∘ P) = f (W'.dblZ P) 
Arguments: {R : Type u} [CommRing R] {W' : WeierstrassCurve.Jacobian R} {S : Type u_1} [CommRing S] (f : R →+* S) (P : Fin 3 → R)
Goal: WeierstrassCurve.Jacobian.dblZ (WeierstrassCurve.map W' f) (⇑f ∘ P) = f (W'.dblZ P)
-- WeierstrassCurve.Jacobian.map_dblU
@[simp] lemma map_dblU : dblU (W'.map f) (f ∘ P) = f (W'.dblU P) 
Arguments: {R : Type u} [CommRing R] {W' : WeierstrassCurve.Jacobian R} {S : Type u_1} [CommRing S] (f : R →+* S) (P : Fin 3 → R)
Goal: WeierstrassCurve.Jacobian.dblU (WeierstrassCurve.map W' f) (⇑f ∘ P) = f (W'.dblU P)
-- WeierstrassCurve.Jacobian.map_dblX
@[simp] lemma map_dblX : dblX (W'.map f) (f ∘ P) = f (W'.dblX P) 
Arguments: {R : Type u} [CommRing R] {W' : WeierstrassCurve.Jacobian R} {S : Type u_1} [CommRing S] (f : R →+* S) (P : Fin 3 → R)
Goal: WeierstrassCurve.Jacobian.dblX (WeierstrassCurve.map W' f) (⇑f ∘ P) = f (W'.dblX P)
-- WeierstrassCurve.Jacobian.map_negDblY
@[simp] lemma map_negDblY : negDblY (W'.map f) (f ∘ P) = f (W'.negDblY P) 
Arguments: {R : Type u} [CommRing R] {W' : WeierstrassCurve.Jacobian R} {S : Type u_1} [CommRing S] (f : R →+* S) (P : Fin 3 → R)
Goal: WeierstrassCurve.Jacobian.negDblY (WeierstrassCurve.map W' f) (⇑f ∘ P) = f (W'.negDblY P)
-- WeierstrassCurve.Jacobian.map_dblY
@[simp] lemma map_dblY : dblY (W'.map f) (f ∘ P) = f (W'.dblY P) 
Arguments: {R : Type u} [CommRing R] {W' : WeierstrassCurve.Jacobian R} {S : Type u_1} [CommRing S] (f : R →+* S) (P : Fin 3 → R)
Goal: WeierstrassCurve.Jacobian.dblY (WeierstrassCurve.map W' f) (⇑f ∘ P) = f (W'.dblY P)
-- WeierstrassCurve.Jacobian.map_dblXYZ
@[simp] lemma map_dblXYZ : dblXYZ (W'.map f) (f ∘ P) = f ∘ dblXYZ W' P 
Arguments: {R : Type u} [CommRing R] {W' : WeierstrassCurve.Jacobian R} {S : Type u_1} [CommRing S] (f : R →+* S) (P : Fin 3 → R)
Goal: WeierstrassCurve.Jacobian.dblXYZ (WeierstrassCurve.map W' f) (⇑f ∘ P) = ⇑f ∘ W'.dblXYZ P
-- irrational_iff_ne_rational
theorem irrational_iff_ne_rational (x : ℝ) : Irrational x ↔ ∀ a b : ℤ, x ≠ a / b 
Arguments: (x : ℝ)
Goal: Irrational x ↔ ∀ (a b : ℤ), x ≠ ↑a / ↑b
-- Transcendental.irrational
theorem Transcendental.irrational {r : ℝ} (tr : Transcendental ℚ r) : Irrational r 
Arguments: {r : ℝ} (tr : Transcendental ℚ r)
Goal: Irrational r
-- irrational_nrt_of_notint_nrt
theorem irrational_nrt_of_notint_nrt {x : ℝ} (n : ℕ) (m : ℤ) (hxr : x ^ n = m)
    (hv : ¬∃ y : ℤ, x = y) (hnpos : 0 < n) : Irrational x 
Arguments: {x : ℝ} (n : ℕ) (m : ℤ) (hxr : x ^ n = ↑m) (hv : ¬∃ y, x = ↑y) (hnpos : 0 < n)
Goal: Irrational x
-- irrational_nrt_of_n_not_dvd_multiplicity
theorem irrational_nrt_of_n_not_dvd_multiplicity {x : ℝ} (n : ℕ) {m : ℤ} (hm : m ≠ 0) (p : ℕ)
    [hp : Fact p.Prime] (hxr : x ^ n = m)
    (hv : (multiplicity (p : ℤ) m).get (finite_int_iff.2 ⟨hp.1.ne_one, hm⟩) % n ≠ 0) :
    Irrational x 
Arguments: {x : ℝ} (n : ℕ) {m : ℤ} (hm : m ≠ 0) (p : ℕ) [hp : Fact (Nat.Prime p)] (hxr : x ^ n = ↑m) (hv : (multiplicity (↑p) m).get ⋯ % n ≠ 0)
Goal: Irrational x
-- irrational_sqrt_of_multiplicity_odd
theorem irrational_sqrt_of_multiplicity_odd (m : ℤ) (hm : 0 < m) (p : ℕ) [hp : Fact p.Prime]
    (Hpv :
      (multiplicity (p : ℤ) m).get (finite_int_iff.2 ⟨hp.1.ne_one, (ne_of_lt hm).symm⟩) % 2 = 1) :
    Irrational (√m) 
Arguments: (m : ℤ) (hm : 0 < m) (p : ℕ) [hp : Fact (Nat.Prime p)] (Hpv : (multiplicity (↑p) m).get ⋯ % 2 = 1)
Goal: Irrational √↑m
-- not_irrational_zero
@[simp] theorem not_irrational_zero : ¬Irrational 0 
Goal: ¬Irrational 0
-- not_irrational_one
@[simp] theorem not_irrational_one : ¬Irrational 1 
Goal: ¬Irrational 1
-- irrational_sqrt_ratCast_iff_of_nonneg
theorem irrational_sqrt_ratCast_iff_of_nonneg {q : ℚ} (hq : 0 ≤ q) :
    Irrational (√q) ↔ ¬IsSquare q 
Arguments: {q : ℚ} (hq : 0 ≤ q)
Goal: Irrational √↑q ↔ ¬IsSquare q
-- irrational_sqrt_ratCast_iff
theorem irrational_sqrt_ratCast_iff {q : ℚ} :
    Irrational (√q) ↔ ¬IsSquare q ∧ 0 ≤ q 
Arguments: {q : ℚ}
Goal: Irrational √↑q ↔ ¬IsSquare q ∧ 0 ≤ q
-- irrational_sqrt_intCast_iff_of_nonneg
theorem irrational_sqrt_intCast_iff_of_nonneg {z : ℤ} (hz : 0 ≤ z) :
    Irrational (√z) ↔ ¬IsSquare z 
Arguments: {z : ℤ} (hz : 0 ≤ z)
Goal: Irrational √↑z ↔ ¬IsSquare z
-- irrational_sqrt_intCast_iff
theorem irrational_sqrt_intCast_iff {z : ℤ} :
    Irrational (√z) ↔ ¬IsSquare z ∧ 0 ≤ z 
Arguments: {z : ℤ}
Goal: Irrational √↑z ↔ ¬IsSquare z ∧ 0 ≤ z
-- irrational_sqrt_natCast_iff
theorem irrational_sqrt_natCast_iff {n : ℕ} : Irrational (√n) ↔ ¬IsSquare n 
Arguments: {n : ℕ}
Goal: Irrational √↑n ↔ ¬IsSquare n
-- irrational_sqrt_ofNat_iff
theorem irrational_sqrt_ofNat_iff {n : ℕ} [n.AtLeastTwo] :
    Irrational (√(no_index (OfNat.ofNat n))) ↔ ¬IsSquare (OfNat.ofNat n) 
Arguments: {n : ℕ} [n.AtLeastTwo]
Goal: Irrational √(OfNat.ofNat n) ↔ ¬IsSquare (OfNat.ofNat n)
-- Nat.Prime.irrational_sqrt
theorem Nat.Prime.irrational_sqrt {p : ℕ} (hp : Nat.Prime p) : Irrational (√p) 
Arguments: {p : ℕ} (hp : Nat.Prime p)
Goal: Irrational √↑p
-- irrational_sqrt_two
theorem irrational_sqrt_two : Irrational (√2) 
Goal: Irrational √2
-- irrational_sqrt_rat_iff
@[deprecated irrational_sqrt_ratCast_iff (since := "2024-06-16")]
theorem irrational_sqrt_rat_iff (q : ℚ) :
    Irrational (√q) ↔ Rat.sqrt q * Rat.sqrt q ≠ q ∧ 0 ≤ q 
Arguments: (q : ℚ)
Goal: Irrational √↑q ↔ Rat.sqrt q * Rat.sqrt q ≠ q ∧ 0 ≤ q
-- Irrational.ne_rat
theorem ne_rat (h : Irrational x) (q : ℚ) : x ≠ q 
Arguments: {x : ℝ} (h : Irrational x) (q : ℚ)
Goal: x ≠ ↑q
-- Irrational.ne_int
theorem ne_int (h : Irrational x) (m : ℤ) : x ≠ m 
Arguments: {x : ℝ} (h : Irrational x) (m : ℤ)
Goal: x ≠ ↑m
-- Irrational.ne_nat
theorem ne_nat (h : Irrational x) (m : ℕ) : x ≠ m 
Arguments: {x : ℝ} (h : Irrational x) (m : ℕ)
Goal: x ≠ ↑m
-- Irrational.ne_zero
theorem ne_zero (h : Irrational x) : x ≠ 0 
Arguments: {x : ℝ} (h : Irrational x)
Goal: x ≠ 0
-- Irrational.ne_one
theorem ne_one (h : Irrational x) : x ≠ 1 
Arguments: {x : ℝ} (h : Irrational x)
Goal: x ≠ 1
-- Irrational.ne_ofNat
@[simp] theorem ne_ofNat (h : Irrational x) (n : ℕ) [n.AtLeastTwo] : x ≠ no_index (OfNat.ofNat n) 
Arguments: {x : ℝ} (h : Irrational x) (n : ℕ) [n.AtLeastTwo]
Goal: x ≠ OfNat.ofNat n
-- Rat.not_irrational
@[simp]
theorem Rat.not_irrational (q : ℚ) : ¬Irrational q 
Arguments: (q : ℚ)
Goal: ¬Irrational ↑q
-- Int.not_irrational
@[simp]
theorem Int.not_irrational (m : ℤ) : ¬Irrational m 
Arguments: (m : ℤ)
Goal: ¬Irrational ↑m
-- Nat.not_irrational
@[simp]
theorem Nat.not_irrational (m : ℕ) : ¬Irrational m 
Arguments: (m : ℕ)
Goal: ¬Irrational ↑m
-- not_irrational_ofNat
@[simp] theorem not_irrational_ofNat (n : ℕ) [n.AtLeastTwo] :
    ¬Irrational (no_index (OfNat.ofNat n)) 
Arguments: (n : ℕ) [n.AtLeastTwo]
Goal: ¬Irrational (OfNat.ofNat n)
-- Irrational.add_cases
theorem add_cases : Irrational (x + y) → Irrational x ∨ Irrational y 
Arguments: {x y : ℝ}
Goal: Irrational (x + y) → Irrational x ∨ Irrational y
-- Irrational.of_rat_add
theorem of_rat_add (h : Irrational (q + x)) : Irrational x 
Arguments: (q : ℚ) {x : ℝ} (h : Irrational (↑q + x))
Goal: Irrational x
-- Irrational.rat_add
theorem rat_add (h : Irrational x) : Irrational (q + x) 
Arguments: (q : ℚ) {x : ℝ} (h : Irrational x)
Goal: Irrational (↑q + x)
-- Irrational.of_add_rat
theorem of_add_rat : Irrational (x + q) → Irrational x 
Arguments: (q : ℚ) {x : ℝ}
Goal: Irrational (x + ↑q) → Irrational x
-- Irrational.add_rat
theorem add_rat (h : Irrational x) : Irrational (x + q) 
Arguments: (q : ℚ) {x : ℝ} (h : Irrational x)
Goal: Irrational (x + ↑q)
-- Irrational.of_int_add
theorem of_int_add (m : ℤ) (h : Irrational (m + x)) : Irrational x 
Arguments: {x : ℝ} (m : ℤ) (h : Irrational (↑m + x))
Goal: Irrational x
-- Irrational.of_add_int
theorem of_add_int (m : ℤ) (h : Irrational (x + m)) : Irrational x 
Arguments: {x : ℝ} (m : ℤ) (h : Irrational (x + ↑m))
Goal: Irrational x
-- Irrational.int_add
theorem int_add (h : Irrational x) (m : ℤ) : Irrational (m + x) 
Arguments: {x : ℝ} (h : Irrational x) (m : ℤ)
Goal: Irrational (↑m + x)
-- Irrational.add_int
theorem add_int (h : Irrational x) (m : ℤ) : Irrational (x + m) 
Arguments: {x : ℝ} (h : Irrational x) (m : ℤ)
Goal: Irrational (x + ↑m)
-- Irrational.of_nat_add
theorem of_nat_add (m : ℕ) (h : Irrational (m + x)) : Irrational x 
Arguments: {x : ℝ} (m : ℕ) (h : Irrational (↑m + x))
Goal: Irrational x
-- Irrational.of_add_nat
theorem of_add_nat (m : ℕ) (h : Irrational (x + m)) : Irrational x 
Arguments: {x : ℝ} (m : ℕ) (h : Irrational (x + ↑m))
Goal: Irrational x
-- Irrational.nat_add
theorem nat_add (h : Irrational x) (m : ℕ) : Irrational (m + x) 
Arguments: {x : ℝ} (h : Irrational x) (m : ℕ)
Goal: Irrational (↑m + x)
-- Irrational.add_nat
theorem add_nat (h : Irrational x) (m : ℕ) : Irrational (x + m) 
Arguments: {x : ℝ} (h : Irrational x) (m : ℕ)
Goal: Irrational (x + ↑m)
-- Irrational.of_neg
theorem of_neg (h : Irrational (-x)) : Irrational x 
Arguments: {x : ℝ} (h : Irrational (-x))
Goal: Irrational x
-- Irrational.neg
protected theorem neg (h : Irrational x) : Irrational (-x) 
Arguments: {x : ℝ} (h : Irrational x)
Goal: Irrational (-x)
-- Irrational.sub_rat
theorem sub_rat (h : Irrational x) : Irrational (x - q) 
Arguments: (q : ℚ) {x : ℝ} (h : Irrational x)
Goal: Irrational (x - ↑q)
-- Irrational.rat_sub
theorem rat_sub (h : Irrational x) : Irrational (q - x) 
Arguments: (q : ℚ) {x : ℝ} (h : Irrational x)
Goal: Irrational (↑q - x)
-- Irrational.of_sub_rat
theorem of_sub_rat (h : Irrational (x - q)) : Irrational x 
Arguments: (q : ℚ) {x : ℝ} (h : Irrational (x - ↑q))
Goal: Irrational x
-- Irrational.of_rat_sub
theorem of_rat_sub (h : Irrational (q - x)) : Irrational x 
Arguments: (q : ℚ) {x : ℝ} (h : Irrational (↑q - x))
Goal: Irrational x
-- Irrational.sub_int
theorem sub_int (h : Irrational x) (m : ℤ) : Irrational (x - m) 
Arguments: {x : ℝ} (h : Irrational x) (m : ℤ)
Goal: Irrational (x - ↑m)
-- Irrational.int_sub
theorem int_sub (h : Irrational x) (m : ℤ) : Irrational (m - x) 
Arguments: {x : ℝ} (h : Irrational x) (m : ℤ)
Goal: Irrational (↑m - x)
-- Irrational.of_sub_int
theorem of_sub_int (m : ℤ) (h : Irrational (x - m)) : Irrational x 
Arguments: {x : ℝ} (m : ℤ) (h : Irrational (x - ↑m))
Goal: Irrational x
-- Irrational.of_int_sub
theorem of_int_sub (m : ℤ) (h : Irrational (m - x)) : Irrational x 
Arguments: {x : ℝ} (m : ℤ) (h : Irrational (↑m - x))
Goal: Irrational x
-- Irrational.sub_nat
theorem sub_nat (h : Irrational x) (m : ℕ) : Irrational (x - m) 
Arguments: {x : ℝ} (h : Irrational x) (m : ℕ)
Goal: Irrational (x - ↑m)
-- Irrational.nat_sub
theorem nat_sub (h : Irrational x) (m : ℕ) : Irrational (m - x) 
Arguments: {x : ℝ} (h : Irrational x) (m : ℕ)
Goal: Irrational (↑m - x)
-- Irrational.of_sub_nat
theorem of_sub_nat (m : ℕ) (h : Irrational (x - m)) : Irrational x 
Arguments: {x : ℝ} (m : ℕ) (h : Irrational (x - ↑m))
Goal: Irrational x
-- Irrational.of_nat_sub
theorem of_nat_sub (m : ℕ) (h : Irrational (m - x)) : Irrational x 
Arguments: {x : ℝ} (m : ℕ) (h : Irrational (↑m - x))
Goal: Irrational x
-- Irrational.mul_cases
theorem mul_cases : Irrational (x * y) → Irrational x ∨ Irrational y 
Arguments: {x y : ℝ}
Goal: Irrational (x * y) → Irrational x ∨ Irrational y
-- Irrational.of_mul_rat
theorem of_mul_rat (h : Irrational (x * q)) : Irrational x 
Arguments: (q : ℚ) {x : ℝ} (h : Irrational (x * ↑q))
Goal: Irrational x
-- Irrational.mul_rat
theorem mul_rat (h : Irrational x) {q : ℚ} (hq : q ≠ 0) : Irrational (x * q) 
Arguments: {x : ℝ} (h : Irrational x) {q : ℚ} (hq : q ≠ 0)
Goal: Irrational (x * ↑q)
-- Irrational.of_rat_mul
theorem of_rat_mul : Irrational (q * x) → Irrational x 
Arguments: (q : ℚ) {x : ℝ}
Goal: Irrational (↑q * x) → Irrational x
-- Irrational.rat_mul
theorem rat_mul (h : Irrational x) {q : ℚ} (hq : q ≠ 0) : Irrational (q * x) 
Arguments: {x : ℝ} (h : Irrational x) {q : ℚ} (hq : q ≠ 0)
Goal: Irrational (↑q * x)
-- Irrational.of_mul_int
theorem of_mul_int (m : ℤ) (h : Irrational (x * m)) : Irrational x 
Arguments: {x : ℝ} (m : ℤ) (h : Irrational (x * ↑m))
Goal: Irrational x
-- Irrational.of_int_mul
theorem of_int_mul (m : ℤ) (h : Irrational (m * x)) : Irrational x 
Arguments: {x : ℝ} (m : ℤ) (h : Irrational (↑m * x))
Goal: Irrational x
-- Irrational.mul_int
theorem mul_int (h : Irrational x) {m : ℤ} (hm : m ≠ 0) : Irrational (x * m) 
Arguments: {x : ℝ} (h : Irrational x) {m : ℤ} (hm : m ≠ 0)
Goal: Irrational (x * ↑m)
-- Irrational.int_mul
theorem int_mul (h : Irrational x) {m : ℤ} (hm : m ≠ 0) : Irrational (m * x) 
Arguments: {x : ℝ} (h : Irrational x) {m : ℤ} (hm : m ≠ 0)
Goal: Irrational (↑m * x)
-- Irrational.of_mul_nat
theorem of_mul_nat (m : ℕ) (h : Irrational (x * m)) : Irrational x 
Arguments: {x : ℝ} (m : ℕ) (h : Irrational (x * ↑m))
Goal: Irrational x
-- Irrational.of_nat_mul
theorem of_nat_mul (m : ℕ) (h : Irrational (m * x)) : Irrational x 
Arguments: {x : ℝ} (m : ℕ) (h : Irrational (↑m * x))
Goal: Irrational x
-- Irrational.mul_nat
theorem mul_nat (h : Irrational x) {m : ℕ} (hm : m ≠ 0) : Irrational (x * m) 
Arguments: {x : ℝ} (h : Irrational x) {m : ℕ} (hm : m ≠ 0)
Goal: Irrational (x * ↑m)
-- Irrational.nat_mul
theorem nat_mul (h : Irrational x) {m : ℕ} (hm : m ≠ 0) : Irrational (m * x) 
Arguments: {x : ℝ} (h : Irrational x) {m : ℕ} (hm : m ≠ 0)
Goal: Irrational (↑m * x)
-- Irrational.of_inv
theorem of_inv (h : Irrational x⁻¹) : Irrational x 
Arguments: {x : ℝ} (h : Irrational x⁻¹)
Goal: Irrational x
-- Irrational.inv
protected theorem inv (h : Irrational x) : Irrational x⁻¹ 
Arguments: {x : ℝ} (h : Irrational x)
Goal: Irrational x⁻¹
-- Irrational.div_cases
theorem div_cases (h : Irrational (x / y)) : Irrational x ∨ Irrational y 
Arguments: {x y : ℝ} (h : Irrational (x / y))
Goal: Irrational x ∨ Irrational y
-- Irrational.of_rat_div
theorem of_rat_div (h : Irrational (q / x)) : Irrational x 
Arguments: (q : ℚ) {x : ℝ} (h : Irrational (↑q / x))
Goal: Irrational x
-- Irrational.of_div_rat
theorem of_div_rat (h : Irrational (x / q)) : Irrational x 
Arguments: (q : ℚ) {x : ℝ} (h : Irrational (x / ↑q))
Goal: Irrational x
-- Irrational.rat_div
theorem rat_div (h : Irrational x) {q : ℚ} (hq : q ≠ 0) : Irrational (q / x) 
Arguments: {x : ℝ} (h : Irrational x) {q : ℚ} (hq : q ≠ 0)
Goal: Irrational (↑q / x)
-- Irrational.div_rat
theorem div_rat (h : Irrational x) {q : ℚ} (hq : q ≠ 0) : Irrational (x / q) 
Arguments: {x : ℝ} (h : Irrational x) {q : ℚ} (hq : q ≠ 0)
Goal: Irrational (x / ↑q)
-- Irrational.of_int_div
theorem of_int_div (m : ℤ) (h : Irrational (m / x)) : Irrational x 
Arguments: {x : ℝ} (m : ℤ) (h : Irrational (↑m / x))
Goal: Irrational x
-- Irrational.of_div_int
theorem of_div_int (m : ℤ) (h : Irrational (x / m)) : Irrational x 
Arguments: {x : ℝ} (m : ℤ) (h : Irrational (x / ↑m))
Goal: Irrational x
-- Irrational.int_div
theorem int_div (h : Irrational x) {m : ℤ} (hm : m ≠ 0) : Irrational (m / x) 
Arguments: {x : ℝ} (h : Irrational x) {m : ℤ} (hm : m ≠ 0)
Goal: Irrational (↑m / x)
-- Irrational.div_int
theorem div_int (h : Irrational x) {m : ℤ} (hm : m ≠ 0) : Irrational (x / m) 
Arguments: {x : ℝ} (h : Irrational x) {m : ℤ} (hm : m ≠ 0)
Goal: Irrational (x / ↑m)
-- Irrational.of_nat_div
theorem of_nat_div (m : ℕ) (h : Irrational (m / x)) : Irrational x 
Arguments: {x : ℝ} (m : ℕ) (h : Irrational (↑m / x))
Goal: Irrational x
-- Irrational.of_div_nat
theorem of_div_nat (m : ℕ) (h : Irrational (x / m)) : Irrational x 
Arguments: {x : ℝ} (m : ℕ) (h : Irrational (x / ↑m))
Goal: Irrational x
-- Irrational.nat_div
theorem nat_div (h : Irrational x) {m : ℕ} (hm : m ≠ 0) : Irrational (m / x) 
Arguments: {x : ℝ} (h : Irrational x) {m : ℕ} (hm : m ≠ 0)
Goal: Irrational (↑m / x)
-- Irrational.div_nat
theorem div_nat (h : Irrational x) {m : ℕ} (hm : m ≠ 0) : Irrational (x / m) 
Arguments: {x : ℝ} (h : Irrational x) {m : ℕ} (hm : m ≠ 0)
Goal: Irrational (x / ↑m)
-- Irrational.of_one_div
theorem of_one_div (h : Irrational (1 / x)) : Irrational x 
Arguments: {x : ℝ} (h : Irrational (1 / x))
Goal: Irrational x
-- Irrational.of_mul_self
theorem of_mul_self (h : Irrational (x * x)) : Irrational x 
Arguments: {x : ℝ} (h : Irrational (x * x))
Goal: Irrational x
-- Irrational.of_pow
theorem of_pow : ∀ n : ℕ, Irrational (x ^ n) → Irrational x
  
Arguments: {x : ℝ} (n : ℕ)
Goal: Irrational (x ^ n) → Irrational x
-- Irrational.of_zpow
theorem of_zpow : ∀ m : ℤ, Irrational (x ^ m) → Irrational x
  
Arguments: {x : ℝ} (m : ℤ)
Goal: Irrational (x ^ m) → Irrational x
-- one_lt_natDegree_of_irrational_root
theorem one_lt_natDegree_of_irrational_root (hx : Irrational x) (p_nonzero : p ≠ 0)
    (x_is_root : aeval x p = 0) : 1 < p.natDegree 
Arguments: (x : ℝ) (p : Polynomial ℤ) (hx : Irrational x) (p_nonzero : p ≠ 0) (x_is_root : (Polynomial.aeval x) p = 0)
Goal: 1 < p.natDegree
-- irrational_rat_add_iff
@[simp]
theorem irrational_rat_add_iff : Irrational (q + x) ↔ Irrational x 
Arguments: {q : ℚ} {x : ℝ}
Goal: Irrational (↑q + x) ↔ Irrational x
-- irrational_int_add_iff
@[simp]
theorem irrational_int_add_iff : Irrational (m + x) ↔ Irrational x 
Arguments: {m : ℤ} {x : ℝ}
Goal: Irrational (↑m + x) ↔ Irrational x
-- irrational_nat_add_iff
@[simp]
theorem irrational_nat_add_iff : Irrational (n + x) ↔ Irrational x 
Arguments: {n : ℕ} {x : ℝ}
Goal: Irrational (↑n + x) ↔ Irrational x
-- irrational_add_rat_iff
@[simp]
theorem irrational_add_rat_iff : Irrational (x + q) ↔ Irrational x 
Arguments: {q : ℚ} {x : ℝ}
Goal: Irrational (x + ↑q) ↔ Irrational x
-- irrational_add_int_iff
@[simp]
theorem irrational_add_int_iff : Irrational (x + m) ↔ Irrational x 
Arguments: {m : ℤ} {x : ℝ}
Goal: Irrational (x + ↑m) ↔ Irrational x
-- irrational_add_nat_iff
@[simp]
theorem irrational_add_nat_iff : Irrational (x + n) ↔ Irrational x 
Arguments: {n : ℕ} {x : ℝ}
Goal: Irrational (x + ↑n) ↔ Irrational x
-- irrational_rat_sub_iff
@[simp]
theorem irrational_rat_sub_iff : Irrational (q - x) ↔ Irrational x 
Arguments: {q : ℚ} {x : ℝ}
Goal: Irrational (↑q - x) ↔ Irrational x
-- irrational_int_sub_iff
@[simp]
theorem irrational_int_sub_iff : Irrational (m - x) ↔ Irrational x 
Arguments: {m : ℤ} {x : ℝ}
Goal: Irrational (↑m - x) ↔ Irrational x
-- irrational_nat_sub_iff
@[simp]
theorem irrational_nat_sub_iff : Irrational (n - x) ↔ Irrational x 
Arguments: {n : ℕ} {x : ℝ}
Goal: Irrational (↑n - x) ↔ Irrational x
-- irrational_sub_rat_iff
@[simp]
theorem irrational_sub_rat_iff : Irrational (x - q) ↔ Irrational x 
Arguments: {q : ℚ} {x : ℝ}
Goal: Irrational (x - ↑q) ↔ Irrational x
-- irrational_sub_int_iff
@[simp]
theorem irrational_sub_int_iff : Irrational (x - m) ↔ Irrational x 
Arguments: {m : ℤ} {x : ℝ}
Goal: Irrational (x - ↑m) ↔ Irrational x
-- irrational_sub_nat_iff
@[simp]
theorem irrational_sub_nat_iff : Irrational (x - n) ↔ Irrational x 
Arguments: {n : ℕ} {x : ℝ}
Goal: Irrational (x - ↑n) ↔ Irrational x
-- irrational_neg_iff
@[simp]
theorem irrational_neg_iff : Irrational (-x) ↔ Irrational x 
Arguments: {x : ℝ}
Goal: Irrational (-x) ↔ Irrational x
-- irrational_inv_iff
@[simp]
theorem irrational_inv_iff : Irrational x⁻¹ ↔ Irrational x 
Arguments: {x : ℝ}
Goal: Irrational x⁻¹ ↔ Irrational x
-- irrational_rat_mul_iff
@[simp]
theorem irrational_rat_mul_iff : Irrational (q * x) ↔ q ≠ 0 ∧ Irrational x 
Arguments: {q : ℚ} {x : ℝ}
Goal: Irrational (↑q * x) ↔ q ≠ 0 ∧ Irrational x
-- irrational_mul_rat_iff
@[simp]
theorem irrational_mul_rat_iff : Irrational (x * q) ↔ q ≠ 0 ∧ Irrational x 
Arguments: {q : ℚ} {x : ℝ}
Goal: Irrational (x * ↑q) ↔ q ≠ 0 ∧ Irrational x
-- irrational_int_mul_iff
@[simp]
theorem irrational_int_mul_iff : Irrational (m * x) ↔ m ≠ 0 ∧ Irrational x 
Arguments: {m : ℤ} {x : ℝ}
Goal: Irrational (↑m * x) ↔ m ≠ 0 ∧ Irrational x
-- irrational_mul_int_iff
@[simp]
theorem irrational_mul_int_iff : Irrational (x * m) ↔ m ≠ 0 ∧ Irrational x 
Arguments: {m : ℤ} {x : ℝ}
Goal: Irrational (x * ↑m) ↔ m ≠ 0 ∧ Irrational x
-- irrational_nat_mul_iff
@[simp]
theorem irrational_nat_mul_iff : Irrational (n * x) ↔ n ≠ 0 ∧ Irrational x 
Arguments: {n : ℕ} {x : ℝ}
Goal: Irrational (↑n * x) ↔ n ≠ 0 ∧ Irrational x
-- irrational_mul_nat_iff
@[simp]
theorem irrational_mul_nat_iff : Irrational (x * n) ↔ n ≠ 0 ∧ Irrational x 
Arguments: {n : ℕ} {x : ℝ}
Goal: Irrational (x * ↑n) ↔ n ≠ 0 ∧ Irrational x
-- irrational_rat_div_iff
@[simp]
theorem irrational_rat_div_iff : Irrational (q / x) ↔ q ≠ 0 ∧ Irrational x 
Arguments: {q : ℚ} {x : ℝ}
Goal: Irrational (↑q / x) ↔ q ≠ 0 ∧ Irrational x