ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- TrivSqZeroExt.algebraMap_eq_inl
theorem algebraMap_eq_inl : ⇑(algebraMap R' (tsze R' M)) = inl 
Arguments: (R' : Type u) (M : Type v) [CommSemiring R'] [AddCommMonoid M] [Module R' M] [Module R'ᵐᵒᵖ M] [IsCentralScalar R' M]
Goal: ⇑(algebraMap R' (tsze R' M)) = TrivSqZeroExt.inl
-- TrivSqZeroExt.algebraMap_eq_inlHom
theorem algebraMap_eq_inlHom : algebraMap R' (tsze R' M) = inlHom R' M 
Arguments: (R' : Type u) (M : Type v) [CommSemiring R'] [AddCommMonoid M] [Module R' M] [Module R'ᵐᵒᵖ M] [IsCentralScalar R' M]
Goal: algebraMap R' (tsze R' M) = TrivSqZeroExt.inlHom R' M
-- TrivSqZeroExt.algebraMap_eq_inl'
theorem algebraMap_eq_inl' (s : S) : algebraMap S (tsze R M) s = inl (algebraMap S R s) 
Arguments: (S : Type u_1) (R : Type u) (M : Type v) [CommSemiring S] [Semiring R] [AddCommMonoid M] [Algebra S R] [Module S M] [Module R M] [Module Rᵐᵒᵖ M] [SMulCommClass R Rᵐᵒᵖ M] [IsScalarTower S R M] [IsScalarTower S Rᵐᵒᵖ M] (s : S)
Goal: (algebraMap S (tsze R M)) s = TrivSqZeroExt.inl ((algebraMap S R) s)
-- TrivSqZeroExt.fstHom_apply
@[simps]
def fstHom : tsze R M →ₐ[S] R 
Arguments: (S : Type u_1) (R : Type u) (M : Type v) [CommSemiring S] [Semiring R] [AddCommMonoid M] [Algebra S R] [Module S M] [Module R M] [Module Rᵐᵒᵖ M] [SMulCommClass R Rᵐᵒᵖ M] [IsScalarTower S R M] [IsScalarTower S Rᵐᵒᵖ M] (x : tsze R M)
Goal: (TrivSqZeroExt.fstHom S R M) x = x.fst
-- TrivSqZeroExt.inlAlgHom_apply
@[simps]
def inlAlgHom : R →ₐ[S] tsze R M 
Arguments: (S : Type u_1) (R : Type u) (M : Type v) [CommSemiring S] [Semiring R] [AddCommMonoid M] [Algebra S R] [Module S M] [Module R M] [Module Rᵐᵒᵖ M] [SMulCommClass R Rᵐᵒᵖ M] [IsScalarTower S R M] [IsScalarTower S Rᵐᵒᵖ M] (r : R)
Goal: (TrivSqZeroExt.inlAlgHom S R M) r = TrivSqZeroExt.inl r
-- TrivSqZeroExt.algHom_ext
theorem algHom_ext {A} [Semiring A] [Algebra R' A] ⦃f g : tsze R' M →ₐ[R'] A⦄
    (h : ∀ m, f (inr m) = g (inr m)) : f = g 
Arguments: {R' : Type u} {M : Type v} [CommSemiring R'] [AddCommMonoid M] [Module R' M] [Module R'ᵐᵒᵖ M] [IsCentralScalar R' M] {A : Type u_2} [Semiring A] [Algebra R' A] ⦃f g : tsze R' M →ₐ[R'] A⦄ (h : ∀ (m : M), f (TrivSqZeroExt.inr m) = g (TrivSqZeroExt.inr m))
Goal: f = g
-- TrivSqZeroExt.algHom_ext'
@[ext]
theorem algHom_ext' {A} [Semiring A] [Algebra S A] ⦃f g : tsze R M →ₐ[S] A⦄
    (hinl : f.comp (inlAlgHom S R M) = g.comp (inlAlgHom S R M))
    (hinr : f.toLinearMap.comp (inrHom R M |>.restrictScalars S) =
      g.toLinearMap.comp (inrHom R M |>.restrictScalars S)) : f = g 
Arguments: {S : Type u_1} {R : Type u} {M : Type v} [CommSemiring S] [Semiring R] [AddCommMonoid M] [Algebra S R] [Module S M] [Module R M] [Module Rᵐᵒᵖ M] [SMulCommClass R Rᵐᵒᵖ M] [IsScalarTower S R M] [IsScalarTower S Rᵐᵒᵖ M] {A : Type u_2} [Semiring A] [Algebra S A] ⦃f g : tsze R M →ₐ[S] A⦄ (hinl : f.comp (TrivSqZeroExt.inlAlgHom S R M) = g.comp (TrivSqZeroExt.inlAlgHom S R M)) (hinr : f.toLinearMap ∘ₗ ↑S (TrivSqZeroExt.inrHom R M) = g.toLinearMap ∘ₗ ↑S (TrivSqZeroExt.inrHom R M))
Goal: f = g
-- TrivSqZeroExt.lift_def
theorem lift_def (f : R →ₐ[S] A) (g : M →ₗ[S] A)
    (hg : ∀ x y, g x * g y = 0)
    (hfg : ∀ r x, g (r • x) = f r * g x)
    (hgf : ∀ r x, g (op r • x) = g x * f r) (x : tsze R M) :
    lift f g hg hfg hgf x = f x.fst + g x.snd 
Arguments: {S : Type u_1} {R : Type u} {M : Type v} [CommSemiring S] [Semiring R] [AddCommMonoid M] [Algebra S R] [Module S M] [Module R M] [Module Rᵐᵒᵖ M] [SMulCommClass R Rᵐᵒᵖ M] [IsScalarTower S R M] [IsScalarTower S Rᵐᵒᵖ M] {A : Type u_2} [Semiring A] [Algebra S A] (f : R →ₐ[S] A) (g : M →ₗ[S] A) (hg : ∀ (x y : M), g x * g y = 0) (hfg : ∀ (r : R) (x : M), g (r •> x) = f r * g x) (hgf : ∀ (r : R) (x : M), g (x <• r) = g x * f r) (x : tsze R M)
Goal: (TrivSqZeroExt.lift f g hg hfg hgf) x = f x.fst + g x.snd
-- TrivSqZeroExt.lift_apply_inl
@[simp]
theorem lift_apply_inl (f : R →ₐ[S] A) (g : M →ₗ[S] A)
    (hg : ∀ x y, g x * g y = 0)
    (hfg : ∀ r x, g (r •> x) = f r * g x)
    (hgf : ∀ r x, g (x <• r) = g x * f r)
    (r : R) :
    lift f g hg hfg hgf (inl r) = f r 
Arguments: {S : Type u_1} {R : Type u} {M : Type v} [CommSemiring S] [Semiring R] [AddCommMonoid M] [Algebra S R] [Module S M] [Module R M] [Module Rᵐᵒᵖ M] [SMulCommClass R Rᵐᵒᵖ M] [IsScalarTower S R M] [IsScalarTower S Rᵐᵒᵖ M] {A : Type u_2} [Semiring A] [Algebra S A] (f : R →ₐ[S] A) (g : M →ₗ[S] A) (hg : ∀ (x y : M), g x * g y = 0) (hfg : ∀ (r : R) (x : M), g (r •> x) = f r * g x) (hgf : ∀ (r : R) (x : M), g (x <• r) = g x * f r) (r : R)
Goal: (TrivSqZeroExt.lift f g hg hfg hgf) (TrivSqZeroExt.inl r) = f r
-- TrivSqZeroExt.lift_apply_inr
@[simp]
theorem lift_apply_inr (f : R →ₐ[S] A) (g : M →ₗ[S] A)
    (hg : ∀ x y, g x * g y = 0)
    (hfg : ∀ r x, g (r •> x) = f r * g x)
    (hgf : ∀ r x, g (x <• r) = g x * f r)
    (m : M) :
    lift f g hg hfg hgf (inr m) = g m 
Arguments: {S : Type u_1} {R : Type u} {M : Type v} [CommSemiring S] [Semiring R] [AddCommMonoid M] [Algebra S R] [Module S M] [Module R M] [Module Rᵐᵒᵖ M] [SMulCommClass R Rᵐᵒᵖ M] [IsScalarTower S R M] [IsScalarTower S Rᵐᵒᵖ M] {A : Type u_2} [Semiring A] [Algebra S A] (f : R →ₐ[S] A) (g : M →ₗ[S] A) (hg : ∀ (x y : M), g x * g y = 0) (hfg : ∀ (r : R) (x : M), g (r •> x) = f r * g x) (hgf : ∀ (r : R) (x : M), g (x <• r) = g x * f r) (m : M)
Goal: (TrivSqZeroExt.lift f g hg hfg hgf) (TrivSqZeroExt.inr m) = g m
-- TrivSqZeroExt.lift_comp_inlHom
@[simp]
theorem lift_comp_inlHom (f : R →ₐ[S] A) (g : M →ₗ[S] A)
    (hg : ∀ x y, g x * g y = 0)
    (hfg : ∀ r x, g (r •> x) = f r * g x)
    (hgf : ∀ r x, g (x <• r) = g x * f r) :
    (lift f g hg hfg hgf).comp (inlAlgHom S R M) = f 
Arguments: {S : Type u_1} {R : Type u} {M : Type v} [CommSemiring S] [Semiring R] [AddCommMonoid M] [Algebra S R] [Module S M] [Module R M] [Module Rᵐᵒᵖ M] [SMulCommClass R Rᵐᵒᵖ M] [IsScalarTower S R M] [IsScalarTower S Rᵐᵒᵖ M] {A : Type u_2} [Semiring A] [Algebra S A] (f : R →ₐ[S] A) (g : M →ₗ[S] A) (hg : ∀ (x y : M), g x * g y = 0) (hfg : ∀ (r : R) (x : M), g (r •> x) = f r * g x) (hgf : ∀ (r : R) (x : M), g (x <• r) = g x * f r)
Goal: (TrivSqZeroExt.lift f g hg hfg hgf).comp (TrivSqZeroExt.inlAlgHom S R M) = f
-- TrivSqZeroExt.lift_comp_inrHom
@[simp]
theorem lift_comp_inrHom (f : R →ₐ[S] A) (g : M →ₗ[S] A)
    (hg : ∀ x y, g x * g y = 0)
    (hfg : ∀ r x, g (r •> x) = f r * g x)
    (hgf : ∀ r x, g (x <• r) = g x * f r) :
    (lift f g hg hfg hgf).toLinearMap.comp (inrHom R M |>.restrictScalars S) = g 
Arguments: {S : Type u_1} {R : Type u} {M : Type v} [CommSemiring S] [Semiring R] [AddCommMonoid M] [Algebra S R] [Module S M] [Module R M] [Module Rᵐᵒᵖ M] [SMulCommClass R Rᵐᵒᵖ M] [IsScalarTower S R M] [IsScalarTower S Rᵐᵒᵖ M] {A : Type u_2} [Semiring A] [Algebra S A] (f : R →ₐ[S] A) (g : M →ₗ[S] A) (hg : ∀ (x y : M), g x * g y = 0) (hfg : ∀ (r : R) (x : M), g (r •> x) = f r * g x) (hgf : ∀ (r : R) (x : M), g (x <• r) = g x * f r)
Goal: (TrivSqZeroExt.lift f g hg hfg hgf).toLinearMap ∘ₗ ↑S (TrivSqZeroExt.inrHom R M) = g
-- TrivSqZeroExt.lift_inlAlgHom_inrHom
@[simp]
theorem lift_inlAlgHom_inrHom :
    lift (inlAlgHom _ _ _) (inrHom R M |>.restrictScalars S)
      (inr_mul_inr R) (fun _ _ => (inl_mul_inr _ _).symm) (fun _ _ => (inr_mul_inl _ _).symm) =
    AlgHom.id S (tsze R M) 
Arguments: {S : Type u_1} {R : Type u} {M : Type v} [CommSemiring S] [Semiring R] [AddCommMonoid M] [Algebra S R] [Module S M] [Module R M] [Module Rᵐᵒᵖ M] [SMulCommClass R Rᵐᵒᵖ M] [IsScalarTower S R M] [IsScalarTower S Rᵐᵒᵖ M]
Goal: TrivSqZeroExt.lift (TrivSqZeroExt.inlAlgHom S R M) (↑S (TrivSqZeroExt.inrHom R M)) ⋯ ⋯ ⋯ = AlgHom.id S (tsze R M)
-- TrivSqZeroExt.liftEquiv_apply
@[simps! apply symm_apply_coe]
def liftEquiv :
    {fg : (R →ₐ[S] A) × (M →ₗ[S] A) //
      (∀ x y, fg.2 x * fg.2 y = 0) ∧
      (∀ r x, fg.2 (r •> x) = fg.1 r * fg.2 x) ∧
      (∀ r x, fg.2 (x <• r) = fg.2 x * fg.1 r)} ≃ (tsze R M →ₐ[S] A) 
Arguments: {S : Type u_1} {R : Type u} {M : Type v} [CommSemiring S] [Semiring R] [AddCommMonoid M] [Algebra S R] [Module S M] [Module R M] [Module Rᵐᵒᵖ M] [SMulCommClass R Rᵐᵒᵖ M] [IsScalarTower S R M] [IsScalarTower S Rᵐᵒᵖ M] {A : Type u_2} [Semiring A] [Algebra S A] (fg :
  { fg //
    (∀ (x y : M), fg.2 x * fg.2 y = 0) ∧
      (∀ (r : R) (x : M), fg.2 (r •> x) = fg.1 r * fg.2 x) ∧ ∀ (r : R) (x : M), fg.2 (x <• r) = fg.2 x * fg.1 r })
Goal: TrivSqZeroExt.liftEquiv fg = TrivSqZeroExt.lift (↑fg).1 (↑fg).2 ⋯ ⋯ ⋯
-- TrivSqZeroExt.liftEquiv_symm_apply_coe
@[simps! apply symm_apply_coe]
def liftEquiv :
    {fg : (R →ₐ[S] A) × (M →ₗ[S] A) //
      (∀ x y, fg.2 x * fg.2 y = 0) ∧
      (∀ r x, fg.2 (r •> x) = fg.1 r * fg.2 x) ∧
      (∀ r x, fg.2 (x <• r) = fg.2 x * fg.1 r)} ≃ (tsze R M →ₐ[S] A) 
Arguments: {S : Type u_1} {R : Type u} {M : Type v} [CommSemiring S] [Semiring R] [AddCommMonoid M] [Algebra S R] [Module S M] [Module R M] [Module Rᵐᵒᵖ M] [SMulCommClass R Rᵐᵒᵖ M] [IsScalarTower S R M] [IsScalarTower S Rᵐᵒᵖ M] {A : Type u_2} [Semiring A] [Algebra S A] (F : tsze R M →ₐ[S] A)
Goal: ↑(TrivSqZeroExt.liftEquiv.symm F) =
  (F.comp (TrivSqZeroExt.inlAlgHom S R M), F.toLinearMap ∘ₗ ↑S (TrivSqZeroExt.inrHom R M))
-- TrivSqZeroExt.liftEquivOfComm_symm_apply_coe
@[simps! apply symm_apply_coe]
def liftEquivOfComm :
    { f : M →ₗ[R'] A // ∀ x y, f x * f y = 0 } ≃ (tsze R' M →ₐ[R'] A) 
Arguments: {R' : Type u} {M : Type v} [CommSemiring R'] [AddCommMonoid M] [Module R' M] [Module R'ᵐᵒᵖ M] [IsCentralScalar R' M] {A : Type u_2} [Semiring A] [Algebra R' A]
Goal: ∀ (a : tsze R' M →ₐ[R'] A), ↑(TrivSqZeroExt.liftEquivOfComm.symm a) = a.toLinearMap ∘ₗ ↑R' (TrivSqZeroExt.inrHom R' M)
-- TrivSqZeroExt.liftEquivOfComm_apply
@[simps! apply symm_apply_coe]
def liftEquivOfComm :
    { f : M →ₗ[R'] A // ∀ x y, f x * f y = 0 } ≃ (tsze R' M →ₐ[R'] A) 
Arguments: {R' : Type u} {M : Type v} [CommSemiring R'] [AddCommMonoid M] [Module R' M] [Module R'ᵐᵒᵖ M] [IsCentralScalar R' M] {A : Type u_2} [Semiring A] [Algebra R' A]
Goal: ∀ (a : { f // ∀ (x y : M), f x * f y = 0 }),
  TrivSqZeroExt.liftEquivOfComm a = TrivSqZeroExt.lift (Algebra.ofId R' A) ↑a ⋯ ⋯ ⋯
-- TrivSqZeroExt.map_inl
@[simp]
theorem map_inl (f : M →ₗ[R'] N) (r : R') : map f (inl r) = inl r 
Arguments: {R' : Type u} {M : Type v} [CommSemiring R'] [AddCommMonoid M] [Module R' M] [Module R'ᵐᵒᵖ M] [IsCentralScalar R' M] {N : Type u_3} [AddCommMonoid N] [Module R' N] [Module R'ᵐᵒᵖ N] [IsCentralScalar R' N] (f : M →ₗ[R'] N) (r : R')
Goal: (TrivSqZeroExt.map f) (TrivSqZeroExt.inl r) = TrivSqZeroExt.inl r
-- TrivSqZeroExt.map_inr
@[simp]
theorem map_inr (f : M →ₗ[R'] N) (x : M) : map f (inr x) = inr (f x) 
Arguments: {R' : Type u} {M : Type v} [CommSemiring R'] [AddCommMonoid M] [Module R' M] [Module R'ᵐᵒᵖ M] [IsCentralScalar R' M] {N : Type u_3} [AddCommMonoid N] [Module R' N] [Module R'ᵐᵒᵖ N] [IsCentralScalar R' N] (f : M →ₗ[R'] N) (x : M)
Goal: (TrivSqZeroExt.map f) (TrivSqZeroExt.inr x) = TrivSqZeroExt.inr (f x)
-- TrivSqZeroExt.fst_map
@[simp]
theorem fst_map (f : M →ₗ[R'] N) (x : TrivSqZeroExt R' M) : fst (map f x) = fst x 
Arguments: {R' : Type u} {M : Type v} [CommSemiring R'] [AddCommMonoid M] [Module R' M] [Module R'ᵐᵒᵖ M] [IsCentralScalar R' M] {N : Type u_3} [AddCommMonoid N] [Module R' N] [Module R'ᵐᵒᵖ N] [IsCentralScalar R' N] (f : M →ₗ[R'] N) (x : tsze R' M)
Goal: ((TrivSqZeroExt.map f) x).fst = x.fst
-- TrivSqZeroExt.snd_map
@[simp]
theorem snd_map (f : M →ₗ[R'] N) (x : TrivSqZeroExt R' M) : snd (map f x) = f (snd x) 
Arguments: {R' : Type u} {M : Type v} [CommSemiring R'] [AddCommMonoid M] [Module R' M] [Module R'ᵐᵒᵖ M] [IsCentralScalar R' M] {N : Type u_3} [AddCommMonoid N] [Module R' N] [Module R'ᵐᵒᵖ N] [IsCentralScalar R' N] (f : M →ₗ[R'] N) (x : tsze R' M)
Goal: ((TrivSqZeroExt.map f) x).snd = f x.snd
-- TrivSqZeroExt.map_comp_inlAlgHom
@[simp]
theorem map_comp_inlAlgHom (f : M →ₗ[R'] N) :
    (map f).comp (inlAlgHom R' R' M) = inlAlgHom R' R' N 
Arguments: {R' : Type u} {M : Type v} [CommSemiring R'] [AddCommMonoid M] [Module R' M] [Module R'ᵐᵒᵖ M] [IsCentralScalar R' M] {N : Type u_3} [AddCommMonoid N] [Module R' N] [Module R'ᵐᵒᵖ N] [IsCentralScalar R' N] (f : M →ₗ[R'] N)
Goal: (TrivSqZeroExt.map f).comp (TrivSqZeroExt.inlAlgHom R' R' M) = TrivSqZeroExt.inlAlgHom R' R' N
-- TrivSqZeroExt.map_comp_inrHom
@[simp]
theorem map_comp_inrHom (f : M →ₗ[R'] N) :
    (map f).toLinearMap ∘ₗ inrHom R' M = inrHom R' N ∘ₗ f 
Arguments: {R' : Type u} {M : Type v} [CommSemiring R'] [AddCommMonoid M] [Module R' M] [Module R'ᵐᵒᵖ M] [IsCentralScalar R' M] {N : Type u_3} [AddCommMonoid N] [Module R' N] [Module R'ᵐᵒᵖ N] [IsCentralScalar R' N] (f : M →ₗ[R'] N)
Goal: (TrivSqZeroExt.map f).toLinearMap ∘ₗ TrivSqZeroExt.inrHom R' M = TrivSqZeroExt.inrHom R' N ∘ₗ f
-- TrivSqZeroExt.fstHom_comp_map
@[simp]
theorem fstHom_comp_map (f : M →ₗ[R'] N) :
    (fstHom R' R' N).comp (map f) = fstHom R' R' M 
Arguments: {R' : Type u} {M : Type v} [CommSemiring R'] [AddCommMonoid M] [Module R' M] [Module R'ᵐᵒᵖ M] [IsCentralScalar R' M] {N : Type u_3} [AddCommMonoid N] [Module R' N] [Module R'ᵐᵒᵖ N] [IsCentralScalar R' N] (f : M →ₗ[R'] N)
Goal: (TrivSqZeroExt.fstHom R' R' N).comp (TrivSqZeroExt.map f) = TrivSqZeroExt.fstHom R' R' M
-- TrivSqZeroExt.sndHom_comp_map
@[simp]
theorem sndHom_comp_map (f : M →ₗ[R'] N) :
    sndHom R' N ∘ₗ (map f).toLinearMap = f ∘ₗ sndHom R' M 
Arguments: {R' : Type u} {M : Type v} [CommSemiring R'] [AddCommMonoid M] [Module R' M] [Module R'ᵐᵒᵖ M] [IsCentralScalar R' M] {N : Type u_3} [AddCommMonoid N] [Module R' N] [Module R'ᵐᵒᵖ N] [IsCentralScalar R' N] (f : M →ₗ[R'] N)
Goal: TrivSqZeroExt.sndHom R' N ∘ₗ (TrivSqZeroExt.map f).toLinearMap = f ∘ₗ TrivSqZeroExt.sndHom R' M
-- TrivSqZeroExt.map_id
@[simp]
theorem map_id : map (LinearMap.id : M →ₗ[R'] M) = AlgHom.id R' _ 
Arguments: {R' : Type u} {M : Type v} [CommSemiring R'] [AddCommMonoid M] [Module R' M] [Module R'ᵐᵒᵖ M] [IsCentralScalar R' M]
Goal: TrivSqZeroExt.map LinearMap.id = AlgHom.id R' (tsze R' M)
-- TrivSqZeroExt.map_comp_map
theorem map_comp_map (f : M →ₗ[R'] N) (g : N →ₗ[R'] P) :
    map (g.comp f) = (map g).comp (map f) 
Arguments: {R' : Type u} {M : Type v} [CommSemiring R'] [AddCommMonoid M] [Module R' M] [Module R'ᵐᵒᵖ M] [IsCentralScalar R' M] {N : Type u_3} {P : Type u_4} [AddCommMonoid N] [Module R' N] [Module R'ᵐᵒᵖ N] [IsCentralScalar R' N] [AddCommMonoid P] [Module R' P] [Module R'ᵐᵒᵖ P] [IsCentralScalar R' P] (f : M →ₗ[R'] N) (g : N →ₗ[R'] P)
Goal: TrivSqZeroExt.map (g ∘ₗ f) = (TrivSqZeroExt.map g).comp (TrivSqZeroExt.map f)
-- AlgebraicGeometry.IsLocallyNoetherian.component_noetherian
class IsLocallyNoetherian (X : Scheme) : Prop where
  component_noetherian : ∀ (U : X.affineOpens),
    IsNoetherianRing Γ(X, U) := by infer_instance
Arguments: {X : AlgebraicGeometry.Scheme} [self : AlgebraicGeometry.IsLocallyNoetherian X] (U : ↑X.affineOpens)
Goal: IsNoetherianRing ↑Γ(X, ↑U)
-- AlgebraicGeometry.isNoetherianRing_of_away
theorem isNoetherianRing_of_away : IsNoetherianRing R 
Arguments: {R : Type u} [CommRing R] (S : Finset R) (hS : Ideal.span ↑S = ⊤) (hN : ∀ (s : { x // x ∈ S }), IsNoetherianRing (Localization.Away ↑s))
Goal: IsNoetherianRing R
-- AlgebraicGeometry.isLocallyNoetherian_of_affine_cover
theorem isLocallyNoetherian_of_affine_cover {ι} {S : ι → X.affineOpens}
    (hS : (⨆ i, S i : X.Opens) = ⊤)
    (hS' : ∀ i, IsNoetherianRing Γ(X, S i)) : IsLocallyNoetherian X 
Arguments: {X : AlgebraicGeometry.Scheme} {ι : Sort u_1} {S : ι → ↑X.affineOpens} (hS : ⨆ i, ↑(S i) = ⊤) (hS' : ∀ (i : ι), IsNoetherianRing ↑Γ(X, ↑(S i)))
Goal: AlgebraicGeometry.IsLocallyNoetherian X
-- AlgebraicGeometry.isLocallyNoetherian_iff_of_iSup_eq_top
theorem isLocallyNoetherian_iff_of_iSup_eq_top {ι} {S : ι → X.affineOpens}
    (hS : (⨆ i, S i : X.Opens) = ⊤) :
    IsLocallyNoetherian X ↔ ∀ i, IsNoetherianRing Γ(X, S i) 
Arguments: {X : AlgebraicGeometry.Scheme} {ι : Sort u_1} {S : ι → ↑X.affineOpens} (hS : ⨆ i, ↑(S i) = ⊤)
Goal: AlgebraicGeometry.IsLocallyNoetherian X ↔ ∀ (i : ι), IsNoetherianRing ↑Γ(X, ↑(S i))
-- AlgebraicGeometry.isLocallyNoetherian_iff_of_affine_openCover
theorem isLocallyNoetherian_iff_of_affine_openCover (𝒰 : Scheme.OpenCover.{v, u} X)
    [∀ i, IsAffine (𝒰.obj i)] :
    IsLocallyNoetherian X ↔ ∀ (i : 𝒰.J), IsNoetherianRing Γ(𝒰.obj i, ⊤) 
Arguments: {X : AlgebraicGeometry.Scheme} (𝒰 : X.OpenCover) [∀ (i : 𝒰.J), AlgebraicGeometry.IsAffine (𝒰.obj i)]
Goal: AlgebraicGeometry.IsLocallyNoetherian X ↔ ∀ (i : 𝒰.J), IsNoetherianRing ↑Γ(𝒰.obj i, ⊤)
-- AlgebraicGeometry.isLocallyNoetherian_of_isOpenImmersion
lemma isLocallyNoetherian_of_isOpenImmersion {Y : Scheme} (f : X ⟶ Y) [IsOpenImmersion f]
    [IsLocallyNoetherian Y] : IsLocallyNoetherian X 
Arguments: {X Y : AlgebraicGeometry.Scheme} (f : X ⟶ Y) [AlgebraicGeometry.IsOpenImmersion f] [AlgebraicGeometry.IsLocallyNoetherian Y]
Goal: AlgebraicGeometry.IsLocallyNoetherian X
-- AlgebraicGeometry.isLocallyNoetherian_iff_openCover
theorem isLocallyNoetherian_iff_openCover (𝒰 : Scheme.OpenCover X) :
    IsLocallyNoetherian X ↔ ∀ (i : 𝒰.J), IsLocallyNoetherian (𝒰.obj i) 
Arguments: {X : AlgebraicGeometry.Scheme} (𝒰 : X.OpenCover)
Goal: AlgebraicGeometry.IsLocallyNoetherian X ↔ ∀ (i : 𝒰.J), AlgebraicGeometry.IsLocallyNoetherian (𝒰.obj i)
-- AlgebraicGeometry.noetherianSpace_of_isAffine
lemma noetherianSpace_of_isAffine [IsAffine X] [IsNoetherianRing Γ(X, ⊤)] :
    NoetherianSpace X 
Arguments: {X : AlgebraicGeometry.Scheme} [AlgebraicGeometry.IsAffine X] [IsNoetherianRing ↑Γ(X, ⊤)]
Goal: TopologicalSpace.NoetherianSpace ↑↑X.toPresheafedSpace
-- AlgebraicGeometry.noetherianSpace_of_isAffineOpen
lemma noetherianSpace_of_isAffineOpen (U : X.Opens) (hU : IsAffineOpen U)
    [IsNoetherianRing Γ(X, U)] :
    NoetherianSpace U 
Arguments: {X : AlgebraicGeometry.Scheme} (U : X.Opens) (hU : AlgebraicGeometry.IsAffineOpen U) [IsNoetherianRing ↑Γ(X, U)]
Goal: TopologicalSpace.NoetherianSpace ↑↑(↑U).toPresheafedSpace
-- AlgebraicGeometry.isNoetherian_iff
@[mk_iff]
class IsNoetherian (X : Scheme) extends IsLocallyNoetherian X, CompactSpace X : Prop
Arguments: (X : AlgebraicGeometry.Scheme)
Goal: AlgebraicGeometry.IsNoetherian X ↔ AlgebraicGeometry.IsLocallyNoetherian X ∧ CompactSpace ↑↑X.toPresheafedSpace
-- AlgebraicGeometry.isNoetherian_iff_of_finite_iSup_eq_top
theorem isNoetherian_iff_of_finite_iSup_eq_top {ι} [Finite ι] {S : ι → X.affineOpens}
    (hS : (⨆ i, S i : X.Opens) = ⊤) :
    IsNoetherian X ↔ ∀ i, IsNoetherianRing Γ(X, S i) 
Arguments: {X : AlgebraicGeometry.Scheme} {ι : Sort u_1} [Finite ι] {S : ι → ↑X.affineOpens} (hS : ⨆ i, ↑(S i) = ⊤)
Goal: AlgebraicGeometry.IsNoetherian X ↔ ∀ (i : ι), IsNoetherianRing ↑Γ(X, ↑(S i))
-- AlgebraicGeometry.isNoetherian_iff_of_finite_affine_openCover
theorem isNoetherian_iff_of_finite_affine_openCover {𝒰 : Scheme.OpenCover.{v, u} X}
    [Finite 𝒰.J] [∀ i, IsAffine (𝒰.obj i)] :
    IsNoetherian X ↔ ∀ (i : 𝒰.J), IsNoetherianRing Γ(𝒰.obj i, ⊤) 
Arguments: {X : AlgebraicGeometry.Scheme} {𝒰 : X.OpenCover} [Finite 𝒰.J] [∀ (i : 𝒰.J), AlgebraicGeometry.IsAffine (𝒰.obj i)]
Goal: AlgebraicGeometry.IsNoetherian X ↔ ∀ (i : 𝒰.J), IsNoetherianRing ↑Γ(𝒰.obj i, ⊤)
-- AlgebraicGeometry.isNoetherian_Spec
theorem isNoetherian_Spec {R : CommRingCat} :
    IsNoetherian (Spec R) ↔ IsNoetherianRing R 
Arguments: {R : CommRingCat}
Goal: AlgebraicGeometry.IsNoetherian (AlgebraicGeometry.Spec R) ↔ IsNoetherianRing ↑R
-- AlgebraicGeometry.finite_irreducibleComponents_of_isNoetherian
theorem finite_irreducibleComponents_of_isNoetherian [IsNoetherian X] :
    (irreducibleComponents X).Finite 
Arguments: {X : AlgebraicGeometry.Scheme} [AlgebraicGeometry.IsNoetherian X]
Goal: (irreducibleComponents ↑↑X.toPresheafedSpace).Finite
-- minimalPrimes.finite_of_isNoetherianRing
lemma _root_.minimalPrimes.finite_of_isNoetherianRing : (minimalPrimes R).Finite 
Arguments: (R : Type u) [CommRing R] [IsNoetherianRing R]
Goal: (minimalPrimes R).Finite
-- MaximalSpectrum.toPrimeSpectrum_range
theorem toPrimeSpectrum_range :
    Set.range (@toPrimeSpectrum R _) = { x | IsClosed ({x} : Set <| PrimeSpectrum R) } 
Arguments: {R : Type u} [CommRing R]
Goal: Set.range MaximalSpectrum.toPrimeSpectrum = {x | IsClosed {x}}
-- MaximalSpectrum.toPrimeSpectrum_continuous
theorem toPrimeSpectrum_continuous : Continuous <| @toPrimeSpectrum R _ 
Arguments: {R : Type u} [CommRing R]
Goal: Continuous MaximalSpectrum.toPrimeSpectrum
-- ModuleCat.Tilde.isLocallyFraction_pred
@[simp]
theorem isLocallyFraction_pred {U : Opens (PrimeSpectrum.Top R)}
    (f : ∀ x : U, Localizations M x) :
    (isLocallyFraction M).pred f =
      ∀ y : U,
        ∃ (V : _) (_ : y.1 ∈ V) (i : V ⟶ U),
          ∃ (m : M) (s: R), ∀ x : V, ¬s ∈ x.1.asIdeal ∧ s • f (i x) =
            LocalizedModule.mkLinearMap x.1.asIdeal.primeCompl M m 
Arguments: {R : Type u} [CommRing R] (M : ModuleCat R) {U : TopologicalSpace.Opens ↑(AlgebraicGeometry.PrimeSpectrum.Top R)} (f : (x : ↥U) → ModuleCat.Tilde.Localizations M ↑x)
Goal: (ModuleCat.Tilde.isLocallyFraction M).pred f =
  ∀ (y : ↥U),
    ∃ V,
      ∃ (_ : ↑y ∈ V),
        ∃ i m s,
          ∀ (x : ↥V),
            s ∉ (↑x).asIdeal ∧ s • f ((fun x => ⟨↑x, ⋯⟩) x) = (LocalizedModule.mkLinearMap (↑x).asIdeal.primeCompl ↑M) m
-- ModuleCat.Tilde.sections_smul_localizations_def
@[simp]
lemma sections_smul_localizations_def
    {U : (Opens (PrimeSpectrum.Top R))ᵒᵖ} (x : U.unop)
    (r : (Spec.structureSheaf R).val.obj U)
    (m : Localizations M ↑x) :
  r • m = r.1 x • m 
Arguments: {R : Type u} [CommRing R] (M : ModuleCat R) {U : (TopologicalSpace.Opens ↑(AlgebraicGeometry.PrimeSpectrum.Top R))ᵒᵖ} (x : ↥(Opposite.unop U)) (r : ↑((AlgebraicGeometry.Spec.structureSheaf R).val.obj U)) (m : ModuleCat.Tilde.Localizations M ↑x)
Goal: r • m = ↑r x • m
-- LocalizedModule.subsingleton_iff_disjoint
lemma LocalizedModule.subsingleton_iff_disjoint {f : R} :
    Subsingleton (LocalizedModule (.powers f) M) ↔
      Disjoint ↑(PrimeSpectrum.basicOpen f) (Module.support R M) 
Arguments: {R : Type u_1} {M : Type u_3} [CommRing R] [AddCommGroup M] [Module R M] {f : R}
Goal: Subsingleton (LocalizedModule (Submonoid.powers f) M) ↔ Disjoint (↑(PrimeSpectrum.basicOpen f)) (Module.support R M)
-- Module.stableUnderSpecialization_support
lemma Module.stableUnderSpecialization_support :
    StableUnderSpecialization (Module.support R M) 
Arguments: {R : Type u_1} {M : Type u_3} [CommRing R] [AddCommGroup M] [Module R M]
Goal: StableUnderSpecialization (Module.support R M)
-- Module.isClosed_support
lemma Module.isClosed_support [Module.Finite R M] :
    IsClosed (Module.support R M) 
Arguments: {R : Type u_1} {M : Type u_3} [CommRing R] [AddCommGroup M] [Module R M] [Module.Finite R M]
Goal: IsClosed (Module.support R M)
-- Module.support_subset_preimage_comap
lemma Module.support_subset_preimage_comap [IsScalarTower R A M] :
    Module.support A M ⊆ PrimeSpectrum.comap (algebraMap R A) ⁻¹' Module.support R M 
Arguments: {R : Type u_1} {A : Type u_2} {M : Type u_3} [CommRing R] [AddCommGroup M] [Module R M] [CommRing A] [Algebra R A] [Module A M] [IsScalarTower R A M]
Goal: Module.support A M ⊆ ⇑(PrimeSpectrum.comap (algebraMap R A)) ⁻¹' Module.support R M
-- AlgebraicGeometry.Polynomial.isOpen_imageOfDf
theorem isOpen_imageOfDf : IsOpen (imageOfDf f) 
Arguments: {R : Type u_1} [CommRing R] {f : R[X]}
Goal: IsOpen (AlgebraicGeometry.Polynomial.imageOfDf f)
-- AlgebraicGeometry.Polynomial.comap_C_mem_imageOfDf
theorem comap_C_mem_imageOfDf {I : PrimeSpectrum R[X]}
    (H : I ∈ (zeroLocus {f} : Set (PrimeSpectrum R[X]))ᶜ) :
    PrimeSpectrum.comap (Polynomial.C : R →+* R[X]) I ∈ imageOfDf f 
Arguments: {R : Type u_1} [CommRing R] {f : R[X]} {I : PrimeSpectrum R[X]} (H : I ∈ (PrimeSpectrum.zeroLocus {f})ᶜ)
Goal: (PrimeSpectrum.comap Polynomial.C) I ∈ AlgebraicGeometry.Polynomial.imageOfDf f
-- AlgebraicGeometry.Polynomial.imageOfDf_eq_comap_C_compl_zeroLocus
theorem imageOfDf_eq_comap_C_compl_zeroLocus :
    imageOfDf f = PrimeSpectrum.comap (C : R →+* R[X]) '' (zeroLocus {f})ᶜ 
Arguments: {R : Type u_1} [CommRing R] {f : R[X]}
Goal: AlgebraicGeometry.Polynomial.imageOfDf f = ⇑(PrimeSpectrum.comap Polynomial.C) '' (PrimeSpectrum.zeroLocus {f})ᶜ
-- AlgebraicGeometry.Polynomial.isOpenMap_comap_C
theorem isOpenMap_comap_C : IsOpenMap (PrimeSpectrum.comap (C : R →+* R[X])) 
Arguments: {R : Type u_1} [CommRing R]
Goal: IsOpenMap ⇑(PrimeSpectrum.comap Polynomial.C)
-- AlgebraicGeometry.Scheme.GlueData.oneHypercover_f
@[simps]
noncomputable def oneHypercover : Scheme.zariskiTopology.OneHypercover D.glued 
Arguments: (D : AlgebraicGeometry.Scheme.GlueData) (i : D.J)
Goal: D.oneHypercover.f i = D.ι i