ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- AddConstMapClass.rel_map_of_Icc
protected theorem rel_map_of_Icc [LinearOrderedAddCommGroup G] [Archimedean G] [AddGroup H]
    [AddConstMapClass F G H a b] {f : F} {R : H → H → Prop} [IsTrans H R]
    [hR : CovariantClass H H (fun x y ↦ y + x) R] (ha : 0 < a) {l : G}
    (hf : ∀ x ∈ Icc l (l + a), ∀ y ∈ Icc l (l + a), x < y → R (f x) (f y)) :
    ((· < ·) ⇒ R) f f 
Arguments: {F : Type u_1} {G : Type u_2} {H : Type u_3} [FunLike F G H] {a : G} {b : H} [LinearOrderedAddCommGroup G] [Archimedean G] [AddGroup H] [AddConstMapClass F G H a b] {f : F} {R : H → H → Prop} [IsTrans H R] [hR : CovariantClass H H (fun x y => y + x) R] (ha : 0 < a) {l : G} (hf : ∀ x ∈ Set.Icc l (l + a), ∀ y ∈ Set.Icc l (l + a), x < y → R (f x) (f y))
Goal: ((fun x x_1 => x < x_1) ⇒ R) ⇑f ⇑f
-- AddConstMapClass.monotone_iff_Icc
theorem monotone_iff_Icc [LinearOrderedAddCommGroup G] [Archimedean G] [OrderedAddCommGroup H]
    [AddConstMapClass F G H a b] {f : F} (ha : 0 < a) (l : G) :
    Monotone f ↔ MonotoneOn f (Icc l (l + a)) 
Arguments: {F : Type u_1} {G : Type u_2} {H : Type u_3} [FunLike F G H] {a : G} {b : H} [LinearOrderedAddCommGroup G] [Archimedean G] [OrderedAddCommGroup H] [AddConstMapClass F G H a b] {f : F} (ha : 0 < a) (l : G)
Goal: Monotone ⇑f ↔ MonotoneOn (⇑f) (Set.Icc l (l + a))
-- AddConstMapClass.antitone_iff_Icc
theorem antitone_iff_Icc [LinearOrderedAddCommGroup G] [Archimedean G] [OrderedAddCommGroup H]
    [AddConstMapClass F G H a b] {f : F} (ha : 0 < a) (l : G) :
    Antitone f ↔ AntitoneOn f (Icc l (l + a)) 
Arguments: {F : Type u_1} {G : Type u_2} {H : Type u_3} [FunLike F G H] {a : G} {b : H} [LinearOrderedAddCommGroup G] [Archimedean G] [OrderedAddCommGroup H] [AddConstMapClass F G H a b] {f : F} (ha : 0 < a) (l : G)
Goal: Antitone ⇑f ↔ AntitoneOn (⇑f) (Set.Icc l (l + a))
-- AddConstMapClass.strictMono_iff_Icc
theorem strictMono_iff_Icc [LinearOrderedAddCommGroup G] [Archimedean G] [OrderedAddCommGroup H]
    [AddConstMapClass F G H a b] {f : F} (ha : 0 < a) (l : G) :
    StrictMono f ↔ StrictMonoOn f (Icc l (l + a)) 
Arguments: {F : Type u_1} {G : Type u_2} {H : Type u_3} [FunLike F G H] {a : G} {b : H} [LinearOrderedAddCommGroup G] [Archimedean G] [OrderedAddCommGroup H] [AddConstMapClass F G H a b] {f : F} (ha : 0 < a) (l : G)
Goal: StrictMono ⇑f ↔ StrictMonoOn (⇑f) (Set.Icc l (l + a))
-- AddConstMapClass.strictAnti_iff_Icc
theorem strictAnti_iff_Icc [LinearOrderedAddCommGroup G] [Archimedean G] [OrderedAddCommGroup H]
    [AddConstMapClass F G H a b] {f : F} (ha : 0 < a) (l : G) :
    StrictAnti f ↔ StrictAntiOn f (Icc l (l + a)) 
Arguments: {F : Type u_1} {G : Type u_2} {H : Type u_3} [FunLike F G H] {a : G} {b : H} [LinearOrderedAddCommGroup G] [Archimedean G] [OrderedAddCommGroup H] [AddConstMapClass F G H a b] {f : F} (ha : 0 < a) (l : G)
Goal: StrictAnti ⇑f ↔ StrictAntiOn (⇑f) (Set.Icc l (l + a))
-- AddConstMap.coe_mk
@[simp, push_cast] theorem coe_mk (f : G → H) (hf) : ⇑(mk f hf : G →+c[a, b] H) = f 
Arguments: {G : Type u_1} {H : Type u_2} [Add G] [Add H] {a : G} {b : H} (f : G → H) (hf : ∀ (x : G), f (x + a) = f x + b)
Goal: ⇑{ toFun := f, map_add_const' := hf } = f
-- AddConstMap.mk_coe
@[simp] theorem mk_coe (f : G →+c[a, b] H) : mk f f.2 = f 
Arguments: {G : Type u_1} {H : Type u_2} [Add G] [Add H] {a : G} {b : H} (f : AddConstMap G H a b)
Goal: { toFun := ⇑f, map_add_const' := ⋯ } = f
-- AddConstMap.toFun_eq_coe
@[simp] theorem toFun_eq_coe (f : G →+c[a, b] H) : f.toFun = f 
Arguments: {G : Type u_1} {H : Type u_2} [Add G] [Add H] {a : G} {b : H} (f : AddConstMap G H a b)
Goal: f.toFun = ⇑f
-- AddConstMap.ext
@[ext] protected theorem ext {f g : G →+c[a, b] H} (h : ∀ x, f x = g x) : f = g 
Arguments: {G : Type u_1} {H : Type u_2} [Add G] [Add H] {a : G} {b : H} {f g : AddConstMap G H a b} (h : ∀ (x : G), f x = g x)
Goal: f = g
-- AddConstMap.coe_id
@[simps (config := .asFn)]
protected def id : G →+c[a, a] G 
Arguments: {G : Type u_1} [Add G] {a : G}
Goal: ⇑AddConstMap.id = id
-- AddConstMap.coe_comp
@[simps (config := .asFn)]
def comp {K : Type*} [Add K] {c : K} (g : H →+c[b, c] K) (f : G →+c[a, b] H) :
    G →+c[a, c] K 
Arguments: {G : Type u_1} {H : Type u_2} [Add G] [Add H] {a : G} {b : H} {K : Type u_3} [Add K] {c : K} (g : AddConstMap H K b c) (f : AddConstMap G H a b)
Goal: ⇑(g.comp f) = ⇑g ∘ ⇑f
-- AddConstMap.comp_id
@[simp] theorem comp_id (f : G →+c[a, b] H) : f.comp .id = f 
Arguments: {G : Type u_1} {H : Type u_2} [Add G] [Add H] {a : G} {b : H} (f : AddConstMap G H a b)
Goal: f.comp AddConstMap.id = f
-- AddConstMap.id_comp
@[simp] theorem id_comp (f : G →+c[a, b] H) : .comp .id f = f 
Arguments: {G : Type u_1} {H : Type u_2} [Add G] [Add H] {a : G} {b : H} (f : AddConstMap G H a b)
Goal: AddConstMap.id.comp f = f
-- AddConstMap.coe_replaceConsts
@[simps (config := .asFn)]
def replaceConsts (f : G →+c[a, b] H) (a' b') (ha : a = a') (hb : b = b') :
    G →+c[a', b'] H 
Arguments: {G : Type u_1} {H : Type u_2} [Add G] [Add H] {a : G} {b : H} (f : AddConstMap G H a b) (a' : G) (b' : H) (ha : a = a') (hb : b = b')
Goal: ⇑(f.replaceConsts a' b' ha hb) = ⇑f
-- AddConstMap.coe_vadd
@[simp, norm_cast]
theorem coe_vadd {K : Type*} [VAdd K H] [VAddAssocClass K H H] (c : K) (f : G →+c[a, b] H) :
    ⇑(c +ᵥ f) = c +ᵥ ⇑f 
Arguments: {G : Type u_1} {H : Type u_2} [Add G] [Add H] {a : G} {b : H} {K : Type u_3} [VAdd K H] [VAddAssocClass K H H] (c : K) (f : AddConstMap G H a b)
Goal: ⇑(c +ᵥ f) = c +ᵥ ⇑f
-- AddConstMap.mul_def
theorem mul_def (f g : G →+c[a, a] G) : f * g = f.comp g 
Arguments: {G : Type u_1} [Add G] {a : G} (f g : AddConstMap G G a a)
Goal: f * g = f.comp g
-- AddConstMap.coe_mul
@[simp, push_cast] theorem coe_mul (f g : G →+c[a, a] G) : ⇑(f * g) = f ∘ g 
Arguments: {G : Type u_1} [Add G] {a : G} (f g : AddConstMap G G a a)
Goal: ⇑(f * g) = ⇑f ∘ ⇑g
-- AddConstMap.one_def
theorem one_def : (1 : G →+c[a, a] G) = .id 
Arguments: {G : Type u_1} [Add G] {a : G}
Goal: 1 = AddConstMap.id
-- AddConstMap.coe_one
@[simp, push_cast] theorem coe_one : ⇑(1 : G →+c[a, a] G) = id 
Arguments: {G : Type u_1} [Add G] {a : G}
Goal: ⇑1 = id
-- AddConstMap.coe_pow
@[simp, push_cast] theorem coe_pow (f : G →+c[a, a] G) (n : ℕ) : ⇑(f ^ n) = f^[n] 
Arguments: {G : Type u_1} [Add G] {a : G} (f : AddConstMap G G a a) (n : ℕ)
Goal: ⇑(f ^ n) = (⇑f)^[n]
-- AddConstMap.pow_apply
theorem pow_apply (f : G →+c[a, a] G) (n : ℕ) (x : G) : (f ^ n) x = f^[n] x 
Arguments: {G : Type u_1} [Add G] {a : G} (f : AddConstMap G G a a) (n : ℕ) (x : G)
Goal: (f ^ n) x = (⇑f)^[n] x
-- AddConstMap.toEnd_apply
@[simps (config := .asFn)]
def toEnd : (G →+c[a, a] G) →* Function.End G 
Arguments: {G : Type u_1} [Add G] {a : G}
Goal: ⇑AddConstMap.toEnd = DFunLike.coe
-- AddConstMap.coe_smul
@[simps (config := .asFn)]
def smul [DistribSMul K H] (c : K) (f : G →+c[a, b] H) : G →+c[a, c • b] H 
Arguments: {G : Type u_1} {H : Type u_2} {K : Type u_3} [Add G] [AddZeroClass H] {a : G} {b : H} [DistribSMul K H] (c : K) (f : AddConstMap G H a b)
Goal: ⇑(AddConstMap.smul c f) = c • ⇑f
-- AddConstMap.coe_addLeftHom_apply
@[simps! (config := .asFn)]
def addLeftHom : Multiplicative G →* (G →+c[a, a] G) 
Arguments: {G : Type u_1} [AddMonoid G] {a : G} (c : Multiplicative G)
Goal: ⇑(AddConstMap.addLeftHom c) = c • id
-- AddConstMap.coe_conjNeg_apply
@[simps! apply_coe]
def conjNeg : (G →+c[a, b] H) ≃ (G →+c[a, b] H) 
Arguments: {G : Type u_1} {H : Type u_2} [AddCommGroup G] [AddCommGroup H] {a : G} {b : H} (f : AddConstMap G H a b) (x : G)
Goal: (AddConstMap.conjNeg f) x = -f (-x)
-- AddConstMap.conjNeg_symm
@[simp] theorem conjNeg_symm : (conjNeg (a := a) (b := b)).symm = conjNeg 
Arguments: {G : Type u_1} {H : Type u_2} [AddCommGroup G] [AddCommGroup H] {a : G} {b : H}
Goal: AddConstMap.conjNeg.symm = AddConstMap.conjNeg
-- Subbimodule.mk_carrier
@[simps]
def mk (p : AddSubmonoid M) (hA : ∀ (a : A) {m : M}, m ∈ p → a • m ∈ p)
    (hB : ∀ (b : B) {m : M}, m ∈ p → b • m ∈ p) : Submodule (A ⊗[R] B) M 
Arguments: {R : Type u_1} {A : Type u_2} {B : Type u_3} {M : Type u_4} [CommSemiring R] [AddCommMonoid M] [Module R M] [Semiring A] [Semiring B] [Module A M] [Module B M] [Algebra R A] [Algebra R B] [IsScalarTower R A M] [IsScalarTower R B M] [SMulCommClass A B M] (p : AddSubmonoid M) (hA : ∀ (a : A) {m : M}, m ∈ p → a • m ∈ p) (hB : ∀ (b : B) {m : M}, m ∈ p → b • m ∈ p)
Goal: ↑(Subbimodule.mk p hA hB) = ↑p
-- Subbimodule.smul_mem
theorem smul_mem (p : Submodule (A ⊗[R] B) M) (a : A) {m : M} (hm : m ∈ p) : a • m ∈ p 
Arguments: {R : Type u_1} {A : Type u_2} {B : Type u_3} {M : Type u_4} [CommSemiring R] [AddCommMonoid M] [Module R M] [Semiring A] [Semiring B] [Module A M] [Module B M] [Algebra R A] [Algebra R B] [IsScalarTower R A M] [IsScalarTower R B M] [SMulCommClass A B M] (p : Submodule (A ⊗[R] B) M) (a : A) {m : M} (hm : m ∈ p)
Goal: a • m ∈ p
-- Subbimodule.smul_mem'
theorem smul_mem' (p : Submodule (A ⊗[R] B) M) (b : B) {m : M} (hm : m ∈ p) : b • m ∈ p 
Arguments: {R : Type u_1} {A : Type u_2} {B : Type u_3} {M : Type u_4} [CommSemiring R] [AddCommMonoid M] [Module R M] [Semiring A] [Semiring B] [Module A M] [Module B M] [Algebra R A] [Algebra R B] [IsScalarTower R A M] [IsScalarTower R B M] [SMulCommClass A B M] (p : Submodule (A ⊗[R] B) M) (b : B) {m : M} (hm : m ∈ p)
Goal: b • m ∈ p
-- Subbimodule.baseChange_carrier
@[simps!]
def baseChange (S : Type*) [CommSemiring S] [Module S M] [Algebra S A] [Algebra S B]
    [IsScalarTower S A M] [IsScalarTower S B M] (p : Submodule (A ⊗[R] B) M) :
    Submodule (A ⊗[S] B) M 
Arguments: {R : Type u_1} {A : Type u_2} {B : Type u_3} {M : Type u_4} [CommSemiring R] [AddCommMonoid M] [Module R M] [Semiring A] [Semiring B] [Module A M] [Module B M] [Algebra R A] [Algebra R B] [IsScalarTower R A M] [IsScalarTower R B M] [SMulCommClass A B M] (S : Type u_5) [CommSemiring S] [Module S M] [Algebra S A] [Algebra S B] [IsScalarTower S A M] [IsScalarTower S B M] (p : Submodule (A ⊗[R] B) M)
Goal: ↑(Subbimodule.baseChange S p) = ↑p
-- Subbimodule.toSubmodule_carrier
@[simps]
def toSubmodule (p : Submodule (A ⊗[R] B) M) : Submodule A M 
Arguments: {R : Type u_1} {A : Type u_2} {B : Type u_3} {M : Type u_4} [CommSemiring R] [AddCommMonoid M] [Module R M] [Semiring A] [Semiring B] [Module A M] [Module B M] [Algebra R A] [Algebra R B] [IsScalarTower R A M] [IsScalarTower R B M] [SMulCommClass A B M] (p : Submodule (A ⊗[R] B) M)
Goal: ↑(Subbimodule.toSubmodule p) = ↑p
-- Subbimodule.toSubmodule'_carrier
@[simps]
def toSubmodule' (p : Submodule (A ⊗[R] B) M) : Submodule B M 
Arguments: {R : Type u_1} {A : Type u_2} {B : Type u_3} {M : Type u_4} [CommSemiring R] [AddCommMonoid M] [Module R M] [Semiring A] [Semiring B] [Module A M] [Module B M] [Algebra R A] [Algebra R B] [IsScalarTower R A M] [IsScalarTower R B M] [SMulCommClass A B M] (p : Submodule (A ⊗[R] B) M)
Goal: ↑(Subbimodule.toSubmodule' p) = ↑p
-- Subbimodule.toSubbimoduleInt_carrier
@[simps!]
def toSubbimoduleInt (p : Submodule (R ⊗[ℕ] S) M) : Submodule (R ⊗[ℤ] S) M 
Arguments: (R : Type u_1) (S : Type u_2) (M : Type u_3) [Ring R] [Ring S] [AddCommGroup M] [Module R M] [Module S M] [SMulCommClass R S M] (p : Submodule (R ⊗[ℕ] S) M)
Goal: ↑(Subbimodule.toSubbimoduleInt R S M p) = ↑p
-- Subbimodule.toSubbimoduleNat_carrier
@[simps!]
def toSubbimoduleNat (p : Submodule (R ⊗[ℤ] S) M) : Submodule (R ⊗[ℕ] S) M 
Arguments: (R : Type u_1) (S : Type u_2) (M : Type u_3) [Ring R] [Ring S] [AddCommGroup M] [Module R M] [Module S M] [SMulCommClass R S M] (p : Submodule (R ⊗[ℤ] S) M)
Goal: ↑(Subbimodule.toSubbimoduleNat R S M p) = ↑p
-- vadd_pi_subset
@[to_additive]
theorem smul_pi_subset [∀ i, SMul K (R i)] (r : K) (s : Set ι) (t : ∀ i, Set (R i)) :
    r • pi s t ⊆ pi s (r • t) 
Arguments: {K : Type u_1} {ι : Type u_2} {R : ι → Type u_3} [(i : ι) → VAdd K (R i)] (r : K) (s : Set ι) (t : (i : ι) → Set (R i))
Goal: r +ᵥ s.pi t ⊆ s.pi (r +ᵥ t)
-- smul_pi_subset
@[to_additive]
theorem smul_pi_subset [∀ i, SMul K (R i)] (r : K) (s : Set ι) (t : ∀ i, Set (R i)) :
    r • pi s t ⊆ pi s (r • t) 
Arguments: {K : Type u_1} {ι : Type u_2} {R : ι → Type u_3} [(i : ι) → SMul K (R i)] (r : K) (s : Set ι) (t : (i : ι) → Set (R i))
Goal: r • s.pi t ⊆ s.pi (r • t)
-- smul_univ_pi
@[to_additive]
theorem smul_univ_pi [∀ i, SMul K (R i)] (r : K) (t : ∀ i, Set (R i)) :
    r • pi (univ : Set ι) t = pi (univ : Set ι) (r • t) 
Arguments: {K : Type u_1} {ι : Type u_2} {R : ι → Type u_3} [(i : ι) → SMul K (R i)] (r : K) (t : (i : ι) → Set (R i))
Goal: r • Set.univ.pi t = Set.univ.pi (r • t)
-- vadd_univ_pi
@[to_additive]
theorem smul_univ_pi [∀ i, SMul K (R i)] (r : K) (t : ∀ i, Set (R i)) :
    r • pi (univ : Set ι) t = pi (univ : Set ι) (r • t) 
Arguments: {K : Type u_1} {ι : Type u_2} {R : ι → Type u_3} [(i : ι) → VAdd K (R i)] (r : K) (t : (i : ι) → Set (R i))
Goal: r +ᵥ Set.univ.pi t = Set.univ.pi (r +ᵥ t)
-- smul_pi
@[to_additive]
theorem smul_pi [Group K] [∀ i, MulAction K (R i)] (r : K) (S : Set ι) (t : ∀ i, Set (R i)) :
    r • S.pi t = S.pi (r • t) 
Arguments: {K : Type u_1} {ι : Type u_2} {R : ι → Type u_3} [Group K] [(i : ι) → MulAction K (R i)] (r : K) (S : Set ι) (t : (i : ι) → Set (R i))
Goal: r • S.pi t = S.pi (r • t)
-- vadd_pi
@[to_additive]
theorem smul_pi [Group K] [∀ i, MulAction K (R i)] (r : K) (S : Set ι) (t : ∀ i, Set (R i)) :
    r • S.pi t = S.pi (r • t) 
Arguments: {K : Type u_1} {ι : Type u_2} {R : ι → Type u_3} [AddGroup K] [(i : ι) → AddAction K (R i)] (r : K) (S : Set ι) (t : (i : ι) → Set (R i))
Goal: r +ᵥ S.pi t = S.pi (r +ᵥ t)
-- smul_pi₀
theorem smul_pi₀ [GroupWithZero K] [∀ i, MulAction K (R i)] {r : K} (S : Set ι) (t : ∀ i, Set (R i))
    (hr : r ≠ 0) : r • S.pi t = S.pi (r • t) 
Arguments: {K : Type u_1} {ι : Type u_2} {R : ι → Type u_3} [GroupWithZero K] [(i : ι) → MulAction K (R i)] {r : K} (S : Set ι) (t : (i : ι) → Set (R i)) (hr : r ≠ 0)
Goal: r • S.pi t = S.pi (r • t)
-- Submodule.isInternal_prime_power_torsion_of_is_torsion_by_ideal
theorem isInternal_prime_power_torsion_of_is_torsion_by_ideal {I : Ideal R} (hI : I ≠ ⊥)
    (hM : Module.IsTorsionBySet R M I) :
    DirectSum.IsInternal fun p : (factors I).toFinset =>
      torsionBySet R M (p ^ (factors I).count ↑p : Ideal R) 
Arguments: {R : Type u} [CommRing R] [IsDomain R] {M : Type v} [AddCommGroup M] [Module R M] [IsDedekindDomain R] {I : Ideal R} (hI : I ≠ ⊥) (hM : Module.IsTorsionBySet R M ↑I)
Goal: DirectSum.IsInternal fun p =>
  Submodule.torsionBySet R M ↑(↑p ^ Multiset.count (↑p) (UniqueFactorizationMonoid.factors I))
-- Submodule.isInternal_prime_power_torsion
theorem isInternal_prime_power_torsion [Module.Finite R M] (hM : Module.IsTorsion R M) :
    DirectSum.IsInternal fun p : (factors (⊤ : Submodule R M).annihilator).toFinset =>
      torsionBySet R M (p ^ (factors (⊤ : Submodule R M).annihilator).count ↑p : Ideal R) 
Arguments: {R : Type u} [CommRing R] [IsDomain R] {M : Type v} [AddCommGroup M] [Module R M] [IsDedekindDomain R] [Module.Finite R M] (hM : Module.IsTorsion R M)
Goal: DirectSum.IsInternal fun p =>
  Submodule.torsionBySet R M ↑(↑p ^ Multiset.count (↑p) (UniqueFactorizationMonoid.factors ⊤.annihilator))
-- Submodule.exists_isInternal_prime_power_torsion
theorem exists_isInternal_prime_power_torsion [Module.Finite R M] (hM : Module.IsTorsion R M) :
    ∃ (P : Finset <| Ideal R) (_ : DecidableEq P) (_ : ∀ p ∈ P, Prime p) (e : P → ℕ),
      DirectSum.IsInternal fun p : P => torsionBySet R M (p ^ e p : Ideal R) 
Arguments: {R : Type u} [CommRing R] [IsDomain R] {M : Type v} [AddCommGroup M] [Module R M] [IsDedekindDomain R] [Module.Finite R M] (hM : Module.IsTorsion R M)
Goal: ∃ P x, ∃ (_ : ∀ p ∈ P, Prime p), ∃ e, DirectSum.IsInternal fun p => Submodule.torsionBySet R M ↑(↑p ^ e p)
-- map_ratCast_smul
theorem map_ratCast_smul [AddCommGroup M] [AddCommGroup M₂] {F : Type*} [FunLike F M M₂]
    [AddMonoidHomClass F M M₂] (f : F) (R S : Type*) [DivisionRing R] [DivisionRing S] [Module R M]
    [Module S M₂] (c : ℚ) (x : M) :
    f ((c : R) • x) = (c : S) • f x 
Arguments: {M : Type u_1} {M₂ : Type u_2} [AddCommGroup M] [AddCommGroup M₂] {F : Type u_3} [FunLike F M M₂] [AddMonoidHomClass F M M₂] (f : F) (R : Type u_4) (S : Type u_5) [DivisionRing R] [DivisionRing S] [Module R M] [Module S M₂] (c : ℚ) (x : M)
Goal: f (↑c • x) = ↑c • f x
-- map_rat_smul
theorem map_rat_smul [AddCommGroup M] [AddCommGroup M₂]
    [_instM : Module ℚ M] [_instM₂ : Module ℚ M₂]
    {F : Type*} [FunLike F M M₂] [AddMonoidHomClass F M M₂]
    (f : F) (c : ℚ) (x : M) : f (c • x) = c • f x 
Arguments: {M : Type u_1} {M₂ : Type u_2} [AddCommGroup M] [AddCommGroup M₂] [_instM : Module ℚ M] [_instM₂ : Module ℚ M₂] {F : Type u_3} [FunLike F M M₂] [AddMonoidHomClass F M M₂] (f : F) (c : ℚ) (x : M)
Goal: f (c • x) = c • f x
-- ratCast_smul_eq
theorem ratCast_smul_eq {E : Type*} (R S : Type*) [AddCommGroup E] [DivisionRing R]
    [DivisionRing S] [Module R E] [Module S E] (r : ℚ) (x : E) : (r : R) • x = (r : S) • x 
Arguments: {E : Type u_3} (R : Type u_4) (S : Type u_5) [AddCommGroup E] [DivisionRing R] [DivisionRing S] [Module R E] [Module S E] (r : ℚ) (x : E)
Goal: ↑r • x = ↑r • x
-- LinearMap.one_eq_id
theorem one_eq_id : (1 : Module.End R M) = id 
Arguments: {R : Type u_1} {M : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M]
Goal: 1 = LinearMap.id
-- LinearMap.mul_eq_comp
theorem mul_eq_comp (f g : Module.End R M) : f * g = f.comp g 
Arguments: {R : Type u_1} {M : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M] (f g : Module.End R M)
Goal: f * g = f ∘ₗ g
-- LinearMap.one_apply
@[simp]
theorem one_apply (x : M) : (1 : Module.End R M) x = x 
Arguments: {R : Type u_1} {M : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M] (x : M)
Goal: 1 x = x
-- LinearMap.mul_apply
@[simp]
theorem mul_apply (f g : Module.End R M) (x : M) : (f * g) x = f (g x) 
Arguments: {R : Type u_1} {M : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M] (f g : Module.End R M) (x : M)
Goal: (f * g) x = f (g x)
-- LinearMap.coe_one
theorem coe_one : ⇑(1 : Module.End R M) = _root_.id 
Arguments: {R : Type u_1} {M : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M]
Goal: ⇑1 = id
-- LinearMap.coe_mul
theorem coe_mul (f g : Module.End R M) : ⇑(f * g) = f ∘ g 
Arguments: {R : Type u_1} {M : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M] (f g : Module.End R M)
Goal: ⇑(f * g) = ⇑f ∘ ⇑g
-- Module.End.natCast_apply
@[simp]
theorem _root_.Module.End.natCast_apply (n : ℕ) (m : M) : (↑n : Module.End R M) m = n • m 
Arguments: {R : Type u_1} {M : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M] (n : ℕ) (m : M)
Goal: ↑n m = n • m
-- Module.End.ofNat_apply
@[simp]
theorem _root_.Module.End.ofNat_apply (n : ℕ) [n.AtLeastTwo] (m : M) :
    (no_index (OfNat.ofNat n) : Module.End R M) m = OfNat.ofNat n • m 
Arguments: {R : Type u_1} {M : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M] (n : ℕ) [n.AtLeastTwo] (m : M)
Goal: (OfNat.ofNat n) m = OfNat.ofNat n • m
-- Module.End.intCast_apply
@[simp]
theorem _root_.Module.End.intCast_apply (z : ℤ) (m : N₁) : (z : Module.End R N₁) m = z • m 
Arguments: {R : Type u_1} {N₁ : Type u_10} [Semiring R] [AddCommGroup N₁] [Module R N₁] (z : ℤ) (m : N₁)
Goal: ↑z m = z • m
-- Module.End_isUnit_apply_inv_apply_of_isUnit
theorem _root_.Module.End_isUnit_apply_inv_apply_of_isUnit
    {f : Module.End R M} (h : IsUnit f) (x : M) :
    f (h.unit.inv x) = x 
Arguments: {R : Type u_1} {M : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M] {f : Module.End R M} (h : IsUnit f) (x : M)
Goal: f (h.unit.inv x) = x
-- Module.End_isUnit_inv_apply_apply_of_isUnit
theorem _root_.Module.End_isUnit_inv_apply_apply_of_isUnit
    {f : Module.End R M} (h : IsUnit f) (x : M) :
    h.unit.inv (f x) = x 
Arguments: {R : Type u_1} {M : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M] {f : Module.End R M} (h : IsUnit f) (x : M)
Goal: h.unit.inv (f x) = x
-- LinearMap.coe_pow
theorem coe_pow (f : M →ₗ[R] M) (n : ℕ) : ⇑(f ^ n) = f^[n] 
Arguments: {R : Type u_1} {M : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M] (f : M →ₗ[R] M) (n : ℕ)
Goal: ⇑(f ^ n) = (⇑f)^[n]
-- LinearMap.pow_apply
theorem pow_apply (f : M →ₗ[R] M) (n : ℕ) (m : M) : (f ^ n) m = f^[n] m 
Arguments: {R : Type u_1} {M : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M] (f : M →ₗ[R] M) (n : ℕ) (m : M)
Goal: (f ^ n) m = (⇑f)^[n] m
-- LinearMap.pow_map_zero_of_le
theorem pow_map_zero_of_le {f : Module.End R M} {m : M} {k l : ℕ} (hk : k ≤ l)
    (hm : (f ^ k) m = 0) : (f ^ l) m = 0 
Arguments: {R : Type u_1} {M : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M] {f : Module.End R M} {m : M} {k l : ℕ} (hk : k ≤ l) (hm : (f ^ k) m = 0)
Goal: (f ^ l) m = 0
-- LinearMap.commute_pow_left_of_commute
theorem commute_pow_left_of_commute
    [Semiring R₂] [AddCommMonoid M₂] [Module R₂ M₂] {σ₁₂ : R →+* R₂}
    {f : M →ₛₗ[σ₁₂] M₂} {g : Module.End R M} {g₂ : Module.End R₂ M₂}
    (h : g₂.comp f = f.comp g) (k : ℕ) : (g₂ ^ k).comp f = f.comp (g ^ k) 
Arguments: {R : Type u_1} {R₂ : Type u_3} {M : Type u_5} {M₂ : Type u_7} [Semiring R] [AddCommMonoid M] [Module R M] [Semiring R₂] [AddCommMonoid M₂] [Module R₂ M₂] {σ₁₂ : R →+* R₂} {f : M →ₛₗ[σ₁₂] M₂} {g : Module.End R M} {g₂ : Module.End R₂ M₂} (h : LinearMap.comp g₂ f = f.comp g) (k : ℕ)
Goal: LinearMap.comp (g₂ ^ k) f = f.comp (g ^ k)
-- LinearMap.id_pow
@[simp]
theorem id_pow (n : ℕ) : (id : M →ₗ[R] M) ^ n = id 
Arguments: {R : Type u_1} {M : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M] (n : ℕ)
Goal: LinearMap.id ^ n = LinearMap.id
-- LinearMap.iterate_succ
theorem iterate_succ (n : ℕ) : f' ^ (n + 1) = comp (f' ^ n) f' 
Arguments: {R : Type u_1} {M : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M] {f' : M →ₗ[R] M} (n : ℕ)
Goal: f' ^ (n + 1) = (f' ^ n) ∘ₗ f'
-- LinearMap.iterate_surjective
theorem iterate_surjective (h : Surjective f') : ∀ n : ℕ, Surjective (f' ^ n)
  
Arguments: {R : Type u_1} {M : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M] {f' : M →ₗ[R] M} (h : Function.Surjective ⇑f') (n : ℕ)
Goal: Function.Surjective ⇑(f' ^ n)
-- LinearMap.iterate_injective
theorem iterate_injective (h : Injective f') : ∀ n : ℕ, Injective (f' ^ n)
  
Arguments: {R : Type u_1} {M : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M] {f' : M →ₗ[R] M} (h : Function.Injective ⇑f') (n : ℕ)
Goal: Function.Injective ⇑(f' ^ n)
-- LinearMap.iterate_bijective
theorem iterate_bijective (h : Bijective f') : ∀ n : ℕ, Bijective (f' ^ n)
  
Arguments: {R : Type u_1} {M : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M] {f' : M →ₗ[R] M} (h : Function.Bijective ⇑f') (n : ℕ)
Goal: Function.Bijective ⇑(f' ^ n)
-- LinearMap.injective_of_iterate_injective
theorem injective_of_iterate_injective {n : ℕ} (hn : n ≠ 0) (h : Injective (f' ^ n)) :
    Injective f' 
Arguments: {R : Type u_1} {M : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M] {f' : M →ₗ[R] M} {n : ℕ} (hn : n ≠ 0) (h : Function.Injective ⇑(f' ^ n))
Goal: Function.Injective ⇑f'
-- LinearMap.surjective_of_iterate_surjective
theorem surjective_of_iterate_surjective {n : ℕ} (hn : n ≠ 0) (h : Surjective (f' ^ n)) :
    Surjective f' 
Arguments: {R : Type u_1} {M : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M] {f' : M →ₗ[R] M} {n : ℕ} (hn : n ≠ 0) (h : Function.Surjective ⇑(f' ^ n))
Goal: Function.Surjective ⇑f'
-- LinearMap.smul_def
@[simp]
protected theorem smul_def (f : Module.End R M) (a : M) : f • a = f a 
Arguments: {R : Type u_1} {M : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M] (f : Module.End R M) (a : M)
Goal: f • a = f a
-- DistribMulAction.toLinearMap_apply
@[simps]
def toLinearMap (s : S) : M →ₗ[R] M 
Arguments: (R : Type u_1) {S : Type u_4} (M : Type u_5) [Semiring R] [AddCommMonoid M] [Module R M] [Monoid S] [DistribMulAction S M] [SMulCommClass S R M] (s : S)
Goal: ∀ (a : M), (DistribMulAction.toLinearMap R M s) a = s • a
-- DistribMulAction.toModuleEnd_apply
@[simps]
def toModuleEnd : S →* Module.End R M 
Arguments: (R : Type u_1) {S : Type u_4} (M : Type u_5) [Semiring R] [AddCommMonoid M] [Module R M] [Monoid S] [DistribMulAction S M] [SMulCommClass S R M] (s : S)
Goal: (DistribMulAction.toModuleEnd R M) s = DistribMulAction.toLinearMap R M s
-- Module.toModuleEnd_apply
@[simps]
def toModuleEnd : S →+* Module.End R M 
Arguments: (R : Type u_1) {S : Type u_4} (M : Type u_5) [Semiring R] [AddCommMonoid M] [Module R M] [Semiring S] [Module S M] [SMulCommClass S R M] (s : S)
Goal: (Module.toModuleEnd R M) s = DistribMulAction.toLinearMap R M s