ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- IsLocalization.ext
protected theorem ext (j k : S → P) (hj1 : j 1 = 1) (hk1 : k 1 = 1)
    (hjm : ∀ a b, j (a * b) = j a * j b) (hkm : ∀ a b, k (a * b) = k a * k b)
    (h : ∀ a, j (algebraMap R S a) = k (algebraMap R S a)) : j = k 
Arguments: {R : Type u_1} [CommSemiring R] (M : Submonoid R) {S : Type u_2} [CommSemiring S] [Algebra R S] {P : Type u_3} [CommSemiring P] [IsLocalization M S] (j k : S → P) (hj1 : j 1 = 1) (hk1 : k 1 = 1) (hjm : ∀ (a b : S), j (a * b) = j a * j b) (hkm : ∀ (a b : S), k (a * b) = k a * k b) (h : ∀ (a : R), j ((algebraMap R S) a) = k ((algebraMap R S) a))
Goal: j = k
-- IsLocalization.lift_unique
theorem lift_unique {j : S →+* P} (hj : ∀ x, j ((algebraMap R S) x) = g x) : lift hg = j 
Arguments: {R : Type u_1} [CommSemiring R] {M : Submonoid R} {S : Type u_2} [CommSemiring S] [Algebra R S] {P : Type u_3} [CommSemiring P] [IsLocalization M S] {g : R →+* P} (hg : ∀ (y : ↥M), IsUnit (g ↑y)) {j : S →+* P} (hj : ∀ (x : R), j ((algebraMap R S) x) = g x)
Goal: IsLocalization.lift hg = j
-- IsLocalization.lift_id
@[simp]
theorem lift_id (x) : lift (map_units S : ∀ _ : M, IsUnit _) x = x 
Arguments: {R : Type u_1} [CommSemiring R] {M : Submonoid R} {S : Type u_2} [CommSemiring S] [Algebra R S] [IsLocalization M S] (x : S)
Goal: (IsLocalization.lift ⋯) x = x
-- IsLocalization.lift_surjective_iff
theorem lift_surjective_iff :
    Surjective (lift hg : S → P) ↔ ∀ v : P, ∃ x : R × M, v * g x.2 = g x.1 
Arguments: {R : Type u_1} [CommSemiring R] {M : Submonoid R} {S : Type u_2} [CommSemiring S] [Algebra R S] {P : Type u_3} [CommSemiring P] [IsLocalization M S] {g : R →+* P} (hg : ∀ (y : ↥M), IsUnit (g ↑y))
Goal: Function.Surjective ⇑(IsLocalization.lift hg) ↔ ∀ (v : P), ∃ x, v * g ↑x.2 = g x.1
-- IsLocalization.lift_injective_iff
theorem lift_injective_iff :
    Injective (lift hg : S → P) ↔ ∀ x y, algebraMap R S x = algebraMap R S y ↔ g x = g y 
Arguments: {R : Type u_1} [CommSemiring R] {M : Submonoid R} {S : Type u_2} [CommSemiring S] [Algebra R S] {P : Type u_3} [CommSemiring P] [IsLocalization M S] {g : R →+* P} (hg : ∀ (y : ↥M), IsUnit (g ↑y))
Goal: Function.Injective ⇑(IsLocalization.lift hg) ↔ ∀ (x y : R), (algebraMap R S) x = (algebraMap R S) y ↔ g x = g y
-- IsLocalization.map_eq
@[simp]
theorem map_eq (x) : map Q g hy ((algebraMap R S) x) = algebraMap P Q (g x) 
Arguments: {R : Type u_1} [CommSemiring R] {M : Submonoid R} {S : Type u_2} [CommSemiring S] [Algebra R S] {P : Type u_3} [CommSemiring P] [IsLocalization M S] {g : R →+* P} {T : Submonoid P} {Q : Type u_4} [CommSemiring Q] (hy : M ≤ Submonoid.comap g T) [Algebra P Q] [IsLocalization T Q] (x : R)
Goal: (IsLocalization.map Q g hy) ((algebraMap R S) x) = (algebraMap P Q) (g x)
-- IsLocalization.map_comp
@[simp]
theorem map_comp : (map Q g hy).comp (algebraMap R S) = (algebraMap P Q).comp g 
Arguments: {R : Type u_1} [CommSemiring R] {M : Submonoid R} {S : Type u_2} [CommSemiring S] [Algebra R S] {P : Type u_3} [CommSemiring P] [IsLocalization M S] {g : R →+* P} {T : Submonoid P} {Q : Type u_4} [CommSemiring Q] (hy : M ≤ Submonoid.comap g T) [Algebra P Q] [IsLocalization T Q]
Goal: (IsLocalization.map Q g hy).comp (algebraMap R S) = (algebraMap P Q).comp g
-- IsLocalization.map_mk'
theorem map_mk' (x) (y : M) : map Q g hy (mk' S x y) = mk' Q (g x) ⟨g y, hy y.2⟩ 
Arguments: {R : Type u_1} [CommSemiring R] {M : Submonoid R} {S : Type u_2} [CommSemiring S] [Algebra R S] {P : Type u_3} [CommSemiring P] [IsLocalization M S] {g : R →+* P} {T : Submonoid P} {Q : Type u_4} [CommSemiring Q] (hy : M ≤ Submonoid.comap g T) [Algebra P Q] [IsLocalization T Q] (x : R) (y : ↥M)
Goal: (IsLocalization.map Q g hy) (IsLocalization.mk' S x y) = IsLocalization.mk' Q (g x) ⟨g ↑y, ⋯⟩
-- IsLocalization.map_id_mk'
@[simp]
theorem map_id_mk' {Q : Type*} [CommSemiring Q] [Algebra R Q] [IsLocalization M Q] (x) (y : M) :
    map Q (RingHom.id R) (le_refl M) (mk' S x y) = mk' Q x y 
Arguments: {R : Type u_1} [CommSemiring R] {M : Submonoid R} {S : Type u_2} [CommSemiring S] [Algebra R S] [IsLocalization M S] {Q : Type u_5} [CommSemiring Q] [Algebra R Q] [IsLocalization M Q] (x : R) (y : ↥M)
Goal: (IsLocalization.map Q (RingHom.id R) ⋯) (IsLocalization.mk' S x y) = IsLocalization.mk' Q x y
-- IsLocalization.map_id
@[simp]
theorem map_id (z : S) (h : M ≤ M.comap (RingHom.id R) := le_refl M) :
    map S (RingHom.id _) h z = z 
Arguments: {R : Type u_1} [CommSemiring R] {M : Submonoid R} {S : Type u_2} [CommSemiring S] [Algebra R S] [IsLocalization M S] (z : S) (h : M ≤ Submonoid.comap (RingHom.id R) M := ⋯)
Goal: (IsLocalization.map S (RingHom.id R) h) z = z
-- IsLocalization.map_unique
theorem map_unique (j : S →+* Q) (hj : ∀ x : R, j (algebraMap R S x) = algebraMap P Q (g x)) :
    map Q g hy = j 
Arguments: {R : Type u_1} [CommSemiring R] {M : Submonoid R} {S : Type u_2} [CommSemiring S] [Algebra R S] {P : Type u_3} [CommSemiring P] [IsLocalization M S] {g : R →+* P} {T : Submonoid P} {Q : Type u_4} [CommSemiring Q] (hy : M ≤ Submonoid.comap g T) [Algebra P Q] [IsLocalization T Q] (j : S →+* Q) (hj : ∀ (x : R), j ((algebraMap R S) x) = (algebraMap P Q) (g x))
Goal: IsLocalization.map Q g hy = j
-- IsLocalization.map_comp_map
theorem map_comp_map {A : Type*} [CommSemiring A] {U : Submonoid A} {W} [CommSemiring W]
    [Algebra A W] [IsLocalization U W] {l : P →+* A} (hl : T ≤ U.comap l) :
    (map W l hl).comp (map Q g hy : S →+* _) = map W (l.comp g) fun _ hx => hl (hy hx) 
Arguments: {R : Type u_1} [CommSemiring R] {M : Submonoid R} {S : Type u_2} [CommSemiring S] [Algebra R S] {P : Type u_3} [CommSemiring P] [IsLocalization M S] {g : R →+* P} {T : Submonoid P} {Q : Type u_4} [CommSemiring Q] (hy : M ≤ Submonoid.comap g T) [Algebra P Q] [IsLocalization T Q] {A : Type u_5} [CommSemiring A] {U : Submonoid A} {W : Type u_6} [CommSemiring W] [Algebra A W] [IsLocalization U W] {l : P →+* A} (hl : T ≤ Submonoid.comap l U)
Goal: (IsLocalization.map W l hl).comp (IsLocalization.map Q g hy) = IsLocalization.map W (l.comp g) ⋯
-- IsLocalization.map_map
theorem map_map {A : Type*} [CommSemiring A] {U : Submonoid A} {W} [CommSemiring W] [Algebra A W]
    [IsLocalization U W] {l : P →+* A} (hl : T ≤ U.comap l) (x : S) :
    map W l hl (map Q g hy x) = map W (l.comp g) (fun x hx => hl (hy hx)) x 
Arguments: {R : Type u_1} [CommSemiring R] {M : Submonoid R} {S : Type u_2} [CommSemiring S] [Algebra R S] {P : Type u_3} [CommSemiring P] [IsLocalization M S] {g : R →+* P} {T : Submonoid P} {Q : Type u_4} [CommSemiring Q] (hy : M ≤ Submonoid.comap g T) [Algebra P Q] [IsLocalization T Q] {A : Type u_5} [CommSemiring A] {U : Submonoid A} {W : Type u_6} [CommSemiring W] [Algebra A W] [IsLocalization U W] {l : P →+* A} (hl : T ≤ Submonoid.comap l U) (x : S)
Goal: (IsLocalization.map W l hl) ((IsLocalization.map Q g hy) x) = (IsLocalization.map W (l.comp g) ⋯) x
-- IsLocalization.map_smul
theorem map_smul (x : S) (z : R) : map Q g hy (z • x : S) = g z • map Q g hy x 
Arguments: {R : Type u_1} [CommSemiring R] {M : Submonoid R} {S : Type u_2} [CommSemiring S] [Algebra R S] {P : Type u_3} [CommSemiring P] [IsLocalization M S] {g : R →+* P} {T : Submonoid P} {Q : Type u_4} [CommSemiring Q] (hy : M ≤ Submonoid.comap g T) [Algebra P Q] [IsLocalization T Q] (x : S) (z : R)
Goal: (IsLocalization.map Q g hy) (z • x) = g z • (IsLocalization.map Q g hy) x
-- IsLocalization.ringEquivOfRingEquiv_apply
@[simps]
noncomputable def ringEquivOfRingEquiv (h : R ≃+* P) (H : M.map h.toMonoidHom = T) : S ≃+* Q 
Arguments: {R : Type u_1} [CommSemiring R] {M : Submonoid R} (S : Type u_2) [CommSemiring S] [Algebra R S] {P : Type u_3} [CommSemiring P] [IsLocalization M S] {T : Submonoid P} (Q : Type u_4) [CommSemiring Q] [Algebra P Q] [IsLocalization T Q] (h : R ≃+* P) (H : Submonoid.map h.toMonoidHom M = T) (a : S)
Goal: (IsLocalization.ringEquivOfRingEquiv S Q h H) a = (IsLocalization.map Q ↑h ⋯) a
-- IsLocalization.ringEquivOfRingEquiv_symm_apply
@[simps]
noncomputable def ringEquivOfRingEquiv (h : R ≃+* P) (H : M.map h.toMonoidHom = T) : S ≃+* Q 
Arguments: {R : Type u_1} [CommSemiring R] {M : Submonoid R} (S : Type u_2) [CommSemiring S] [Algebra R S] {P : Type u_3} [CommSemiring P] [IsLocalization M S] {T : Submonoid P} (Q : Type u_4) [CommSemiring Q] [Algebra P Q] [IsLocalization T Q] (h : R ≃+* P) (H : Submonoid.map h.toMonoidHom M = T) (a : Q)
Goal: (IsLocalization.ringEquivOfRingEquiv S Q h H).symm a = (IsLocalization.map S ↑h.symm ⋯) a
-- IsLocalization.ringEquivOfRingEquiv_eq_map
theorem ringEquivOfRingEquiv_eq_map {j : R ≃+* P} (H : M.map j.toMonoidHom = T) :
    (ringEquivOfRingEquiv S Q j H : S →+* Q) =
      map Q (j : R →+* P) (M.le_comap_of_map_le (le_of_eq H)) 
Arguments: {R : Type u_1} [CommSemiring R] {M : Submonoid R} {S : Type u_2} [CommSemiring S] [Algebra R S] {P : Type u_3} [CommSemiring P] [IsLocalization M S] {T : Submonoid P} {Q : Type u_4} [CommSemiring Q] [Algebra P Q] [IsLocalization T Q] {j : R ≃+* P} (H : Submonoid.map j.toMonoidHom M = T)
Goal: ↑(IsLocalization.ringEquivOfRingEquiv S Q j H) = IsLocalization.map Q ↑j ⋯
-- IsLocalization.ringEquivOfRingEquiv_eq
theorem ringEquivOfRingEquiv_eq {j : R ≃+* P} (H : M.map j.toMonoidHom = T) (x) :
    ringEquivOfRingEquiv S Q j H ((algebraMap R S) x) = algebraMap P Q (j x) 
Arguments: {R : Type u_1} [CommSemiring R] {M : Submonoid R} {S : Type u_2} [CommSemiring S] [Algebra R S] {P : Type u_3} [CommSemiring P] [IsLocalization M S] {T : Submonoid P} {Q : Type u_4} [CommSemiring Q] [Algebra P Q] [IsLocalization T Q] {j : R ≃+* P} (H : Submonoid.map j.toMonoidHom M = T) (x : R)
Goal: (IsLocalization.ringEquivOfRingEquiv S Q j H) ((algebraMap R S) x) = (algebraMap P Q) (j x)
-- IsLocalization.ringEquivOfRingEquiv_mk'
theorem ringEquivOfRingEquiv_mk' {j : R ≃+* P} (H : M.map j.toMonoidHom = T) (x : R) (y : M) :
    ringEquivOfRingEquiv S Q j H (mk' S x y) =
      mk' Q (j x) ⟨j y, show j y ∈ T from H ▸ Set.mem_image_of_mem j y.2⟩ 
Arguments: {R : Type u_1} [CommSemiring R] {M : Submonoid R} {S : Type u_2} [CommSemiring S] [Algebra R S] {P : Type u_3} [CommSemiring P] [IsLocalization M S] {T : Submonoid P} {Q : Type u_4} [CommSemiring Q] [Algebra P Q] [IsLocalization T Q] {j : R ≃+* P} (H : Submonoid.map j.toMonoidHom M = T) (x : R) (y : ↥M)
Goal: (IsLocalization.ringEquivOfRingEquiv S Q j H) (IsLocalization.mk' S x y) = IsLocalization.mk' Q (j x) ⟨j ↑y, ⋯⟩
-- IsLocalization.algEquiv_apply
@[simps!]
noncomputable def algEquiv : S ≃ₐ[R] Q 
Arguments: {R : Type u_1} [CommSemiring R] (M : Submonoid R) (S : Type u_2) [CommSemiring S] [Algebra R S] [IsLocalization M S] (Q : Type u_4) [CommSemiring Q] [Algebra R Q] [IsLocalization M Q] (a : S)
Goal: (IsLocalization.algEquiv M S Q) a = (IsLocalization.map Q (RingHom.id R) ⋯) a
-- IsLocalization.algEquiv_symm_apply
@[simps!]
noncomputable def algEquiv : S ≃ₐ[R] Q 
Arguments: {R : Type u_1} [CommSemiring R] (M : Submonoid R) (S : Type u_2) [CommSemiring S] [Algebra R S] [IsLocalization M S] (Q : Type u_4) [CommSemiring Q] [Algebra R Q] [IsLocalization M Q] (a : Q)
Goal: (IsLocalization.algEquiv M S Q).symm a = (IsLocalization.map S (RingHom.id R) ⋯) a
-- IsLocalization.algEquiv_mk'
theorem algEquiv_mk' (x : R) (y : M) : algEquiv M S Q (mk' S x y) = mk' Q x y 
Arguments: {R : Type u_1} [CommSemiring R] {M : Submonoid R} {S : Type u_2} [CommSemiring S] [Algebra R S] [IsLocalization M S] {Q : Type u_4} [CommSemiring Q] [Algebra R Q] [IsLocalization M Q] (x : R) (y : ↥M)
Goal: (IsLocalization.algEquiv M S Q) (IsLocalization.mk' S x y) = IsLocalization.mk' Q x y
-- IsLocalization.algEquiv_symm_mk'
theorem algEquiv_symm_mk' (x : R) (y : M) : (algEquiv M S Q).symm (mk' Q x y) = mk' S x y 
Arguments: {R : Type u_1} [CommSemiring R] {M : Submonoid R} {S : Type u_2} [CommSemiring S] [Algebra R S] [IsLocalization M S] {Q : Type u_4} [CommSemiring Q] [Algebra R Q] [IsLocalization M Q] (x : R) (y : ↥M)
Goal: (IsLocalization.algEquiv M S Q).symm (IsLocalization.mk' Q x y) = IsLocalization.mk' S x y
-- IsLocalization.bijective
protected lemma bijective (f : S →+* Q) (hf : f.comp (algebraMap R S) = algebraMap R Q) :
    Function.Bijective f 
Arguments: {R : Type u_1} [CommSemiring R] (M : Submonoid R) {S : Type u_2} [CommSemiring S] [Algebra R S] [IsLocalization M S] {Q : Type u_4} [CommSemiring Q] [Algebra R Q] [IsLocalization M Q] (f : S →+* Q) (hf : f.comp (algebraMap R S) = algebraMap R Q)
Goal: Function.Bijective ⇑f
-- IsLocalization.at_units
lemma at_units {R : Type*} [CommSemiring R] (S : Submonoid R)
    (hS : S ≤ IsUnit.submonoid R) : IsLocalization S R 
Arguments: {R : Type u_4} [CommSemiring R] (S : Submonoid R) (hS : S ≤ IsUnit.submonoid R)
Goal: IsLocalization S R
-- IsLocalization.map_injective_of_injective
theorem map_injective_of_injective (h : Function.Injective g) [IsLocalization (M.map g) Q] :
    Function.Injective (map Q g M.le_comap_map : S → Q) 
Arguments: {R : Type u_1} [CommSemiring R] (M : Submonoid R) (S : Type u_2) [CommSemiring S] [Algebra R S] {P : Type u_3} [CommSemiring P] [IsLocalization M S] {g : R →+* P} (Q : Type u_4) [CommSemiring Q] [Algebra P Q] (h : Function.Injective ⇑g) [IsLocalization (Submonoid.map g M) Q]
Goal: Function.Injective ⇑(IsLocalization.map Q g ⋯)
-- IsLocalization.map_surjective_of_surjective
theorem map_surjective_of_surjective (h : Function.Surjective g) [IsLocalization (M.map g) Q] :
    Function.Surjective (map Q g M.le_comap_map : S → Q) 
Arguments: {R : Type u_1} [CommSemiring R] (M : Submonoid R) (S : Type u_2) [CommSemiring S] [Algebra R S] {P : Type u_3} [CommSemiring P] [IsLocalization M S] {g : R →+* P} (Q : Type u_4) [CommSemiring Q] [Algebra P Q] (h : Function.Surjective ⇑g) [IsLocalization (Submonoid.map g M) Q]
Goal: Function.Surjective ⇑(IsLocalization.map Q g ⋯)
-- IsLocalization.isLocalization_of_algEquiv
theorem isLocalization_of_algEquiv [Algebra R P] [IsLocalization M S] (h : S ≃ₐ[R] P) :
    IsLocalization M P 
Arguments: {R : Type u_1} [CommSemiring R] (M : Submonoid R) {S : Type u_2} [CommSemiring S] [Algebra R S] {P : Type u_3} [CommSemiring P] [Algebra R P] [IsLocalization M S] (h : S ≃ₐ[R] P)
Goal: IsLocalization M P
-- IsLocalization.isLocalization_iff_of_algEquiv
theorem isLocalization_iff_of_algEquiv [Algebra R P] (h : S ≃ₐ[R] P) :
    IsLocalization M S ↔ IsLocalization M P 
Arguments: {R : Type u_1} [CommSemiring R] (M : Submonoid R) {S : Type u_2} [CommSemiring S] [Algebra R S] {P : Type u_3} [CommSemiring P] [Algebra R P] (h : S ≃ₐ[R] P)
Goal: IsLocalization M S ↔ IsLocalization M P
-- IsLocalization.isLocalization_iff_of_ringEquiv
theorem isLocalization_iff_of_ringEquiv (h : S ≃+* P) :
    IsLocalization M S ↔
      haveI := (h.toRingHom.comp <| algebraMap R S).toAlgebra; IsLocalization M P 
Arguments: {R : Type u_1} [CommSemiring R] (M : Submonoid R) {S : Type u_2} [CommSemiring S] [Algebra R S] {P : Type u_3} [CommSemiring P] (h : S ≃+* P)
Goal: IsLocalization M S ↔ IsLocalization M P
-- IsLocalization.isLocalization_of_base_ringEquiv
theorem isLocalization_of_base_ringEquiv [IsLocalization M S] (h : R ≃+* P) :
    haveI := ((algebraMap R S).comp h.symm.toRingHom).toAlgebra
    IsLocalization (M.map h.toMonoidHom) S 
Arguments: {R : Type u_1} [CommSemiring R] (M : Submonoid R) (S : Type u_2) [CommSemiring S] [Algebra R S] {P : Type u_3} [CommSemiring P] [IsLocalization M S] (h : R ≃+* P)
Goal: IsLocalization (Submonoid.map h.toMonoidHom M) S
-- IsLocalization.isLocalization_iff_of_base_ringEquiv
theorem isLocalization_iff_of_base_ringEquiv (h : R ≃+* P) :
    IsLocalization M S ↔
      haveI := ((algebraMap R S).comp h.symm.toRingHom).toAlgebra
      IsLocalization (M.map h.toMonoidHom) S 
Arguments: {R : Type u_1} [CommSemiring R] (M : Submonoid R) (S : Type u_2) [CommSemiring S] [Algebra R S] {P : Type u_3} [CommSemiring P] (h : R ≃+* P)
Goal: IsLocalization M S ↔ IsLocalization (Submonoid.map h.toMonoidHom M) S
-- IsLocalization.nonZeroDivisors_le_comap
theorem nonZeroDivisors_le_comap [IsLocalization M S] :
    nonZeroDivisors R ≤ (nonZeroDivisors S).comap (algebraMap R S) 
Arguments: {R : Type u_1} [CommSemiring R] (M : Submonoid R) (S : Type u_2) [CommSemiring S] [Algebra R S] [IsLocalization M S]
Goal: nonZeroDivisors R ≤ Submonoid.comap (algebraMap R S) (nonZeroDivisors S)
-- IsLocalization.map_nonZeroDivisors_le
theorem map_nonZeroDivisors_le [IsLocalization M S] :
    (nonZeroDivisors R).map (algebraMap R S) ≤ nonZeroDivisors S 
Arguments: {R : Type u_1} [CommSemiring R] (M : Submonoid R) (S : Type u_2) [CommSemiring S] [Algebra R S] [IsLocalization M S]
Goal: Submonoid.map (algebraMap R S) (nonZeroDivisors R) ≤ nonZeroDivisors S
-- Localization.add_mk
theorem add_mk (a b c d) : (mk a b : Localization M) + mk c d =
    mk ((b : R) * c + (d : R) * a) (b * d) 
Arguments: {R : Type u_1} [CommSemiring R] {M : Submonoid R} (a : R) (b : ↥M) (c : R) (d : ↥M)
Goal: Localization.mk a b + Localization.mk c d = Localization.mk (↑b * c + ↑d * a) (b * d)
-- Localization.add_mk_self
theorem add_mk_self (a b c) : (mk a b : Localization M) + mk c b = mk (a + c) b 
Arguments: {R : Type u_1} [CommSemiring R] {M : Submonoid R} (a : R) (b : ↥M) (c : R)
Goal: Localization.mk a b + Localization.mk c b = Localization.mk (a + c) b
-- Localization.mkAddMonoidHom_apply
@[simps]
def mkAddMonoidHom (b : M) : R →+ Localization M 
Arguments: {R : Type u_1} [CommSemiring R] {M : Submonoid R} (b : ↥M) (a : R)
Goal: (Localization.mkAddMonoidHom b) a = Localization.mk a b
-- Localization.mk_sum
theorem mk_sum {ι : Type*} (f : ι → R) (s : Finset ι) (b : M) :
    mk (∑ i ∈ s, f i) b = ∑ i ∈ s, mk (f i) b 
Arguments: {R : Type u_1} [CommSemiring R] {M : Submonoid R} {ι : Type u_4} (f : ι → R) (s : Finset ι) (b : ↥M)
Goal: Localization.mk (∑ i ∈ s, f i) b = ∑ i ∈ s, Localization.mk (f i) b
-- Localization.mk_list_sum
theorem mk_list_sum (l : List R) (b : M) : mk l.sum b = (l.map fun a => mk a b).sum 
Arguments: {R : Type u_1} [CommSemiring R] {M : Submonoid R} (l : List R) (b : ↥M)
Goal: Localization.mk l.sum b = (List.map (fun a => Localization.mk a b) l).sum
-- Localization.mk_multiset_sum
theorem mk_multiset_sum (l : Multiset R) (b : M) : mk l.sum b = (l.map fun a => mk a b).sum 
Arguments: {R : Type u_1} [CommSemiring R] {M : Submonoid R} (l : Multiset R) (b : ↥M)
Goal: Localization.mk l.sum b = (Multiset.map (fun a => Localization.mk a b) l).sum
-- Localization.toLocalizationMap_eq_monoidOf
@[simp]
theorem toLocalizationMap_eq_monoidOf : toLocalizationMap M (Localization M) = monoidOf M 
Arguments: {R : Type u_1} [CommSemiring R] {M : Submonoid R}
Goal: IsLocalization.toLocalizationMap M (Localization M) = Localization.monoidOf M
-- Localization.monoidOf_eq_algebraMap
theorem monoidOf_eq_algebraMap (x) : (monoidOf M).toMap x = algebraMap R (Localization M) x 
Arguments: {R : Type u_1} [CommSemiring R] {M : Submonoid R} (x : R)
Goal: (Localization.monoidOf M).toMap x = (algebraMap R (Localization M)) x
-- Localization.mk_one_eq_algebraMap
theorem mk_one_eq_algebraMap (x) : mk x 1 = algebraMap R (Localization M) x 
Arguments: {R : Type u_1} [CommSemiring R] {M : Submonoid R} (x : R)
Goal: Localization.mk x 1 = (algebraMap R (Localization M)) x
-- Localization.mk_eq_mk'_apply
theorem mk_eq_mk'_apply (x y) : mk x y = IsLocalization.mk' (Localization M) x y 
Arguments: {R : Type u_1} [CommSemiring R] {M : Submonoid R} (x : R) (y : ↥M)
Goal: Localization.mk x y = IsLocalization.mk' (Localization M) x y
-- Localization.mk_eq_mk'
theorem mk_eq_mk' : (mk : R → M → Localization M) = IsLocalization.mk' (Localization M) 
Arguments: {R : Type u_1} [CommSemiring R] {M : Submonoid R}
Goal: Localization.mk = IsLocalization.mk' (Localization M)
-- Localization.mk_algebraMap
theorem mk_algebraMap {A : Type*} [CommSemiring A] [Algebra A R] (m : A) :
    mk (algebraMap A R m) 1 = algebraMap A (Localization M) m 
Arguments: {R : Type u_1} [CommSemiring R] {M : Submonoid R} {A : Type u_4} [CommSemiring A] [Algebra A R] (m : A)
Goal: Localization.mk ((algebraMap A R) m) 1 = (algebraMap A (Localization M)) m
-- Localization.mk_natCast
theorem mk_natCast (m : ℕ) : (mk m 1 : Localization M) = m 
Arguments: {R : Type u_1} [CommSemiring R] {M : Submonoid R} (m : ℕ)
Goal: Localization.mk (↑m) 1 = ↑m
-- Localization.algEquiv_apply
@[simps!]
noncomputable def algEquiv : Localization M ≃ₐ[R] S 
Arguments: {R : Type u_1} [CommSemiring R] (M : Submonoid R) (S : Type u_2) [CommSemiring S] [Algebra R S] [IsLocalization M S] (a : Localization M)
Goal: (Localization.algEquiv M S) a = (IsLocalization.map S (RingHom.id R) ⋯) a
-- Localization.algEquiv_symm_apply
@[simps!]
noncomputable def algEquiv : Localization M ≃ₐ[R] S 
Arguments: {R : Type u_1} [CommSemiring R] (M : Submonoid R) (S : Type u_2) [CommSemiring S] [Algebra R S] [IsLocalization M S] (a : S)
Goal: (Localization.algEquiv M S).symm a = (IsLocalization.map (Localization M) (RingHom.id R) ⋯) a
-- Localization.algEquiv_mk'
nonrec theorem algEquiv_mk' (x : R) (y : M) : algEquiv M S (mk' (Localization M) x y) = mk' S x y 
Arguments: {R : Type u_1} [CommSemiring R] {M : Submonoid R} (S : Type u_2) [CommSemiring S] [Algebra R S] [IsLocalization M S] (x : R) (y : ↥M)
Goal: (Localization.algEquiv M S) (IsLocalization.mk' (Localization M) x y) = IsLocalization.mk' S x y
-- Localization.algEquiv_symm_mk'
nonrec theorem algEquiv_symm_mk' (x : R) (y : M) :
    (algEquiv M S).symm (mk' S x y) = mk' (Localization M) x y 
Arguments: {R : Type u_1} [CommSemiring R] {M : Submonoid R} (S : Type u_2) [CommSemiring S] [Algebra R S] [IsLocalization M S] (x : R) (y : ↥M)
Goal: (Localization.algEquiv M S).symm (IsLocalization.mk' S x y) = IsLocalization.mk' (Localization M) x y
-- Localization.algEquiv_mk
theorem algEquiv_mk (x y) : algEquiv M S (mk x y) = mk' S x y 
Arguments: {R : Type u_1} [CommSemiring R] {M : Submonoid R} (S : Type u_2) [CommSemiring S] [Algebra R S] [IsLocalization M S] (x : R) (y : ↥M)
Goal: (Localization.algEquiv M S) (Localization.mk x y) = IsLocalization.mk' S x y
-- Localization.algEquiv_symm_mk
theorem algEquiv_symm_mk (x : R) (y : M) : (algEquiv M S).symm (mk' S x y) = mk x y 
Arguments: {R : Type u_1} [CommSemiring R] {M : Submonoid R} (S : Type u_2) [CommSemiring S] [Algebra R S] [IsLocalization M S] (x : R) (y : ↥M)
Goal: (Localization.algEquiv M S).symm (IsLocalization.mk' S x y) = Localization.mk x y
-- Localization.coe_algEquiv
lemma coe_algEquiv :
    (Localization.algEquiv M S : Localization M →+* S) =
    IsLocalization.map (M := M) (T := M) _ (RingHom.id R) le_rfl 
Arguments: {R : Type u_1} [CommSemiring R] {M : Submonoid R} (S : Type u_2) [CommSemiring S] [Algebra R S] [IsLocalization M S]
Goal: ↑(Localization.algEquiv M S) = IsLocalization.map S (RingHom.id R) ⋯
-- Localization.coe_algEquiv_symm
lemma coe_algEquiv_symm :
    ((Localization.algEquiv M S).symm : S →+* Localization M) =
    IsLocalization.map (M := M) (T := M) _ (RingHom.id R) le_rfl 
Arguments: {R : Type u_1} [CommSemiring R] {M : Submonoid R} (S : Type u_2) [CommSemiring S] [Algebra R S] [IsLocalization M S]
Goal: ↑(Localization.algEquiv M S).symm = IsLocalization.map (Localization M) (RingHom.id R) ⋯
-- Localization.neg_mk
theorem neg_mk (a b) : -(mk a b : Localization M) = mk (-a) b 
Arguments: {R : Type u_1} [CommRing R] {M : Submonoid R} (a : R) (b : ↥M)
Goal: -Localization.mk a b = Localization.mk (-a) b
-- Localization.sub_mk
theorem sub_mk (a c) (b d) : (mk a b : Localization M) - mk c d =
    mk ((d : R) * a - b * c) (b * d) 
Arguments: {R : Type u_1} [CommRing R] {M : Submonoid R} (a c : R) (b d : ↥M)
Goal: Localization.mk a b - Localization.mk c d = Localization.mk (↑d * a - ↑b * c) (b * d)
-- Localization.mk_intCast
theorem mk_intCast (m : ℤ) : (mk m 1 : Localization M) = m 
Arguments: {R : Type u_1} [CommRing R] {M : Submonoid R} (m : ℤ)
Goal: Localization.mk (↑m) 1 = ↑m
-- IsLocalization.to_map_eq_zero_iff
theorem to_map_eq_zero_iff {x : R} (hM : M ≤ nonZeroDivisors R) : algebraMap R S x = 0 ↔ x = 0 
Arguments: {R : Type u_1} [CommRing R] {M : Submonoid R} (S : Type u_2) [CommRing S] [Algebra R S] [IsLocalization M S] {x : R} (hM : M ≤ nonZeroDivisors R)
Goal: (algebraMap R S) x = 0 ↔ x = 0
-- IsLocalization.injective
protected theorem injective (hM : M ≤ nonZeroDivisors R) : Injective (algebraMap R S) 
Arguments: {R : Type u_1} [CommRing R] {M : Submonoid R} (S : Type u_2) [CommRing S] [Algebra R S] [IsLocalization M S] (hM : M ≤ nonZeroDivisors R)
Goal: Function.Injective ⇑(algebraMap R S)