ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- algebraMap.coe_add
@[norm_cast]
theorem coe_add (a b : R) : (↑(a + b : R) : A) = ↑a + ↑b 
Arguments: {R : Type u_1} {A : Type u_2} [CommSemiring R] [Semiring A] [Algebra R A] (a b : R)
Goal: ↑(a + b) = ↑a + ↑b
-- algebraMap.coe_mul
@[norm_cast]
theorem coe_mul (a b : R) : (↑(a * b : R) : A) = ↑a * ↑b 
Arguments: {R : Type u_1} {A : Type u_2} [CommSemiring R] [Semiring A] [Algebra R A] (a b : R)
Goal: ↑(a * b) = ↑a * ↑b
-- algebraMap.coe_pow
@[norm_cast]
theorem coe_pow (a : R) (n : ℕ) : (↑(a ^ n : R) : A) = (a : A) ^ n 
Arguments: {R : Type u_1} {A : Type u_2} [CommSemiring R] [Semiring A] [Algebra R A] (a : R) (n : ℕ)
Goal: ↑(a ^ n) = ↑a ^ n
-- algebraMap.coe_neg
@[norm_cast]
theorem coe_neg (x : R) : (↑(-x : R) : A) = -↑x 
Arguments: {R : Type u_1} {A : Type u_2} [CommRing R] [Ring A] [Algebra R A] (x : R)
Goal: ↑(-x) = -↑x
-- algebraMap.coe_sub
@[norm_cast]
theorem coe_sub (a b : R) :
    (↑(a - b : R) : A) = ↑a - ↑b 
Arguments: {R : Type u_1} {A : Type u_2} [CommRing R] [Ring A] [Algebra R A] (a b : R)
Goal: ↑(a - b) = ↑a - ↑b
-- algebraMap.coe_prod
@[norm_cast]
theorem coe_prod {ι : Type*} {s : Finset ι} (a : ι → R) :
    (↑(∏ i ∈ s, a i : R) : A) = ∏ i ∈ s, (↑(a i) : A) 
Arguments: {R : Type u_1} {A : Type u_2} [CommSemiring R] [CommSemiring A] [Algebra R A] {ι : Type u_3} {s : Finset ι} (a : ι → R)
Goal: ↑(∏ i ∈ s, a i) = ∏ i ∈ s, ↑(a i)
-- algebraMap.coe_sum
@[norm_cast]
theorem coe_sum {ι : Type*} {s : Finset ι} (a : ι → R) :
    ↑(∑ i ∈ s, a i) = ∑ i ∈ s, (↑(a i) : A) 
Arguments: {R : Type u_1} {A : Type u_2} [CommSemiring R] [CommSemiring A] [Algebra R A] {ι : Type u_3} {s : Finset ι} (a : ι → R)
Goal: ↑(∑ i ∈ s, a i) = ∑ i ∈ s, ↑(a i)
-- RingHom.algebraMap_toAlgebra
theorem RingHom.algebraMap_toAlgebra {R S} [CommSemiring R] [CommSemiring S] (i : R →+* S) :
    @algebraMap R S _ _ i.toAlgebra = i 
Arguments: {R : Type u_1} {S : Type u_2} [CommSemiring R] [CommSemiring S] (i : R →+* S)
Goal: algebraMap R S = i
-- Algebra.algebra_ext
@[ext]
theorem algebra_ext {R : Type*} [CommSemiring R] {A : Type*} [Semiring A] (P Q : Algebra R A)
    (h : ∀ r : R, (haveI := P; algebraMap R A r) = haveI := Q; algebraMap R A r) :
    P = Q 
Arguments: {R : Type u_2} [CommSemiring R] {A : Type u_3} [Semiring A] (P Q : Algebra R A) (h : ∀ (r : R), (algebraMap R A) r = (algebraMap R A) r)
Goal: P = Q
-- Algebra.smul_def
theorem smul_def (r : R) (x : A) : r • x = algebraMap R A r * x 
Arguments: {R : Type u} {A : Type w} [CommSemiring R] [Semiring A] [Algebra R A] (r : R) (x : A)
Goal: r • x = (algebraMap R A) r * x
-- Algebra.algebraMap_eq_smul_one
theorem algebraMap_eq_smul_one (r : R) : algebraMap R A r = r • (1 : A) 
Arguments: {R : Type u} {A : Type w} [CommSemiring R] [Semiring A] [Algebra R A] (r : R)
Goal: (algebraMap R A) r = r • 1
-- Algebra.algebraMap_eq_smul_one'
theorem algebraMap_eq_smul_one' : ⇑(algebraMap R A) = fun r => r • (1 : A) 
Arguments: {R : Type u} {A : Type w} [CommSemiring R] [Semiring A] [Algebra R A]
Goal: ⇑(algebraMap R A) = fun r => r • 1
-- Algebra.commutes
theorem commutes (r : R) (x : A) : algebraMap R A r * x = x * algebraMap R A r 
Arguments: {R : Type u} {A : Type w} [CommSemiring R] [Semiring A] [Algebra R A] (r : R) (x : A)
Goal: (algebraMap R A) r * x = x * (algebraMap R A) r
-- Algebra.commute_algebraMap_left
lemma commute_algebraMap_left (r : R) (x : A) : Commute (algebraMap R A r) x 
Arguments: {R : Type u} {A : Type w} [CommSemiring R] [Semiring A] [Algebra R A] (r : R) (x : A)
Goal: Commute ((algebraMap R A) r) x
-- Algebra.commute_algebraMap_right
lemma commute_algebraMap_right (r : R) (x : A) : Commute x (algebraMap R A r) 
Arguments: {R : Type u} {A : Type w} [CommSemiring R] [Semiring A] [Algebra R A] (r : R) (x : A)
Goal: Commute x ((algebraMap R A) r)
-- Algebra.left_comm
theorem left_comm (x : A) (r : R) (y : A) :
    x * (algebraMap R A r * y) = algebraMap R A r * (x * y) 
Arguments: {R : Type u} {A : Type w} [CommSemiring R] [Semiring A] [Algebra R A] (x : A) (r : R) (y : A)
Goal: x * ((algebraMap R A) r * y) = (algebraMap R A) r * (x * y)
-- Algebra.right_comm
theorem right_comm (x : A) (r : R) (y : A) :
    x * algebraMap R A r * y = x * y * algebraMap R A r 
Arguments: {R : Type u} {A : Type w} [CommSemiring R] [Semiring A] [Algebra R A] (x : A) (r : R) (y : A)
Goal: x * (algebraMap R A) r * y = x * y * (algebraMap R A) r
-- RingHom.smulOneHom_eq_algebraMap
@[simp]
theorem _root_.RingHom.smulOneHom_eq_algebraMap : RingHom.smulOneHom = algebraMap R A 
Arguments: {R : Type u} {A : Type w} [CommSemiring R] [Semiring A] [Algebra R A]
Goal: RingHom.smulOneHom = algebraMap R A
-- Algebra.mul_smul_comm
@[simp]
protected theorem mul_smul_comm (s : R) (x y : A) : x * s • y = s • (x * y) 
Arguments: {R : Type u} {A : Type w} [CommSemiring R] [Semiring A] [Algebra R A] (s : R) (x y : A)
Goal: x * s • y = s • (x * y)
-- Algebra.smul_mul_assoc
@[simp]
protected theorem smul_mul_assoc (r : R) (x y : A) : r • x * y = r • (x * y) 
Arguments: {R : Type u} {A : Type w} [CommSemiring R] [Semiring A] [Algebra R A] (r : R) (x y : A)
Goal: r • x * y = r • (x * y)
-- smul_algebraMap
@[simp]
theorem _root_.smul_algebraMap {α : Type*} [Monoid α] [MulDistribMulAction α A]
    [SMulCommClass α R A] (a : α) (r : R) : a • algebraMap R A r = algebraMap R A r 
Arguments: {R : Type u} {A : Type w} [CommSemiring R] [Semiring A] [Algebra R A] {α : Type u_2} [Monoid α] [MulDistribMulAction α A] [SMulCommClass α R A] (a : α) (r : R)
Goal: a • (algebraMap R A) r = (algebraMap R A) r
-- Algebra.linearMap_apply
@[simp]
theorem linearMap_apply (r : R) : Algebra.linearMap R A r = algebraMap R A r 
Arguments: (R : Type u) (A : Type w) [CommSemiring R] [Semiring A] [Algebra R A] (r : R)
Goal: (Algebra.linearMap R A) r = (algebraMap R A) r
-- Algebra.coe_linearMap
theorem coe_linearMap : ⇑(Algebra.linearMap R A) = algebraMap R A 
Arguments: (R : Type u) (A : Type w) [CommSemiring R] [Semiring A] [Algebra R A]
Goal: ⇑(Algebra.linearMap R A) = ⇑(algebraMap R A)
-- Algebra.id.map_eq_id
@[simp]
theorem map_eq_id : algebraMap R R = RingHom.id _ 
Arguments: {R : Type u} [CommSemiring R]
Goal: algebraMap R R = RingHom.id R
-- Algebra.id.map_eq_self
theorem map_eq_self (x : R) : algebraMap R R x = x 
Arguments: {R : Type u} [CommSemiring R] (x : R)
Goal: (algebraMap R R) x = x
-- Algebra.id.smul_eq_mul
@[simp]
theorem smul_eq_mul (x y : R) : x • y = x * y 
Arguments: {R : Type u} [CommSemiring R] (x y : R)
Goal: x • y = x * y
-- algebraMap.coe_smul
@[norm_cast]
theorem algebraMap.coe_smul (A B C : Type*) [SMul A B] [CommSemiring B] [Semiring C] [Algebra B C]
    [SMul A C] [IsScalarTower A B C] (a : A) (b : B) : (a • b : B) = a • (b : C) 
Arguments: (A : Type u_1) (B : Type u_2) (C : Type u_3) [SMul A B] [CommSemiring B] [Semiring C] [Algebra B C] [SMul A C] [IsScalarTower A B C] (a : A) (b : B)
Goal: ↑(a • b) = a • ↑b
-- RingHom.codomain_trivial_iff_range_eq_singleton_zero
theorem codomain_trivial_iff_range_eq_singleton_zero : (0 : β) = 1 ↔ Set.range f = {0} 
Arguments: {α : Type u_2} {β : Type u_3}
Goal: ∀ {x : NonAssocSemiring α} {x_1 : NonAssocSemiring β} (f : α →+* β), 0 = 1 ↔ Set.range ⇑f = {0}
-- RingHom.isUnit_map
theorem isUnit_map (f : α →+* β) {a : α} : IsUnit a → IsUnit (f a) 
Arguments: {α : Type u_2} {β : Type u_3} [Semiring α] [Semiring β] (f : α →+* β) {a : α}
Goal: IsUnit a → IsUnit (f a)
-- RingHom.map_dvd
protected theorem map_dvd (f : α →+* β) {a b : α} : a ∣ b → f a ∣ f b 
Arguments: {α : Type u_2} {β : Type u_3} [Semiring α] [Semiring β] (f : α →+* β) {a b : α}
Goal: a ∣ b → f a ∣ f b
-- Function.Injective.isDomain
protected theorem Function.Injective.isDomain [Ring α] [IsDomain α] [Ring β] (f : β →+* α)
    (hf : Injective f) : IsDomain β 
Arguments: {α : Type u_2} {β : Type u_3} [Ring α] [IsDomain α] [Ring β] (f : β →+* α) (hf : Function.Injective ⇑f)
Goal: IsDomain β
-- Subalgebra.mul_toSubmodule_le
theorem mul_toSubmodule_le (S T : Subalgebra R A) :
    (Subalgebra.toSubmodule S)* (Subalgebra.toSubmodule T) ≤ Subalgebra.toSubmodule (S ⊔ T) 
Arguments: {R : Type u_1} {A : Type u_2} [CommSemiring R] [Semiring A] [Algebra R A] (S T : Subalgebra R A)
Goal: Subalgebra.toSubmodule S * Subalgebra.toSubmodule T ≤ Subalgebra.toSubmodule (S ⊔ T)
-- Subalgebra.mul_self
@[simp]
theorem mul_self (S : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule S)
    = (Subalgebra.toSubmodule S) 
Arguments: {R : Type u_1} {A : Type u_2} [CommSemiring R] [Semiring A] [Algebra R A] (S : Subalgebra R A)
Goal: Subalgebra.toSubmodule S * Subalgebra.toSubmodule S = Subalgebra.toSubmodule S
-- Subalgebra.mul_toSubmodule
theorem mul_toSubmodule {R : Type*} {A : Type*} [CommSemiring R] [CommSemiring A] [Algebra R A]
    (S T : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule T)
        = Subalgebra.toSubmodule (S ⊔ T) 
Arguments: {R : Type u_3} {A : Type u_4} [CommSemiring R] [CommSemiring A] [Algebra R A] (S T : Subalgebra R A)
Goal: Subalgebra.toSubmodule S * Subalgebra.toSubmodule T = Subalgebra.toSubmodule (S ⊔ T)
-- Subalgebra.coe_pointwise_smul
@[simp]
theorem coe_pointwise_smul (m : R') (S : Subalgebra R A) : ↑(m • S) = m • (S : Set A) 
Arguments: {R : Type u_1} {A : Type u_2} [CommSemiring R] [Semiring A] [Algebra R A] {R' : Type u_3} [Semiring R'] [MulSemiringAction R' A] [SMulCommClass R' R A] (m : R') (S : Subalgebra R A)
Goal: ↑(m • S) = m • ↑S
-- Subalgebra.pointwise_smul_toSubsemiring
@[simp]
theorem pointwise_smul_toSubsemiring (m : R') (S : Subalgebra R A) :
    (m • S).toSubsemiring = m • S.toSubsemiring 
Arguments: {R : Type u_1} {A : Type u_2} [CommSemiring R] [Semiring A] [Algebra R A] {R' : Type u_3} [Semiring R'] [MulSemiringAction R' A] [SMulCommClass R' R A] (m : R') (S : Subalgebra R A)
Goal: (m • S).toSubsemiring = m • S.toSubsemiring
-- Subalgebra.pointwise_smul_toSubmodule
@[simp]
theorem pointwise_smul_toSubmodule (m : R') (S : Subalgebra R A) :
    Subalgebra.toSubmodule (m • S) = m • Subalgebra.toSubmodule S 
Arguments: {R : Type u_1} {A : Type u_2} [CommSemiring R] [Semiring A] [Algebra R A] {R' : Type u_3} [Semiring R'] [MulSemiringAction R' A] [SMulCommClass R' R A] (m : R') (S : Subalgebra R A)
Goal: Subalgebra.toSubmodule (m • S) = m • Subalgebra.toSubmodule S
-- Subalgebra.pointwise_smul_toSubring
@[simp]
theorem pointwise_smul_toSubring {R' R A : Type*} [Semiring R'] [CommRing R] [Ring A]
    [MulSemiringAction R' A] [Algebra R A] [SMulCommClass R' R A] (m : R') (S : Subalgebra R A) :
    (m • S).toSubring = m • S.toSubring 
Arguments: {R' : Type u_4} {R : Type u_5} {A : Type u_6} [Semiring R'] [CommRing R] [Ring A] [MulSemiringAction R' A] [Algebra R A] [SMulCommClass R' R A] (m : R') (S : Subalgebra R A)
Goal: (m • S).toSubring = m • S.toSubring
-- Subalgebra.smul_mem_pointwise_smul
theorem smul_mem_pointwise_smul (m : R') (r : A) (S : Subalgebra R A) : r ∈ S → m • r ∈ m • S 
Arguments: {R : Type u_1} {A : Type u_2} [CommSemiring R] [Semiring A] [Algebra R A] {R' : Type u_3} [Semiring R'] [MulSemiringAction R' A] [SMulCommClass R' R A] (m : R') (r : A) (S : Subalgebra R A)
Goal: r ∈ S → m • r ∈ m • S
-- Algebra.lsmul_coe
@[simp]
theorem lsmul_coe (a : A) : (lsmul R B M a : M → M) = (a • ·) 
Arguments: (R : Type u) {A : Type w} (B : Type u₁) (M : Type v₁) [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] [AddCommMonoid M] [Module R M] [Module A M] [Module B M] [IsScalarTower R A M] [IsScalarTower R B M] [SMulCommClass A B M] (a : A)
Goal: ⇑((Algebra.lsmul R B M) a) = fun x => a • x
-- IsScalarTower.algebraMap_smul
theorem algebraMap_smul [SMul R M] [IsScalarTower R A M] (r : R) (x : M) :
    algebraMap R A r • x = r • x 
Arguments: {R : Type u} (A : Type w) {M : Type v₁} [CommSemiring R] [Semiring A] [Algebra R A] [MulAction A M] [SMul R M] [IsScalarTower R A M] (r : R) (x : M)
Goal: (algebraMap R A) r • x = r • x
-- IsScalarTower.of_algebraMap_smul
theorem of_algebraMap_smul [SMul R M] (h : ∀ (r : R) (x : M), algebraMap R A r • x = r • x) :
    IsScalarTower R A M 
Arguments: {R : Type u} {A : Type w} {M : Type v₁} [CommSemiring R] [Semiring A] [Algebra R A] [MulAction A M] [SMul R M] (h : ∀ (r : R) (x : M), (algebraMap R A) r • x = r • x)
Goal: IsScalarTower R A M
-- IsScalarTower.of_compHom
theorem of_compHom : letI := MulAction.compHom M (algebraMap R A : R →* A); IsScalarTower R A M 
Arguments: (R : Type u) (A : Type w) (M : Type v₁) [CommSemiring R] [Semiring A] [Algebra R A] [MulAction A M]
Goal: IsScalarTower R A M
-- IsScalarTower.of_algebraMap_eq
theorem of_algebraMap_eq [Algebra R A]
    (h : ∀ x, algebraMap R A x = algebraMap S A (algebraMap R S x)) : IsScalarTower R S A 
Arguments: {R : Type u} {S : Type v} {A : Type w} [CommSemiring R] [CommSemiring S] [Semiring A] [Algebra R S] [Algebra S A] [Algebra R A] (h : ∀ (x : R), (algebraMap R A) x = (algebraMap S A) ((algebraMap R S) x))
Goal: IsScalarTower R S A
-- IsScalarTower.of_algebraMap_eq'
theorem of_algebraMap_eq' [Algebra R A]
    (h : algebraMap R A = (algebraMap S A).comp (algebraMap R S)) : IsScalarTower R S A 
Arguments: {R : Type u} {S : Type v} {A : Type w} [CommSemiring R] [CommSemiring S] [Semiring A] [Algebra R S] [Algebra S A] [Algebra R A] (h : algebraMap R A = (algebraMap S A).comp (algebraMap R S))
Goal: IsScalarTower R S A
-- IsScalarTower.algebraMap_eq
theorem algebraMap_eq : algebraMap R A = (algebraMap S A).comp (algebraMap R S) 
Arguments: (R : Type u) (S : Type v) (A : Type w) [CommSemiring R] [CommSemiring S] [Semiring A] [Algebra R S] [Algebra S A] [Algebra R A] [IsScalarTower R S A]
Goal: algebraMap R A = (algebraMap S A).comp (algebraMap R S)
-- IsScalarTower.algebraMap_apply
theorem algebraMap_apply (x : R) : algebraMap R A x = algebraMap S A (algebraMap R S x) 
Arguments: (R : Type u) (S : Type v) (A : Type w) [CommSemiring R] [CommSemiring S] [Semiring A] [Algebra R S] [Algebra S A] [Algebra R A] [IsScalarTower R S A] (x : R)
Goal: (algebraMap R A) x = (algebraMap S A) ((algebraMap R S) x)
-- IsScalarTower.Algebra.ext
@[ext]
theorem Algebra.ext {S : Type u} {A : Type v} [CommSemiring S] [Semiring A] (h1 h2 : Algebra S A)
    (h : ∀ (r : S) (x : A), (by have I := h1; exact r • x) = r • x) : h1 = h2 
Arguments: {S : Type u} {A : Type v} [CommSemiring S] [Semiring A] (h1 h2 : Algebra S A) (h :
  ∀ (r : S) (x : A),
    (let_fun I := h1;
      r • x) =
      r • x)
Goal: h1 = h2
-- IsScalarTower.toAlgHom_apply
theorem toAlgHom_apply (y : S) : toAlgHom R S A y = algebraMap S A y 
Arguments: (R : Type u) (S : Type v) (A : Type w) [CommSemiring R] [CommSemiring S] [Semiring A] [Algebra R S] [Algebra S A] [Algebra R A] [IsScalarTower R S A] (y : S)
Goal: (IsScalarTower.toAlgHom R S A) y = (algebraMap S A) y
-- IsScalarTower.coe_toAlgHom
@[simp]
theorem coe_toAlgHom : ↑(toAlgHom R S A) = algebraMap S A 
Arguments: (R : Type u) (S : Type v) (A : Type w) [CommSemiring R] [CommSemiring S] [Semiring A] [Algebra R S] [Algebra S A] [Algebra R A] [IsScalarTower R S A]
Goal: ↑(IsScalarTower.toAlgHom R S A) = algebraMap S A
-- IsScalarTower.coe_toAlgHom'
@[simp]
theorem coe_toAlgHom' : (toAlgHom R S A : S → A) = algebraMap S A 
Arguments: (R : Type u) (S : Type v) (A : Type w) [CommSemiring R] [CommSemiring S] [Semiring A] [Algebra R S] [Algebra S A] [Algebra R A] [IsScalarTower R S A]
Goal: ⇑(IsScalarTower.toAlgHom R S A) = ⇑(algebraMap S A)
-- AlgHom.map_algebraMap
@[simp]
theorem _root_.AlgHom.map_algebraMap (f : A →ₐ[S] B) (r : R) :
    f (algebraMap R A r) = algebraMap R B r 
Arguments: {R : Type u} {S : Type v} {A : Type w} {B : Type u₁} [CommSemiring R] [CommSemiring S] [Semiring A] [Semiring B] [Algebra R S] [Algebra S A] [Algebra S B] [Algebra R A] [Algebra R B] [IsScalarTower R S A] [IsScalarTower R S B] (f : A →ₐ[S] B) (r : R)
Goal: f ((algebraMap R A) r) = (algebraMap R B) r
-- AlgHom.comp_algebraMap_of_tower
@[simp]
theorem _root_.AlgHom.comp_algebraMap_of_tower (f : A →ₐ[S] B) :
    (f : A →+* B).comp (algebraMap R A) = algebraMap R B 
Arguments: (R : Type u) {S : Type v} {A : Type w} {B : Type u₁} [CommSemiring R] [CommSemiring S] [Semiring A] [Semiring B] [Algebra R S] [Algebra S A] [Algebra S B] [Algebra R A] [Algebra R B] [IsScalarTower R S A] [IsScalarTower R S B] (f : A →ₐ[S] B)
Goal: (↑f).comp (algebraMap R A) = algebraMap R B
-- AlgHom.restrictScalars_apply
theorem restrictScalars_apply (f : A →ₐ[S] B) (x : A) : f.restrictScalars R x = f x 
Arguments: (R : Type u) {S : Type v} {A : Type w} {B : Type u₁} [CommSemiring R] [CommSemiring S] [Semiring A] [Semiring B] [Algebra R S] [Algebra S A] [Algebra S B] [Algebra R A] [Algebra R B] [IsScalarTower R S A] [IsScalarTower R S B] (f : A →ₐ[S] B) (x : A)
Goal: (AlgHom.restrictScalars R f) x = f x
-- AlgHom.coe_restrictScalars
@[simp]
theorem coe_restrictScalars (f : A →ₐ[S] B) : (f.restrictScalars R : A →+* B) = f 
Arguments: (R : Type u) {S : Type v} {A : Type w} {B : Type u₁} [CommSemiring R] [CommSemiring S] [Semiring A] [Semiring B] [Algebra R S] [Algebra S A] [Algebra S B] [Algebra R A] [Algebra R B] [IsScalarTower R S A] [IsScalarTower R S B] (f : A →ₐ[S] B)
Goal: ↑(AlgHom.restrictScalars R f) = ↑f
-- AlgHom.coe_restrictScalars'
@[simp]
theorem coe_restrictScalars' (f : A →ₐ[S] B) : (restrictScalars R f : A → B) = f 
Arguments: (R : Type u) {S : Type v} {A : Type w} {B : Type u₁} [CommSemiring R] [CommSemiring S] [Semiring A] [Semiring B] [Algebra R S] [Algebra S A] [Algebra S B] [Algebra R A] [Algebra R B] [IsScalarTower R S A] [IsScalarTower R S B] (f : A →ₐ[S] B)
Goal: ⇑(AlgHom.restrictScalars R f) = ⇑f
-- AlgHom.restrictScalars_injective
theorem restrictScalars_injective :
    Function.Injective (restrictScalars R : (A →ₐ[S] B) → A →ₐ[R] B) 
Arguments: (R : Type u) {S : Type v} {A : Type w} {B : Type u₁} [CommSemiring R] [CommSemiring S] [Semiring A] [Semiring B] [Algebra R S] [Algebra S A] [Algebra S B] [Algebra R A] [Algebra R B] [IsScalarTower R S A] [IsScalarTower R S B]
Goal: Function.Injective (AlgHom.restrictScalars R)
-- AlgEquiv.restrictScalars_apply
theorem restrictScalars_apply (f : A ≃ₐ[S] B) (x : A) : f.restrictScalars R x = f x 
Arguments: (R : Type u) {S : Type v} {A : Type w} {B : Type u₁} [CommSemiring R] [CommSemiring S] [Semiring A] [Semiring B] [Algebra R S] [Algebra S A] [Algebra S B] [Algebra R A] [Algebra R B] [IsScalarTower R S A] [IsScalarTower R S B] (f : A ≃ₐ[S] B) (x : A)
Goal: (AlgEquiv.restrictScalars R f) x = f x
-- AlgEquiv.coe_restrictScalars
@[simp]
theorem coe_restrictScalars (f : A ≃ₐ[S] B) : (f.restrictScalars R : A ≃+* B) = f 
Arguments: (R : Type u) {S : Type v} {A : Type w} {B : Type u₁} [CommSemiring R] [CommSemiring S] [Semiring A] [Semiring B] [Algebra R S] [Algebra S A] [Algebra S B] [Algebra R A] [Algebra R B] [IsScalarTower R S A] [IsScalarTower R S B] (f : A ≃ₐ[S] B)
Goal: ↑(AlgEquiv.restrictScalars R f) = ↑f
-- AlgEquiv.coe_restrictScalars'
@[simp]
theorem coe_restrictScalars' (f : A ≃ₐ[S] B) : (restrictScalars R f : A → B) = f 
Arguments: (R : Type u) {S : Type v} {A : Type w} {B : Type u₁} [CommSemiring R] [CommSemiring S] [Semiring A] [Semiring B] [Algebra R S] [Algebra S A] [Algebra S B] [Algebra R A] [Algebra R B] [IsScalarTower R S A] [IsScalarTower R S B] (f : A ≃ₐ[S] B)
Goal: ⇑(AlgEquiv.restrictScalars R f) = ⇑f
-- AlgEquiv.restrictScalars_injective
theorem restrictScalars_injective :
    Function.Injective (restrictScalars R : (A ≃ₐ[S] B) → A ≃ₐ[R] B) 
Arguments: (R : Type u) {S : Type v} {A : Type w} {B : Type u₁} [CommSemiring R] [CommSemiring S] [Semiring A] [Semiring B] [Algebra R S] [Algebra S A] [Algebra S B] [Algebra R A] [Algebra R B] [IsScalarTower R S A] [IsScalarTower R S B]
Goal: Function.Injective (AlgEquiv.restrictScalars R)
-- Submodule.restrictScalars_span
theorem restrictScalars_span (hsur : Function.Surjective (algebraMap R A)) (X : Set M) :
    restrictScalars R (span A X) = span R X 
Arguments: (R : Type u) (A : Type w) {M : Type v₁} [CommSemiring R] [Semiring A] [Algebra R A] [AddCommMonoid M] [Module R M] [Module A M] [IsScalarTower R A M] (hsur : Function.Surjective ⇑(algebraMap R A)) (X : Set M)
Goal: Submodule.restrictScalars R (Submodule.span A X) = Submodule.span R X
-- Submodule.coe_span_eq_span_of_surjective
theorem coe_span_eq_span_of_surjective (h : Function.Surjective (algebraMap R A)) (s : Set M) :
    (Submodule.span A s : Set M) = Submodule.span R s 
Arguments: (R : Type u) (A : Type w) {M : Type v₁} [CommSemiring R] [Semiring A] [Algebra R A] [AddCommMonoid M] [Module R M] [Module A M] [IsScalarTower R A M] (h : Function.Surjective ⇑(algebraMap R A)) (s : Set M)
Goal: ↑(Submodule.span A s) = ↑(Submodule.span R s)
-- Submodule.smul_mem_span_smul_of_mem
theorem smul_mem_span_smul_of_mem {s : Set S} {t : Set A} {k : S} (hks : k ∈ span R s) {x : A}
    (hx : x ∈ t) : k • x ∈ span R (s • t) 
Arguments: {R : Type u} {S : Type v} {A : Type w} [Semiring R] [Semiring S] [AddCommMonoid A] [Module R S] [Module S A] [Module R A] [IsScalarTower R S A] {s : Set S} {t : Set A} {k : S} (hks : k ∈ Submodule.span R s) {x : A} (hx : x ∈ t)
Goal: k • x ∈ Submodule.span R (s • t)
-- Submodule.span_smul_of_span_eq_top
theorem span_smul_of_span_eq_top {s : Set S} (hs : span R s = ⊤) (t : Set A) :
    span R (s • t) = (span S t).restrictScalars R 
Arguments: {R : Type u} {S : Type v} {A : Type w} [Semiring R] [Semiring S] [AddCommMonoid A] [Module R S] [Module S A] [Module R A] [IsScalarTower R S A] {s : Set S} (hs : Submodule.span R s = ⊤) (t : Set A)
Goal: Submodule.span R (s • t) = Submodule.restrictScalars R (Submodule.span S t)
-- Submodule.smul_mem_span_smul'
theorem smul_mem_span_smul' {s : Set S} (hs : span R s = ⊤) {t : Set A} {k : S} {x : A}
    (hx : x ∈ span R (s • t)) : k • x ∈ span R (s • t) 
Arguments: {R : Type u} {S : Type v} {A : Type w} [Semiring R] [Semiring S] [AddCommMonoid A] [Module R S] [Module S A] [Module R A] [IsScalarTower R S A] {s : Set S} (hs : Submodule.span R s = ⊤) {t : Set A} {k : S} {x : A} (hx : x ∈ Submodule.span R (s • t))
Goal: k • x ∈ Submodule.span R (s • t)
-- Submodule.smul_mem_span_smul
theorem smul_mem_span_smul {s : Set S} (hs : span R s = ⊤) {t : Set A} {k : S} {x : A}
    (hx : x ∈ span R t) : k • x ∈ span R (s • t) 
Arguments: {R : Type u} {S : Type v} {A : Type w} [Semiring R] [Semiring S] [AddCommMonoid A] [Module R S] [Module S A] [Module R A] [IsScalarTower R S A] {s : Set S} (hs : Submodule.span R s = ⊤) {t : Set A} {k : S} {x : A} (hx : x ∈ Submodule.span R t)
Goal: k • x ∈ Submodule.span R (s • t)
-- Submodule.span_algebraMap_image
theorem span_algebraMap_image (a : Set R) :
    Submodule.span R (algebraMap R S '' a) = (Submodule.span R a).map (Algebra.linearMap R S) 
Arguments: {R : Type u} {S : Type v} [CommSemiring R] [Semiring S] [Algebra R S] (a : Set R)
Goal: Submodule.span R (⇑(algebraMap R S) '' a) = Submodule.map (Algebra.linearMap R S) (Submodule.span R a)
-- Submodule.span_algebraMap_image_of_tower
theorem span_algebraMap_image_of_tower {S T : Type*} [CommSemiring S] [Semiring T] [Module R S]
    [Algebra R T] [Algebra S T] [IsScalarTower R S T] (a : Set S) :
    Submodule.span R (algebraMap S T '' a) =
      (Submodule.span R a).map ((Algebra.linearMap S T).restrictScalars R) 
Arguments: {R : Type u} [CommSemiring R] {S : Type u_1} {T : Type u_2} [CommSemiring S] [Semiring T] [Module R S] [Algebra R T] [Algebra S T] [IsScalarTower R S T] (a : Set S)
Goal: Submodule.span R (⇑(algebraMap S T) '' a) = Submodule.map (↑R (Algebra.linearMap S T)) (Submodule.span R a)
-- Submodule.map_mem_span_algebraMap_image
theorem map_mem_span_algebraMap_image {S T : Type*} [CommSemiring S] [Semiring T] [Algebra R S]
    [Algebra R T] [Algebra S T] [IsScalarTower R S T] (x : S) (a : Set S)
    (hx : x ∈ Submodule.span R a) : algebraMap S T x ∈ Submodule.span R (algebraMap S T '' a) 
Arguments: {R : Type u} [CommSemiring R] {S : Type u_1} {T : Type u_2} [CommSemiring S] [Semiring T] [Algebra R S] [Algebra R T] [Algebra S T] [IsScalarTower R S T] (x : S) (a : Set S) (hx : x ∈ Submodule.span R a)
Goal: (algebraMap S T) x ∈ Submodule.span R (⇑(algebraMap S T) '' a)