ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- mul_mem_nonZeroDivisors
theorem mul_mem_nonZeroDivisors {a b : M₁} : a * b ∈ M₁⁰ ↔ a ∈ M₁⁰ ∧ b ∈ M₁⁰ 
Arguments: {M₁ : Type u_4} [CommMonoidWithZero M₁] {a b : M₁}
Goal: a * b ∈ M₁⁰ ↔ a ∈ M₁⁰ ∧ b ∈ M₁⁰
-- isUnit_of_mem_nonZeroDivisors
theorem isUnit_of_mem_nonZeroDivisors {G₀ : Type*} [GroupWithZero G₀] {x : G₀}
    (hx : x ∈ nonZeroDivisors G₀) : IsUnit x 
Arguments: {G₀ : Type u_8} [GroupWithZero G₀] {x : G₀} (hx : x ∈ G₀⁰)
Goal: IsUnit x
-- IsUnit.mem_nonZeroDivisors
lemma IsUnit.mem_nonZeroDivisors {a : M} (ha : IsUnit a) : a ∈ M⁰ 
Arguments: {M : Type u_2} [MonoidWithZero M] {a : M} (ha : IsUnit a)
Goal: a ∈ M⁰
-- eq_zero_of_ne_zero_of_mul_right_eq_zero
theorem eq_zero_of_ne_zero_of_mul_right_eq_zero [NoZeroDivisors M] {x y : M} (hnx : x ≠ 0)
    (hxy : y * x = 0) : y = 0 
Arguments: {M : Type u_2} [MonoidWithZero M] [NoZeroDivisors M] {x y : M} (hnx : x ≠ 0) (hxy : y * x = 0)
Goal: y = 0
-- eq_zero_of_ne_zero_of_mul_left_eq_zero
theorem eq_zero_of_ne_zero_of_mul_left_eq_zero [NoZeroDivisors M] {x y : M} (hnx : x ≠ 0)
    (hxy : x * y = 0) : y = 0 
Arguments: {M : Type u_2} [MonoidWithZero M] [NoZeroDivisors M] {x y : M} (hnx : x ≠ 0) (hxy : x * y = 0)
Goal: y = 0
-- mem_nonZeroDivisors_of_ne_zero
theorem mem_nonZeroDivisors_of_ne_zero [NoZeroDivisors M] {x : M} (hx : x ≠ 0) : x ∈ M⁰ 
Arguments: {M : Type u_2} [MonoidWithZero M] [NoZeroDivisors M] {x : M} (hx : x ≠ 0)
Goal: x ∈ M⁰
-- mem_nonZeroDivisors_iff_ne_zero
theorem mem_nonZeroDivisors_iff_ne_zero [NoZeroDivisors M] [Nontrivial M] {x : M} :
    x ∈ M⁰ ↔ x ≠ 0 
Arguments: {M : Type u_2} [MonoidWithZero M] [NoZeroDivisors M] [Nontrivial M] {x : M}
Goal: x ∈ M⁰ ↔ x ≠ 0
-- map_ne_zero_of_mem_nonZeroDivisors
theorem map_ne_zero_of_mem_nonZeroDivisors [Nontrivial M] [ZeroHomClass F M M'] (g : F)
    (hg : Function.Injective (g : M → M')) {x : M} (h : x ∈ M⁰) : g x ≠ 0 
Arguments: {M : Type u_2} {M' : Type u_3} {F : Type u_7} [MonoidWithZero M] [MonoidWithZero M'] [FunLike F M M'] [Nontrivial M] [ZeroHomClass F M M'] (g : F) (hg : Function.Injective ⇑g) {x : M} (h : x ∈ M⁰)
Goal: g x ≠ 0
-- map_mem_nonZeroDivisors
theorem map_mem_nonZeroDivisors [Nontrivial M] [NoZeroDivisors M'] [ZeroHomClass F M M'] (g : F)
    (hg : Function.Injective g) {x : M} (h : x ∈ M⁰) : g x ∈ M'⁰ 
Arguments: {M : Type u_2} {M' : Type u_3} {F : Type u_7} [MonoidWithZero M] [MonoidWithZero M'] [FunLike F M M'] [Nontrivial M] [NoZeroDivisors M'] [ZeroHomClass F M M'] (g : F) (hg : Function.Injective ⇑g) {x : M} (h : x ∈ M⁰)
Goal: g x ∈ M'⁰
-- le_nonZeroDivisors_of_noZeroDivisors
theorem le_nonZeroDivisors_of_noZeroDivisors [NoZeroDivisors M] {S : Submonoid M}
    (hS : (0 : M) ∉ S) : S ≤ M⁰ 
Arguments: {M : Type u_2} [MonoidWithZero M] [NoZeroDivisors M] {S : Submonoid M} (hS : 0 ∉ S)
Goal: S ≤ M⁰
-- powers_le_nonZeroDivisors_of_noZeroDivisors
theorem powers_le_nonZeroDivisors_of_noZeroDivisors [NoZeroDivisors M] {a : M} (ha : a ≠ 0) :
    Submonoid.powers a ≤ M⁰ 
Arguments: {M : Type u_2} [MonoidWithZero M] [NoZeroDivisors M] {a : M} (ha : a ≠ 0)
Goal: Submonoid.powers a ≤ M⁰
-- map_le_nonZeroDivisors_of_injective
theorem map_le_nonZeroDivisors_of_injective [NoZeroDivisors M'] [MonoidWithZeroHomClass F M M']
    (f : F) (hf : Function.Injective f) {S : Submonoid M} (hS : S ≤ M⁰) : S.map f ≤ M'⁰ 
Arguments: {M : Type u_2} {M' : Type u_3} {F : Type u_7} [MonoidWithZero M] [MonoidWithZero M'] [FunLike F M M'] [NoZeroDivisors M'] [MonoidWithZeroHomClass F M M'] (f : F) (hf : Function.Injective ⇑f) {S : Submonoid M} (hS : S ≤ M⁰)
Goal: Submonoid.map f S ≤ M'⁰
-- nonZeroDivisors_le_comap_nonZeroDivisors_of_injective
theorem nonZeroDivisors_le_comap_nonZeroDivisors_of_injective [NoZeroDivisors M']
    [MonoidWithZeroHomClass F M M'] (f : F) (hf : Function.Injective f) : M⁰ ≤ M'⁰.comap f 
Arguments: {M : Type u_2} {M' : Type u_3} {F : Type u_7} [MonoidWithZero M] [MonoidWithZero M'] [FunLike F M M'] [NoZeroDivisors M'] [MonoidWithZeroHomClass F M M'] (f : F) (hf : Function.Injective ⇑f)
Goal: M⁰ ≤ Submonoid.comap f M'⁰
-- isUnit_iff_mem_nonZeroDivisors_of_finite
lemma isUnit_iff_mem_nonZeroDivisors_of_finite [Finite R] {a : R} :
    IsUnit a ↔ a ∈ nonZeroDivisors R 
Arguments: {R : Type u_5} [Ring R] [Finite R] {a : R}
Goal: IsUnit a ↔ a ∈ R⁰
-- mem_nonZeroSMulDivisors_iff
lemma mem_nonZeroSMulDivisors_iff {x : R} : x ∈ R⁰[M] ↔ ∀ (m : M), x • m = 0 → m = 0 
Arguments: {R : Type u_2} {M : Type u_3} [MonoidWithZero R] [Zero M] [MulAction R M] {x : R}
Goal: x ∈ nonZeroSMulDivisors R M ↔ ∀ (m : M), x • m = 0 → m = 0
-- unop_nonZeroSMulDivisors_mulOpposite_eq_nonZeroDivisors
@[simp]
lemma unop_nonZeroSMulDivisors_mulOpposite_eq_nonZeroDivisors :
    (Rᵐᵒᵖ ⁰[R]).unop = R⁰ 
Arguments: (R : Type u_2) [MonoidWithZero R]
Goal: (nonZeroSMulDivisors Rᵐᵒᵖ R).unop = R⁰
-- nonZeroSMulDivisors_mulOpposite_eq_op_nonZeroDivisors
lemma nonZeroSMulDivisors_mulOpposite_eq_op_nonZeroDivisors :
    Rᵐᵒᵖ ⁰[R] = R⁰.op 
Arguments: (R : Type u_2) [MonoidWithZero R]
Goal: nonZeroSMulDivisors Rᵐᵒᵖ R = R⁰.op
-- unitsNonZeroDivisorsEquiv_apply
@[simps]
def unitsNonZeroDivisorsEquiv : M₀⁰ˣ ≃* M₀ˣ 
Arguments: {M₀ : Type u_1} [MonoidWithZero M₀]
Goal: ∀ (a : (↥M₀⁰)ˣ), unitsNonZeroDivisorsEquiv a = (↑(Units.map M₀⁰.subtype)).toFun a
-- val_inv_unitsNonZeroDivisorsEquiv_symm_apply_coe
@[simps]
def unitsNonZeroDivisorsEquiv : M₀⁰ˣ ≃* M₀ˣ 
Arguments: {M₀ : Type u_1} [MonoidWithZero M₀] (u : M₀ˣ)
Goal: ↑↑(unitsNonZeroDivisorsEquiv.symm u)⁻¹ = ↑u⁻¹
-- val_unitsNonZeroDivisorsEquiv_symm_apply_coe
@[simps]
def unitsNonZeroDivisorsEquiv : M₀⁰ˣ ≃* M₀ˣ 
Arguments: {M₀ : Type u_1} [MonoidWithZero M₀] (u : M₀ˣ)
Goal: ↑↑(unitsNonZeroDivisorsEquiv.symm u) = ↑u
-- nonZeroDivisors.associated_coe
@[simp, norm_cast] lemma nonZeroDivisors.associated_coe : Associated (a : M₀) b ↔ Associated a b 
Arguments: {M₀ : Type u_1} [MonoidWithZero M₀] {a b : ↥M₀⁰}
Goal: Associated ↑a ↑b ↔ Associated a b
-- mk_mem_nonZeroDivisors_associates
theorem mk_mem_nonZeroDivisors_associates : Associates.mk a ∈ (Associates M₀)⁰ ↔ a ∈ M₀⁰ 
Arguments: {M₀ : Type u_2} [CommMonoidWithZero M₀] {a : M₀}
Goal: Associates.mk a ∈ (Associates M₀)⁰ ↔ a ∈ M₀⁰
-- associatesNonZeroDivisorsEquiv_mk_mk
@[simp]
lemma associatesNonZeroDivisorsEquiv_mk_mk (a : M₀) (ha) :
    associatesNonZeroDivisorsEquiv ⟨⟦a⟧, ha⟩ = ⟦⟨a, mk_mem_nonZeroDivisors_associates.1 ha⟩⟧ 
Arguments: {M₀ : Type u_2} [CommMonoidWithZero M₀] (a : M₀) (ha : ⟦a⟧ ∈ (Associates M₀)⁰)
Goal: associatesNonZeroDivisorsEquiv ⟨⟦a⟧, ha⟩ = ⟦⟨a, ⋯⟩⟧
-- associatesNonZeroDivisorsEquiv_symm_mk_mk
@[simp]
lemma associatesNonZeroDivisorsEquiv_symm_mk_mk (a : M₀) (ha) :
    associatesNonZeroDivisorsEquiv.symm ⟦⟨a, ha⟩⟧ = ⟨⟦a⟧, mk_mem_nonZeroDivisors_associates.2 ha⟩ 
Arguments: {M₀ : Type u_2} [CommMonoidWithZero M₀] (a : M₀) (ha : a ∈ M₀⁰)
Goal: associatesNonZeroDivisorsEquiv.symm ⟦⟨a, ha⟩⟧ = ⟨⟦a⟧, ⋯⟩
-- Zlattice.covolume_eq_measure_fundamentalDomain
theorem covolume_eq_measure_fundamentalDomain {F : Set E} (h : IsAddFundamentalDomain L F μ) :
    covolume L μ = (μ F).toReal 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] (L : AddSubgroup E) [DiscreteTopology ↥L] [IsZlattice ℝ L] (μ : MeasureTheory.Measure E := by volume_tac) [MeasureTheory.Measure.IsAddHaarMeasure μ] {F : Set E} (h : MeasureTheory.IsAddFundamentalDomain (↥L) F μ)
Goal: Zlattice.covolume L μ = (μ F).toReal
-- Zlattice.covolume_ne_zero
theorem covolume_ne_zero : covolume L μ ≠ 0 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] (L : AddSubgroup E) [DiscreteTopology ↥L] [IsZlattice ℝ L] (μ : MeasureTheory.Measure E := by volume_tac) [MeasureTheory.Measure.IsAddHaarMeasure μ]
Goal: Zlattice.covolume L μ ≠ 0
-- Zlattice.covolume_pos
theorem covolume_pos : 0 < covolume L μ 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] (L : AddSubgroup E) [DiscreteTopology ↥L] [IsZlattice ℝ L] (μ : MeasureTheory.Measure E := by volume_tac) [MeasureTheory.Measure.IsAddHaarMeasure μ]
Goal: 0 < Zlattice.covolume L μ
-- Zlattice.covolume_eq_det_mul_measure
theorem covolume_eq_det_mul_measure {ι : Type*} [Fintype ι] [DecidableEq ι] (b : Basis ι ℤ L)
    (b₀ : Basis ι ℝ E) :
    covolume L μ = |b₀.det ((↑) ∘ b)| * (μ (Zspan.fundamentalDomain b₀)).toReal 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] (L : AddSubgroup E) [DiscreteTopology ↥L] [IsZlattice ℝ L] (μ : MeasureTheory.Measure E := by volume_tac) [MeasureTheory.Measure.IsAddHaarMeasure μ] {ι : Type u_2} [Fintype ι] [DecidableEq ι] (b : Basis ι ℤ ↥L) (b₀ : Basis ι ℝ E)
Goal: Zlattice.covolume L μ = |b₀.det (Subtype.val ∘ ⇑b)| * (μ (Zspan.fundamentalDomain b₀)).toReal
-- Zlattice.covolume_eq_det
theorem covolume_eq_det {ι : Type*} [Fintype ι] [DecidableEq ι] (L : AddSubgroup (ι → ℝ))
    [DiscreteTopology L] [IsZlattice ℝ L] (b : Basis ι ℤ L) :
    covolume L = |(Matrix.of ((↑) ∘ b)).det| 
Arguments: {ι : Type u_2} [Fintype ι] [DecidableEq ι] (L : AddSubgroup (ι → ℝ)) [DiscreteTopology ↥L] [IsZlattice ℝ L] (b : Basis ι ℤ ↥L)
Goal: Zlattice.covolume L MeasureTheory.volume = |(Matrix.of (Subtype.val ∘ ⇑b)).det|
-- AddConstMap.map_add_const'
structure AddConstMap (G H : Type*) [Add G] [Add H] (a : G) (b : H) where
    protected toFun : G → H
    map_add_const' (x : G) : toFun (x + a) = toFun x + b
Arguments: {G : Type u_1} {H : Type u_2} [Add G] [Add H] {a : G} {b : H} (self : AddConstMap G H a b) (x : G)
Goal: self.toFun (x + a) = self.toFun x + b
-- AddConstMapClass.map_add_const
class AddConstMapClass (F : Type*) (G H : outParam Type*) [Add G] [Add H]
    (a : outParam G) (b : outParam H) [FunLike F G H] : Prop where
    map_add_const (f : F) (x : G) : f (x + a) = f x + b
Arguments: {F : Type u_1} {G : outParam (Type u_2)} {H : outParam (Type u_3)} [Add G] [Add H] {a : outParam G} {b : outParam H} [FunLike F G H] [self : AddConstMapClass F G H a b] (f : F) (x : G)
Goal: f (x + a) = f x + b
-- AddConstMapClass.semiconj
protected theorem semiconj [Add G] [Add H] [AddConstMapClass F G H a b] (f : F) :
    Semiconj f (· + a) (· + b) 
Arguments: {F : Type u_1} {G : Type u_2} {H : Type u_3} [FunLike F G H] {a : G} {b : H} [Add G] [Add H] [AddConstMapClass F G H a b] (f : F)
Goal: Function.Semiconj (⇑f) (fun x => x + a) fun x => x + b
-- AddConstMapClass.map_add_nsmul
@[scoped simp]
theorem map_add_nsmul [AddMonoid G] [AddMonoid H] [AddConstMapClass F G H a b]
    (f : F) (x : G) (n : ℕ) : f (x + n • a) = f x + n • b 
Arguments: {F : Type u_1} {G : Type u_2} {H : Type u_3} [FunLike F G H] {a : G} {b : H} [AddMonoid G] [AddMonoid H] [AddConstMapClass F G H a b] (f : F) (x : G) (n : ℕ)
Goal: f (x + n • a) = f x + n • b
-- AddConstMapClass.map_add_nat'
@[scoped simp]
theorem map_add_nat' [AddMonoidWithOne G] [AddMonoid H] [AddConstMapClass F G H 1 b]
    (f : F) (x : G) (n : ℕ) : f (x + n) = f x + n • b 
Arguments: {F : Type u_1} {G : Type u_2} {H : Type u_3} [FunLike F G H] {b : H} [AddMonoidWithOne G] [AddMonoid H] [AddConstMapClass F G H 1 b] (f : F) (x : G) (n : ℕ)
Goal: f (x + ↑n) = f x + n • b
-- AddConstMapClass.map_add_one
theorem map_add_one [AddMonoidWithOne G] [Add H] [AddConstMapClass F G H 1 b]
    (f : F) (x : G) : f (x + 1) = f x + b 
Arguments: {F : Type u_1} {G : Type u_2} {H : Type u_3} [FunLike F G H] {b : H} [AddMonoidWithOne G] [Add H] [AddConstMapClass F G H 1 b] (f : F) (x : G)
Goal: f (x + 1) = f x + b
-- AddConstMapClass.map_add_ofNat'
@[scoped simp]
theorem map_add_ofNat' [AddMonoidWithOne G] [AddMonoid H] [AddConstMapClass F G H 1 b]
    (f : F) (x : G) (n : ℕ) [n.AtLeastTwo] :
    f (x + no_index (OfNat.ofNat n)) = f x + (OfNat.ofNat n : ℕ) • b 
Arguments: {F : Type u_1} {G : Type u_2} {H : Type u_3} [FunLike F G H] {b : H} [AddMonoidWithOne G] [AddMonoid H] [AddConstMapClass F G H 1 b] (f : F) (x : G) (n : ℕ) [n.AtLeastTwo]
Goal: f (x + OfNat.ofNat n) = f x + OfNat.ofNat n • b
-- AddConstMapClass.map_add_nat
theorem map_add_nat [AddMonoidWithOne G] [AddMonoidWithOne H] [AddConstMapClass F G H 1 1]
    (f : F) (x : G) (n : ℕ) : f (x + n) = f x + n 
Arguments: {F : Type u_1} {G : Type u_2} {H : Type u_3} [FunLike F G H] [AddMonoidWithOne G] [AddMonoidWithOne H] [AddConstMapClass F G H 1 1] (f : F) (x : G) (n : ℕ)
Goal: f (x + ↑n) = f x + ↑n
-- AddConstMapClass.map_add_ofNat
theorem map_add_ofNat [AddMonoidWithOne G] [AddMonoidWithOne H] [AddConstMapClass F G H 1 1]
    (f : F) (x : G) (n : ℕ) [n.AtLeastTwo] :
    f (x + OfNat.ofNat n) = f x + OfNat.ofNat n 
Arguments: {F : Type u_1} {G : Type u_2} {H : Type u_3} [FunLike F G H] [AddMonoidWithOne G] [AddMonoidWithOne H] [AddConstMapClass F G H 1 1] (f : F) (x : G) (n : ℕ) [n.AtLeastTwo]
Goal: f (x + OfNat.ofNat n) = f x + OfNat.ofNat n
-- AddConstMapClass.map_const
@[scoped simp]
theorem map_const [AddZeroClass G] [Add H] [AddConstMapClass F G H a b] (f : F) :
    f a = f 0 + b 
Arguments: {F : Type u_1} {G : Type u_2} {H : Type u_3} [FunLike F G H] {a : G} {b : H} [AddZeroClass G] [Add H] [AddConstMapClass F G H a b] (f : F)
Goal: f a = f 0 + b
-- AddConstMapClass.map_one
theorem map_one [AddZeroClass G] [One G] [Add H] [AddConstMapClass F G H 1 b] (f : F) :
    f 1 = f 0 + b 
Arguments: {F : Type u_1} {G : Type u_2} {H : Type u_3} [FunLike F G H] {b : H} [AddZeroClass G] [One G] [Add H] [AddConstMapClass F G H 1 b] (f : F)
Goal: f 1 = f 0 + b
-- AddConstMapClass.map_nsmul_const
@[scoped simp]
theorem map_nsmul_const [AddMonoid G] [AddMonoid H] [AddConstMapClass F G H a b]
    (f : F) (n : ℕ) : f (n • a) = f 0 + n • b 
Arguments: {F : Type u_1} {G : Type u_2} {H : Type u_3} [FunLike F G H] {a : G} {b : H} [AddMonoid G] [AddMonoid H] [AddConstMapClass F G H a b] (f : F) (n : ℕ)
Goal: f (n • a) = f 0 + n • b
-- AddConstMapClass.map_nat'
@[scoped simp]
theorem map_nat' [AddMonoidWithOne G] [AddMonoid H] [AddConstMapClass F G H 1 b]
    (f : F) (n : ℕ) : f n = f 0 + n • b 
Arguments: {F : Type u_1} {G : Type u_2} {H : Type u_3} [FunLike F G H] {b : H} [AddMonoidWithOne G] [AddMonoid H] [AddConstMapClass F G H 1 b] (f : F) (n : ℕ)
Goal: f ↑n = f 0 + n • b
-- AddConstMapClass.map_ofNat'
theorem map_ofNat' [AddMonoidWithOne G] [AddMonoid H] [AddConstMapClass F G H 1 b]
    (f : F) (n : ℕ) [n.AtLeastTwo] :
    f (OfNat.ofNat n) = f 0 + (OfNat.ofNat n : ℕ) • b 
Arguments: {F : Type u_1} {G : Type u_2} {H : Type u_3} [FunLike F G H] {b : H} [AddMonoidWithOne G] [AddMonoid H] [AddConstMapClass F G H 1 b] (f : F) (n : ℕ) [n.AtLeastTwo]
Goal: f (OfNat.ofNat n) = f 0 + OfNat.ofNat n • b
-- AddConstMapClass.map_nat
theorem map_nat [AddMonoidWithOne G] [AddMonoidWithOne H] [AddConstMapClass F G H 1 1]
    (f : F) (n : ℕ) : f n = f 0 + n 
Arguments: {F : Type u_1} {G : Type u_2} {H : Type u_3} [FunLike F G H] [AddMonoidWithOne G] [AddMonoidWithOne H] [AddConstMapClass F G H 1 1] (f : F) (n : ℕ)
Goal: f ↑n = f 0 + ↑n
-- AddConstMapClass.map_ofNat
theorem map_ofNat [AddMonoidWithOne G] [AddMonoidWithOne H] [AddConstMapClass F G H 1 1]
    (f : F) (n : ℕ) [n.AtLeastTwo] :
    f (OfNat.ofNat n) = f 0 + OfNat.ofNat n 
Arguments: {F : Type u_1} {G : Type u_2} {H : Type u_3} [FunLike F G H] [AddMonoidWithOne G] [AddMonoidWithOne H] [AddConstMapClass F G H 1 1] (f : F) (n : ℕ) [n.AtLeastTwo]
Goal: f (OfNat.ofNat n) = f 0 + OfNat.ofNat n
-- AddConstMapClass.map_const_add
@[scoped simp]
theorem map_const_add [AddCommSemigroup G] [Add H] [AddConstMapClass F G H a b]
    (f : F) (x : G) : f (a + x) = f x + b 
Arguments: {F : Type u_1} {G : Type u_2} {H : Type u_3} [FunLike F G H] {a : G} {b : H} [AddCommSemigroup G] [Add H] [AddConstMapClass F G H a b] (f : F) (x : G)
Goal: f (a + x) = f x + b
-- AddConstMapClass.map_one_add
theorem map_one_add [AddCommMonoidWithOne G] [Add H] [AddConstMapClass F G H 1 b]
    (f : F) (x : G) : f (1 + x) = f x + b 
Arguments: {F : Type u_1} {G : Type u_2} {H : Type u_3} [FunLike F G H] {b : H} [AddCommMonoidWithOne G] [Add H] [AddConstMapClass F G H 1 b] (f : F) (x : G)
Goal: f (1 + x) = f x + b
-- AddConstMapClass.map_nsmul_add
@[scoped simp]
theorem map_nsmul_add [AddCommMonoid G] [AddMonoid H] [AddConstMapClass F G H a b]
    (f : F) (n : ℕ) (x : G) : f (n • a + x) = f x + n • b 
Arguments: {F : Type u_1} {G : Type u_2} {H : Type u_3} [FunLike F G H] {a : G} {b : H} [AddCommMonoid G] [AddMonoid H] [AddConstMapClass F G H a b] (f : F) (n : ℕ) (x : G)
Goal: f (n • a + x) = f x + n • b
-- AddConstMapClass.map_nat_add'
@[scoped simp]
theorem map_nat_add' [AddCommMonoidWithOne G] [AddMonoid H] [AddConstMapClass F G H 1 b]
    (f : F) (n : ℕ) (x : G) : f (↑n + x) = f x + n • b 
Arguments: {F : Type u_1} {G : Type u_2} {H : Type u_3} [FunLike F G H] {b : H} [AddCommMonoidWithOne G] [AddMonoid H] [AddConstMapClass F G H 1 b] (f : F) (n : ℕ) (x : G)
Goal: f (↑n + x) = f x + n • b
-- AddConstMapClass.map_ofNat_add'
theorem map_ofNat_add' [AddCommMonoidWithOne G] [AddMonoid H] [AddConstMapClass F G H 1 b]
    (f : F) (n : ℕ) [n.AtLeastTwo] (x : G) :
    f (OfNat.ofNat n + x) = f x + OfNat.ofNat n • b 
Arguments: {F : Type u_1} {G : Type u_2} {H : Type u_3} [FunLike F G H] {b : H} [AddCommMonoidWithOne G] [AddMonoid H] [AddConstMapClass F G H 1 b] (f : F) (n : ℕ) [n.AtLeastTwo] (x : G)
Goal: f (OfNat.ofNat n + x) = f x + OfNat.ofNat n • b
-- AddConstMapClass.map_nat_add
theorem map_nat_add [AddCommMonoidWithOne G] [AddMonoidWithOne H] [AddConstMapClass F G H 1 1]
    (f : F) (n : ℕ) (x : G) : f (↑n + x) = f x + n 
Arguments: {F : Type u_1} {G : Type u_2} {H : Type u_3} [FunLike F G H] [AddCommMonoidWithOne G] [AddMonoidWithOne H] [AddConstMapClass F G H 1 1] (f : F) (n : ℕ) (x : G)
Goal: f (↑n + x) = f x + ↑n
-- AddConstMapClass.map_ofNat_add
theorem map_ofNat_add [AddCommMonoidWithOne G] [AddMonoidWithOne H] [AddConstMapClass F G H 1 1]
    (f : F) (n : ℕ) [n.AtLeastTwo] (x : G) :
    f (OfNat.ofNat n + x) = f x + OfNat.ofNat n 
Arguments: {F : Type u_1} {G : Type u_2} {H : Type u_3} [FunLike F G H] [AddCommMonoidWithOne G] [AddMonoidWithOne H] [AddConstMapClass F G H 1 1] (f : F) (n : ℕ) [n.AtLeastTwo] (x : G)
Goal: f (OfNat.ofNat n + x) = f x + OfNat.ofNat n
-- AddConstMapClass.map_sub_nsmul
@[scoped simp]
theorem map_sub_nsmul [AddGroup G] [AddGroup H] [AddConstMapClass F G H a b]
    (f : F) (x : G) (n : ℕ) : f (x - n • a) = f x - n • b 
Arguments: {F : Type u_1} {G : Type u_2} {H : Type u_3} [FunLike F G H] {a : G} {b : H} [AddGroup G] [AddGroup H] [AddConstMapClass F G H a b] (f : F) (x : G) (n : ℕ)
Goal: f (x - n • a) = f x - n • b
-- AddConstMapClass.map_sub_const
@[scoped simp]
theorem map_sub_const [AddGroup G] [AddGroup H] [AddConstMapClass F G H a b]
    (f : F) (x : G) : f (x - a) = f x - b 
Arguments: {F : Type u_1} {G : Type u_2} {H : Type u_3} [FunLike F G H] {a : G} {b : H} [AddGroup G] [AddGroup H] [AddConstMapClass F G H a b] (f : F) (x : G)
Goal: f (x - a) = f x - b
-- AddConstMapClass.map_sub_one
theorem map_sub_one [AddGroup G] [One G] [AddGroup H] [AddConstMapClass F G H 1 b]
    (f : F) (x : G) : f (x - 1) = f x - b 
Arguments: {F : Type u_1} {G : Type u_2} {H : Type u_3} [FunLike F G H] {b : H} [AddGroup G] [One G] [AddGroup H] [AddConstMapClass F G H 1 b] (f : F) (x : G)
Goal: f (x - 1) = f x - b
-- AddConstMapClass.map_sub_nat'
@[scoped simp]
theorem map_sub_nat' [AddGroupWithOne G] [AddGroup H] [AddConstMapClass F G H 1 b]
    (f : F) (x : G) (n : ℕ) : f (x - n) = f x - n • b 
Arguments: {F : Type u_1} {G : Type u_2} {H : Type u_3} [FunLike F G H] {b : H} [AddGroupWithOne G] [AddGroup H] [AddConstMapClass F G H 1 b] (f : F) (x : G) (n : ℕ)
Goal: f (x - ↑n) = f x - n • b
-- AddConstMapClass.map_sub_ofNat'
@[scoped simp]
theorem map_sub_ofNat' [AddGroupWithOne G] [AddGroup H] [AddConstMapClass F G H 1 b]
    (f : F) (x : G) (n : ℕ) [n.AtLeastTwo] :
    f (x - no_index (OfNat.ofNat n)) = f x - OfNat.ofNat n • b 
Arguments: {F : Type u_1} {G : Type u_2} {H : Type u_3} [FunLike F G H] {b : H} [AddGroupWithOne G] [AddGroup H] [AddConstMapClass F G H 1 b] (f : F) (x : G) (n : ℕ) [n.AtLeastTwo]
Goal: f (x - OfNat.ofNat n) = f x - OfNat.ofNat n • b
-- AddConstMapClass.map_add_zsmul
@[scoped simp]
theorem map_add_zsmul [AddGroup G] [AddGroup H] [AddConstMapClass F G H a b]
    (f : F) (x : G) : ∀ n : ℤ, f (x + n • a) = f x + n • b
  
Arguments: {F : Type u_1} {G : Type u_2} {H : Type u_3} [FunLike F G H] {a : G} {b : H} [AddGroup G] [AddGroup H] [AddConstMapClass F G H a b] (f : F) (x : G) (n : ℤ)
Goal: f (x + n • a) = f x + n • b
-- AddConstMapClass.map_zsmul_const
@[scoped simp]
theorem map_zsmul_const [AddGroup G] [AddGroup H] [AddConstMapClass F G H a b]
    (f : F) (n : ℤ) : f (n • a) = f 0 + n • b 
Arguments: {F : Type u_1} {G : Type u_2} {H : Type u_3} [FunLike F G H] {a : G} {b : H} [AddGroup G] [AddGroup H] [AddConstMapClass F G H a b] (f : F) (n : ℤ)
Goal: f (n • a) = f 0 + n • b
-- AddConstMapClass.map_add_int'
@[scoped simp]
theorem map_add_int' [AddGroupWithOne G] [AddGroup H] [AddConstMapClass F G H 1 b]
    (f : F) (x : G) (n : ℤ) : f (x + n) = f x + n • b 
Arguments: {F : Type u_1} {G : Type u_2} {H : Type u_3} [FunLike F G H] {b : H} [AddGroupWithOne G] [AddGroup H] [AddConstMapClass F G H 1 b] (f : F) (x : G) (n : ℤ)
Goal: f (x + ↑n) = f x + n • b
-- AddConstMapClass.map_add_int
theorem map_add_int [AddGroupWithOne G] [AddGroupWithOne H] [AddConstMapClass F G H 1 1]
    (f : F) (x : G) (n : ℤ) : f (x + n) = f x + n 
Arguments: {F : Type u_1} {G : Type u_2} {H : Type u_3} [FunLike F G H] [AddGroupWithOne G] [AddGroupWithOne H] [AddConstMapClass F G H 1 1] (f : F) (x : G) (n : ℤ)
Goal: f (x + ↑n) = f x + ↑n
-- AddConstMapClass.map_sub_zsmul
@[scoped simp]
theorem map_sub_zsmul [AddGroup G] [AddGroup H] [AddConstMapClass F G H a b]
    (f : F) (x : G) (n : ℤ) : f (x - n • a) = f x - n • b 
Arguments: {F : Type u_1} {G : Type u_2} {H : Type u_3} [FunLike F G H] {a : G} {b : H} [AddGroup G] [AddGroup H] [AddConstMapClass F G H a b] (f : F) (x : G) (n : ℤ)
Goal: f (x - n • a) = f x - n • b
-- AddConstMapClass.map_sub_int'
@[scoped simp]
theorem map_sub_int' [AddGroupWithOne G] [AddGroup H] [AddConstMapClass F G H 1 b]
    (f : F) (x : G) (n : ℤ) : f (x - n) = f x - n • b 
Arguments: {F : Type u_1} {G : Type u_2} {H : Type u_3} [FunLike F G H] {b : H} [AddGroupWithOne G] [AddGroup H] [AddConstMapClass F G H 1 b] (f : F) (x : G) (n : ℤ)
Goal: f (x - ↑n) = f x - n • b
-- AddConstMapClass.map_sub_int
theorem map_sub_int [AddGroupWithOne G] [AddGroupWithOne H] [AddConstMapClass F G H 1 1]
    (f : F) (x : G) (n : ℤ) : f (x - n) = f x - n 
Arguments: {F : Type u_1} {G : Type u_2} {H : Type u_3} [FunLike F G H] [AddGroupWithOne G] [AddGroupWithOne H] [AddConstMapClass F G H 1 1] (f : F) (x : G) (n : ℤ)
Goal: f (x - ↑n) = f x - ↑n
-- AddConstMapClass.map_zsmul_add
@[scoped simp]
theorem map_zsmul_add [AddCommGroup G] [AddGroup H] [AddConstMapClass F G H a b]
    (f : F) (n : ℤ) (x : G) : f (n • a + x) = f x + n • b 
Arguments: {F : Type u_1} {G : Type u_2} {H : Type u_3} [FunLike F G H] {a : G} {b : H} [AddCommGroup G] [AddGroup H] [AddConstMapClass F G H a b] (f : F) (n : ℤ) (x : G)
Goal: f (n • a + x) = f x + n • b
-- AddConstMapClass.map_int_add'
@[scoped simp]
theorem map_int_add' [AddCommGroupWithOne G] [AddGroup H] [AddConstMapClass F G H 1 b]
    (f : F) (n : ℤ) (x : G) : f (↑n + x) = f x + n • b 
Arguments: {F : Type u_1} {G : Type u_2} {H : Type u_3} [FunLike F G H] {b : H} [AddCommGroupWithOne G] [AddGroup H] [AddConstMapClass F G H 1 b] (f : F) (n : ℤ) (x : G)
Goal: f (↑n + x) = f x + n • b
-- AddConstMapClass.map_int_add
theorem map_int_add [AddCommGroupWithOne G] [AddGroupWithOne H] [AddConstMapClass F G H 1 1]
    (f : F) (n : ℤ) (x : G) : f (↑n + x) = f x + n 
Arguments: {F : Type u_1} {G : Type u_2} {H : Type u_3} [FunLike F G H] [AddCommGroupWithOne G] [AddGroupWithOne H] [AddConstMapClass F G H 1 1] (f : F) (n : ℤ) (x : G)
Goal: f (↑n + x) = f x + ↑n
-- AddConstMapClass.map_fract
theorem map_fract {R : Type*} [LinearOrderedRing R] [FloorRing R] [AddGroup H]
    [FunLike F R H] [AddConstMapClass F R H 1 b] (f : F) (x : R) :
    f (Int.fract x) = f x - ⌊x⌋ • b 
Arguments: {F : Type u_1} {H : Type u_3} {b : H} {R : Type u_4} [LinearOrderedRing R] [FloorRing R] [AddGroup H] [FunLike F R H] [AddConstMapClass F R H 1 b] (f : F) (x : R)
Goal: f (Int.fract x) = f x - ⌊x⌋ • b