ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- AlgebraicGeometry.PresheafedSpace.GlueData.ι_image_preimage_eq
theorem ι_image_preimage_eq (i j : D.J) (U : Opens (D.U i).carrier) :
    (Opens.map (𝖣.ι j).base).obj ((D.ι_openEmbedding i).isOpenMap.functor.obj U) =
      (opensFunctor (D.f j i)).obj
        ((Opens.map (𝖣.t j i).base).obj ((Opens.map (𝖣.f i j).base).obj U)) 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] (D : AlgebraicGeometry.PresheafedSpace.GlueData C) [CategoryTheory.Limits.HasLimits C] (i j : D.J) (U : TopologicalSpace.Opens ↑↑(D.U i))
Goal: (TopologicalSpace.Opens.map (D.ι j).base).obj (⋯.functor.obj U) =
  (AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.opensFunctor (D.f j i)).obj
    ((TopologicalSpace.Opens.map (D.t j i).base).obj ((TopologicalSpace.Opens.map (D.f i j).base).obj U))
-- AlgebraicGeometry.PresheafedSpace.GlueData.opensImagePreimageMap_app'
theorem opensImagePreimageMap_app' (i j k : D.J) (U : Opens (D.U i).carrier) :
    ∃ eq,
      D.opensImagePreimageMap i j U ≫ (D.f j k).c.app _ =
        ((π₁ j, i, k) ≫ D.t j i ≫ D.f i j).c.app (op U) ≫
          (π₂⁻¹ j, i, k) (unop _) ≫ (D.V (j, k)).presheaf.map (eqToHom eq) 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] (D : AlgebraicGeometry.PresheafedSpace.GlueData C) [CategoryTheory.Limits.HasLimits C] (i j k : D.J) (U : TopologicalSpace.Opens ↑↑(D.U i))
Goal: ∃ (eq :
  Opposite.op
      ((AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.opensFunctor
            (CategoryTheory.Limits.pullback.snd (D.f j i) (D.f j k))).obj
        (Opposite.unop
          (Opposite.op
            ((TopologicalSpace.Opens.map
                    (CategoryTheory.Limits.pullback.fst (D.f j i) (D.f j k) ≫ D.t j i ≫ D.f i j).base).toPrefunctor.1
              (Opposite.unop (Opposite.op U)))))) =
    (TopologicalSpace.Opens.map (D.f j k).base).op.obj
      (Opposite.op ((TopologicalSpace.Opens.map (D.ι j).base).obj (⋯.functor.obj U)))),
  D.opensImagePreimageMap i j U ≫
      (D.f j k).c.app (Opposite.op ((TopologicalSpace.Opens.map (D.ι j).base).obj (⋯.functor.obj U))) =
    (CategoryTheory.Limits.pullback.fst (D.f j i) (D.f j k) ≫ D.t j i ≫ D.f i j).c.app (Opposite.op U) ≫
      AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.invApp (CategoryTheory.Limits.pullback.snd (D.f j i) (D.f j k))
          (Opposite.unop
            (Opposite.op
              ((TopologicalSpace.Opens.map
                      (CategoryTheory.Limits.pullback.fst (D.f j i) (D.f j k) ≫ D.t j i ≫ D.f i j).base).toPrefunctor.1
                (Opposite.unop (Opposite.op U))))) ≫
        (D.V (j, k)).presheaf.map (CategoryTheory.eqToHom eq)
-- AlgebraicGeometry.PresheafedSpace.GlueData.opensImagePreimageMap_app
theorem opensImagePreimageMap_app (i j k : D.J) (U : Opens (D.U i).carrier) :
    D.opensImagePreimageMap i j U ≫ (D.f j k).c.app _ =
      ((π₁ j, i, k) ≫ D.t j i ≫ D.f i j).c.app (op U) ≫
        (π₂⁻¹ j, i, k) (unop _) ≫
          (D.V (j, k)).presheaf.map (eqToHom (opensImagePreimageMap_app' D i j k U).choose) 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] (D : AlgebraicGeometry.PresheafedSpace.GlueData C) [CategoryTheory.Limits.HasLimits C] (i j k : D.J) (U : TopologicalSpace.Opens ↑↑(D.U i))
Goal: D.opensImagePreimageMap i j U ≫
    (D.f j k).c.app (Opposite.op ((TopologicalSpace.Opens.map (D.ι j).base).obj (⋯.functor.obj U))) =
  (CategoryTheory.Limits.pullback.fst (D.f j i) (D.f j k) ≫ D.t j i ≫ D.f i j).c.app (Opposite.op U) ≫
    AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.invApp (CategoryTheory.Limits.pullback.snd (D.f j i) (D.f j k))
        (Opposite.unop
          (Opposite.op
            ((TopologicalSpace.Opens.map
                    (CategoryTheory.Limits.pullback.fst (D.f j i) (D.f j k) ≫ D.t j i ≫ D.f i j).base).toPrefunctor.1
              (Opposite.unop (Opposite.op U))))) ≫
      (D.V (j, k)).presheaf.map (CategoryTheory.eqToHom ⋯)
-- AlgebraicGeometry.PresheafedSpace.GlueData.opensImagePreimageMap_app_assoc
theorem opensImagePreimageMap_app_assoc (i j k : D.J) (U : Opens (D.U i).carrier) {X' : C}
    (f' : _ ⟶ X') :
    D.opensImagePreimageMap i j U ≫ (D.f j k).c.app _ ≫ f' =
      ((π₁ j, i, k) ≫ D.t j i ≫ D.f i j).c.app (op U) ≫
        (π₂⁻¹ j, i, k) (unop _) ≫
          (D.V (j, k)).presheaf.map
            (eqToHom (opensImagePreimageMap_app' D i j k U).choose) ≫ f' 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] (D : AlgebraicGeometry.PresheafedSpace.GlueData C) [CategoryTheory.Limits.HasLimits C] (i j k : D.J) (U : TopologicalSpace.Opens ↑↑(D.U i)) {X' : C} (f' :
  ((TopCat.Presheaf.pushforward C (D.f j k).base).obj (D.V (j, k)).presheaf).obj
      (Opposite.op ((TopologicalSpace.Opens.map (D.ι j).base).obj (⋯.functor.obj U))) ⟶
    X')
Goal: D.opensImagePreimageMap i j U ≫
    (D.f j k).c.app (Opposite.op ((TopologicalSpace.Opens.map (D.ι j).base).obj (⋯.functor.obj U))) ≫ f' =
  (CategoryTheory.Limits.pullback.fst (D.f j i) (D.f j k) ≫ D.t j i ≫ D.f i j).c.app (Opposite.op U) ≫
    AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.invApp (CategoryTheory.Limits.pullback.snd (D.f j i) (D.f j k))
        (Opposite.unop
          (Opposite.op
            ((TopologicalSpace.Opens.map
                    (CategoryTheory.Limits.pullback.fst (D.f j i) (D.f j k) ≫ D.t j i ≫ D.f i j).base).toPrefunctor.1
              (Opposite.unop (Opposite.op U))))) ≫
      (D.V (j, k)).presheaf.map (CategoryTheory.eqToHom ⋯) ≫ f'
-- AlgebraicGeometry.PresheafedSpace.GlueData.ιInvApp_π
theorem ιInvApp_π {i : D.J} (U : Opens (D.U i).carrier) :
    ∃ eq, D.ιInvApp U ≫ D.diagramOverOpenπ U i = (D.U i).presheaf.map (eqToHom eq) 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] (D : AlgebraicGeometry.PresheafedSpace.GlueData C) [CategoryTheory.Limits.HasLimits C] {i : D.J} (U : TopologicalSpace.Opens ↑↑(D.U i))
Goal: ∃ (eq :
  Opposite.op U =
    Opposite.op
      ((TopologicalSpace.Opens.map
            (CategoryTheory.Limits.colimit.ι D.diagram.multispan
                (Opposite.unop (Opposite.op (CategoryTheory.Limits.WalkingMultispan.right i)))).base).obj
        (⋯.functor.obj U))),
  D.ιInvApp U ≫ D.diagramOverOpenπ U i = (D.U i).presheaf.map (CategoryTheory.eqToHom eq)
-- AlgebraicGeometry.PresheafedSpace.GlueData.π_ιInvApp_π
theorem π_ιInvApp_π (i j : D.J) (U : Opens (D.U i).carrier) :
    D.diagramOverOpenπ U i ≫ D.ιInvAppπEqMap U ≫ D.ιInvApp U ≫ D.diagramOverOpenπ U j =
      D.diagramOverOpenπ U j 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] (D : AlgebraicGeometry.PresheafedSpace.GlueData C) [CategoryTheory.Limits.HasLimits C] (i j : D.J) (U : TopologicalSpace.Opens ↑↑(D.U i))
Goal: D.diagramOverOpenπ U i ≫ D.ιInvAppπEqMap U ≫ D.ιInvApp U ≫ D.diagramOverOpenπ U j = D.diagramOverOpenπ U j
-- AlgebraicGeometry.PresheafedSpace.GlueData.π_ιInvApp_eq_id
theorem π_ιInvApp_eq_id (i : D.J) (U : Opens (D.U i).carrier) :
    D.diagramOverOpenπ U i ≫ D.ιInvAppπEqMap U ≫ D.ιInvApp U = 𝟙 _ 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] (D : AlgebraicGeometry.PresheafedSpace.GlueData C) [CategoryTheory.Limits.HasLimits C] (i : D.J) (U : TopologicalSpace.Opens ↑↑(D.U i))
Goal: D.diagramOverOpenπ U i ≫ D.ιInvAppπEqMap U ≫ D.ιInvApp U = 𝟙 (CategoryTheory.Limits.limit (D.diagramOverOpen U))
-- AlgebraicGeometry.PresheafedSpace.GlueData.ι_jointly_surjective
theorem ι_jointly_surjective (x : 𝖣.glued) : ∃ (i : D.J) (y : D.U i), (𝖣.ι i).base y = x 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] (D : AlgebraicGeometry.PresheafedSpace.GlueData C) [CategoryTheory.Limits.HasLimits C] (x : ↑↑D.glued)
Goal: ∃ i y, (D.ι i).base y = x
-- AlgebraicGeometry.SheafedSpace.GlueData.f_open
-- Porting note(#5171): this linter isn't ported yet.
-- @[nolint has_nonempty_instance]
structure GlueData extends CategoryTheory.GlueData (SheafedSpace.{u, v, v} C) where
  f_open : ∀ i j, SheafedSpace.IsOpenImmersion (f i j)
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] (self : AlgebraicGeometry.SheafedSpace.GlueData C) (i j : self.J)
Goal: AlgebraicGeometry.SheafedSpace.IsOpenImmersion (self.f i j)
-- AlgebraicGeometry.SheafedSpace.GlueData.ι_isoPresheafedSpace_inv
theorem ι_isoPresheafedSpace_inv (i : D.J) :
    D.toPresheafedSpaceGlueData.toGlueData.ι i ≫ D.isoPresheafedSpace.inv = 𝖣.ι i 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] (D : AlgebraicGeometry.SheafedSpace.GlueData C) [CategoryTheory.Limits.HasLimits C] (i : D.J)
Goal: D.toPresheafedSpaceGlueData.ι i ≫ D.isoPresheafedSpace.inv = D.ι i
-- AlgebraicGeometry.SheafedSpace.GlueData.ι_jointly_surjective
theorem ι_jointly_surjective (x : 𝖣.glued) : ∃ (i : D.J) (y : D.U i), (𝖣.ι i).base y = x 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] (D : AlgebraicGeometry.SheafedSpace.GlueData C) [CategoryTheory.Limits.HasLimits C] (x : ↑↑D.glued.toPresheafedSpace)
Goal: ∃ i y, (D.ι i).base y = x
-- AlgebraicGeometry.LocallyRingedSpace.GlueData.f_open
-- Porting note(#5171): this linter isn't ported yet.
-- @[nolint has_nonempty_instance]
structure GlueData extends CategoryTheory.GlueData LocallyRingedSpace where
  f_open : ∀ i j, LocallyRingedSpace.IsOpenImmersion (f i j)
Arguments: (self : AlgebraicGeometry.LocallyRingedSpace.GlueData) (i j : self.J)
Goal: AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion (self.f i j)
-- AlgebraicGeometry.LocallyRingedSpace.GlueData.ι_isoSheafedSpace_inv
theorem ι_isoSheafedSpace_inv (i : D.J) :
    D.toSheafedSpaceGlueData.toGlueData.ι i ≫ D.isoSheafedSpace.inv = (𝖣.ι i).1 
Arguments: (D : AlgebraicGeometry.LocallyRingedSpace.GlueData) (i : D.J)
Goal: D.toSheafedSpaceGlueData.ι i ≫ D.isoSheafedSpace.inv = (D.ι i).val
-- AlgebraicGeometry.LocallyRingedSpace.GlueData.ι_jointly_surjective
theorem ι_jointly_surjective (x : 𝖣.glued) : ∃ (i : D.J) (y : D.U i), (𝖣.ι i).1.base y = x 
Arguments: (D : AlgebraicGeometry.LocallyRingedSpace.GlueData) (x : ↑D.glued.toTopCat)
Goal: ∃ i y, (D.ι i).val.base y = x
-- Polynomial.Monic.irreducible_of_irreducible_map_of_isPrime_nilradical
theorem Polynomial.Monic.irreducible_of_irreducible_map_of_isPrime_nilradical
    {R S : Type*} [CommRing R] [(nilradical R).IsPrime] [CommRing S] [IsDomain S]
    (φ : R →+* S) (f : R[X]) (hm : f.Monic) (hi : Irreducible (f.map φ)) : Irreducible f 
Arguments: {R : Type u_1} {S : Type u_2} [CommRing R] [(nilradical R).IsPrime] [CommRing S] [IsDomain S] (φ : R →+* S) (f : R[X]) (hm : f.Monic) (hi : Irreducible (Polynomial.map φ f))
Goal: Irreducible f
-- IsCoprime.symm
@[symm]
theorem IsCoprime.symm (H : IsCoprime x y) : IsCoprime y x 
Arguments: {R : Type u} [CommSemiring R] {x y : R} (H : IsCoprime x y)
Goal: IsCoprime y x
-- isCoprime_comm
theorem isCoprime_comm : IsCoprime x y ↔ IsCoprime y x 
Arguments: {R : Type u} [CommSemiring R] {x y : R}
Goal: IsCoprime x y ↔ IsCoprime y x
-- isCoprime_self
theorem isCoprime_self : IsCoprime x x ↔ IsUnit x 
Arguments: {R : Type u} [CommSemiring R] {x : R}
Goal: IsCoprime x x ↔ IsUnit x
-- isCoprime_zero_left
theorem isCoprime_zero_left : IsCoprime 0 x ↔ IsUnit x 
Arguments: {R : Type u} [CommSemiring R] {x : R}
Goal: IsCoprime 0 x ↔ IsUnit x
-- isCoprime_zero_right
theorem isCoprime_zero_right : IsCoprime x 0 ↔ IsUnit x 
Arguments: {R : Type u} [CommSemiring R] {x : R}
Goal: IsCoprime x 0 ↔ IsUnit x
-- not_isCoprime_zero_zero
theorem not_isCoprime_zero_zero [Nontrivial R] : ¬IsCoprime (0 : R) 0 
Arguments: {R : Type u} [CommSemiring R] [Nontrivial R]
Goal: ¬IsCoprime 0 0
-- IsCoprime.intCast
lemma IsCoprime.intCast {R : Type*} [CommRing R] {a b : ℤ} (h : IsCoprime a b) :
    IsCoprime (a : R) (b : R) 
Arguments: {R : Type u_1} [CommRing R] {a b : ℤ} (h : IsCoprime a b)
Goal: IsCoprime ↑a ↑b
-- IsCoprime.ne_zero
theorem IsCoprime.ne_zero [Nontrivial R] {p : Fin 2 → R} (h : IsCoprime (p 0) (p 1)) : p ≠ 0 
Arguments: {R : Type u} [CommSemiring R] [Nontrivial R] {p : Fin 2 → R} (h : IsCoprime (p 0) (p 1))
Goal: p ≠ 0
-- IsCoprime.ne_zero_or_ne_zero
theorem IsCoprime.ne_zero_or_ne_zero [Nontrivial R] (h : IsCoprime x y) : x ≠ 0 ∨ y ≠ 0 
Arguments: {R : Type u} [CommSemiring R] {x y : R} [Nontrivial R] (h : IsCoprime x y)
Goal: x ≠ 0 ∨ y ≠ 0
-- isCoprime_one_left
theorem isCoprime_one_left : IsCoprime 1 x 
Arguments: {R : Type u} [CommSemiring R] {x : R}
Goal: IsCoprime 1 x
-- isCoprime_one_right
theorem isCoprime_one_right : IsCoprime x 1 
Arguments: {R : Type u} [CommSemiring R] {x : R}
Goal: IsCoprime x 1
-- IsCoprime.dvd_of_dvd_mul_right
theorem IsCoprime.dvd_of_dvd_mul_right (H1 : IsCoprime x z) (H2 : x ∣ y * z) : x ∣ y 
Arguments: {R : Type u} [CommSemiring R] {x y z : R} (H1 : IsCoprime x z) (H2 : x ∣ y * z)
Goal: x ∣ y
-- IsCoprime.dvd_of_dvd_mul_left
theorem IsCoprime.dvd_of_dvd_mul_left (H1 : IsCoprime x y) (H2 : x ∣ y * z) : x ∣ z 
Arguments: {R : Type u} [CommSemiring R] {x y z : R} (H1 : IsCoprime x y) (H2 : x ∣ y * z)
Goal: x ∣ z
-- IsCoprime.mul_left
theorem IsCoprime.mul_left (H1 : IsCoprime x z) (H2 : IsCoprime y z) : IsCoprime (x * y) z 
Arguments: {R : Type u} [CommSemiring R] {x y z : R} (H1 : IsCoprime x z) (H2 : IsCoprime y z)
Goal: IsCoprime (x * y) z
-- IsCoprime.mul_right
theorem IsCoprime.mul_right (H1 : IsCoprime x y) (H2 : IsCoprime x z) : IsCoprime x (y * z) 
Arguments: {R : Type u} [CommSemiring R] {x y z : R} (H1 : IsCoprime x y) (H2 : IsCoprime x z)
Goal: IsCoprime x (y * z)
-- IsCoprime.mul_dvd
theorem IsCoprime.mul_dvd (H : IsCoprime x y) (H1 : x ∣ z) (H2 : y ∣ z) : x * y ∣ z 
Arguments: {R : Type u} [CommSemiring R] {x y z : R} (H : IsCoprime x y) (H1 : x ∣ z) (H2 : y ∣ z)
Goal: x * y ∣ z
-- IsCoprime.of_mul_left_left
theorem IsCoprime.of_mul_left_left (H : IsCoprime (x * y) z) : IsCoprime x z 
Arguments: {R : Type u} [CommSemiring R] {x y z : R} (H : IsCoprime (x * y) z)
Goal: IsCoprime x z
-- IsCoprime.of_mul_left_right
theorem IsCoprime.of_mul_left_right (H : IsCoprime (x * y) z) : IsCoprime y z 
Arguments: {R : Type u} [CommSemiring R] {x y z : R} (H : IsCoprime (x * y) z)
Goal: IsCoprime y z
-- IsCoprime.of_mul_right_left
theorem IsCoprime.of_mul_right_left (H : IsCoprime x (y * z)) : IsCoprime x y 
Arguments: {R : Type u} [CommSemiring R] {x y z : R} (H : IsCoprime x (y * z))
Goal: IsCoprime x y
-- IsCoprime.of_mul_right_right
theorem IsCoprime.of_mul_right_right (H : IsCoprime x (y * z)) : IsCoprime x z 
Arguments: {R : Type u} [CommSemiring R] {x y z : R} (H : IsCoprime x (y * z))
Goal: IsCoprime x z
-- IsCoprime.mul_left_iff
theorem IsCoprime.mul_left_iff : IsCoprime (x * y) z ↔ IsCoprime x z ∧ IsCoprime y z 
Arguments: {R : Type u} [CommSemiring R] {x y z : R}
Goal: IsCoprime (x * y) z ↔ IsCoprime x z ∧ IsCoprime y z
-- IsCoprime.mul_right_iff
theorem IsCoprime.mul_right_iff : IsCoprime x (y * z) ↔ IsCoprime x y ∧ IsCoprime x z 
Arguments: {R : Type u} [CommSemiring R] {x y z : R}
Goal: IsCoprime x (y * z) ↔ IsCoprime x y ∧ IsCoprime x z
-- IsCoprime.of_isCoprime_of_dvd_left
theorem IsCoprime.of_isCoprime_of_dvd_left (h : IsCoprime y z) (hdvd : x ∣ y) : IsCoprime x z 
Arguments: {R : Type u} [CommSemiring R] {x y z : R} (h : IsCoprime y z) (hdvd : x ∣ y)
Goal: IsCoprime x z
-- IsCoprime.of_isCoprime_of_dvd_right
theorem IsCoprime.of_isCoprime_of_dvd_right (h : IsCoprime z y) (hdvd : x ∣ y) : IsCoprime z x 
Arguments: {R : Type u} [CommSemiring R] {x y z : R} (h : IsCoprime z y) (hdvd : x ∣ y)
Goal: IsCoprime z x
-- IsCoprime.isUnit_of_dvd
theorem IsCoprime.isUnit_of_dvd (H : IsCoprime x y) (d : x ∣ y) : IsUnit x 
Arguments: {R : Type u} [CommSemiring R] {x y : R} (H : IsCoprime x y) (d : x ∣ y)
Goal: IsUnit x
-- IsCoprime.isUnit_of_dvd'
theorem IsCoprime.isUnit_of_dvd' {a b x : R} (h : IsCoprime a b) (ha : x ∣ a) (hb : x ∣ b) :
    IsUnit x 
Arguments: {R : Type u} [CommSemiring R] {a b x : R} (h : IsCoprime a b) (ha : x ∣ a) (hb : x ∣ b)
Goal: IsUnit x
-- IsCoprime.isRelPrime
theorem IsCoprime.isRelPrime {a b : R} (h : IsCoprime a b) : IsRelPrime a b 
Arguments: {R : Type u} [CommSemiring R] {a b : R} (h : IsCoprime a b)
Goal: IsRelPrime a b
-- IsCoprime.map
theorem IsCoprime.map (H : IsCoprime x y) {S : Type v} [CommSemiring S] (f : R →+* S) :
    IsCoprime (f x) (f y) 
Arguments: {R : Type u} [CommSemiring R] {x y : R} (H : IsCoprime x y) {S : Type v} [CommSemiring S] (f : R →+* S)
Goal: IsCoprime (f x) (f y)
-- IsCoprime.of_add_mul_left_left
theorem IsCoprime.of_add_mul_left_left (h : IsCoprime (x + y * z) y) : IsCoprime x y 
Arguments: {R : Type u} [CommSemiring R] {x y z : R} (h : IsCoprime (x + y * z) y)
Goal: IsCoprime x y
-- IsCoprime.of_add_mul_right_left
theorem IsCoprime.of_add_mul_right_left (h : IsCoprime (x + z * y) y) : IsCoprime x y 
Arguments: {R : Type u} [CommSemiring R] {x y z : R} (h : IsCoprime (x + z * y) y)
Goal: IsCoprime x y
-- IsCoprime.of_add_mul_left_right
theorem IsCoprime.of_add_mul_left_right (h : IsCoprime x (y + x * z)) : IsCoprime x y 
Arguments: {R : Type u} [CommSemiring R] {x y z : R} (h : IsCoprime x (y + x * z))
Goal: IsCoprime x y
-- IsCoprime.of_add_mul_right_right
theorem IsCoprime.of_add_mul_right_right (h : IsCoprime x (y + z * x)) : IsCoprime x y 
Arguments: {R : Type u} [CommSemiring R] {x y z : R} (h : IsCoprime x (y + z * x))
Goal: IsCoprime x y
-- IsCoprime.of_mul_add_left_left
theorem IsCoprime.of_mul_add_left_left (h : IsCoprime (y * z + x) y) : IsCoprime x y 
Arguments: {R : Type u} [CommSemiring R] {x y z : R} (h : IsCoprime (y * z + x) y)
Goal: IsCoprime x y
-- IsCoprime.of_mul_add_right_left
theorem IsCoprime.of_mul_add_right_left (h : IsCoprime (z * y + x) y) : IsCoprime x y 
Arguments: {R : Type u} [CommSemiring R] {x y z : R} (h : IsCoprime (z * y + x) y)
Goal: IsCoprime x y
-- IsCoprime.of_mul_add_left_right
theorem IsCoprime.of_mul_add_left_right (h : IsCoprime x (x * z + y)) : IsCoprime x y 
Arguments: {R : Type u} [CommSemiring R] {x y z : R} (h : IsCoprime x (x * z + y))
Goal: IsCoprime x y
-- IsCoprime.of_mul_add_right_right
theorem IsCoprime.of_mul_add_right_right (h : IsCoprime x (z * x + y)) : IsCoprime x y 
Arguments: {R : Type u} [CommSemiring R] {x y z : R} (h : IsCoprime x (z * x + y))
Goal: IsCoprime x y
-- IsRelPrime.of_add_mul_left_left
theorem IsRelPrime.of_add_mul_left_left (h : IsRelPrime (x + y * z) y) : IsRelPrime x y 
Arguments: {R : Type u} [CommSemiring R] {x y z : R} (h : IsRelPrime (x + y * z) y)
Goal: IsRelPrime x y
-- IsRelPrime.of_add_mul_right_left
theorem IsRelPrime.of_add_mul_right_left (h : IsRelPrime (x + z * y) y) : IsRelPrime x y 
Arguments: {R : Type u} [CommSemiring R] {x y z : R} (h : IsRelPrime (x + z * y) y)
Goal: IsRelPrime x y
-- IsRelPrime.of_add_mul_left_right
theorem IsRelPrime.of_add_mul_left_right (h : IsRelPrime x (y + x * z)) : IsRelPrime x y 
Arguments: {R : Type u} [CommSemiring R] {x y z : R} (h : IsRelPrime x (y + x * z))
Goal: IsRelPrime x y
-- IsRelPrime.of_add_mul_right_right
theorem IsRelPrime.of_add_mul_right_right (h : IsRelPrime x (y + z * x)) : IsRelPrime x y 
Arguments: {R : Type u} [CommSemiring R] {x y z : R} (h : IsRelPrime x (y + z * x))
Goal: IsRelPrime x y
-- IsRelPrime.of_mul_add_left_left
theorem IsRelPrime.of_mul_add_left_left (h : IsRelPrime (y * z + x) y) : IsRelPrime x y 
Arguments: {R : Type u} [CommSemiring R] {x y z : R} (h : IsRelPrime (y * z + x) y)
Goal: IsRelPrime x y
-- IsRelPrime.of_mul_add_right_left
theorem IsRelPrime.of_mul_add_right_left (h : IsRelPrime (z * y + x) y) : IsRelPrime x y 
Arguments: {R : Type u} [CommSemiring R] {x y z : R} (h : IsRelPrime (z * y + x) y)
Goal: IsRelPrime x y
-- IsRelPrime.of_mul_add_left_right
theorem IsRelPrime.of_mul_add_left_right (h : IsRelPrime x (x * z + y)) : IsRelPrime x y 
Arguments: {R : Type u} [CommSemiring R] {x y z : R} (h : IsRelPrime x (x * z + y))
Goal: IsRelPrime x y
-- IsRelPrime.of_mul_add_right_right
theorem IsRelPrime.of_mul_add_right_right (h : IsRelPrime x (z * x + y)) : IsRelPrime x y 
Arguments: {R : Type u} [CommSemiring R] {x y z : R} (h : IsRelPrime x (z * x + y))
Goal: IsRelPrime x y
-- isCoprime_group_smul_left
theorem isCoprime_group_smul_left : IsCoprime (x • y) z ↔ IsCoprime y z 
Arguments: {R : Type u_1} {G : Type u_2} [CommSemiring R] [Group G] [MulAction G R] [SMulCommClass G R R] [IsScalarTower G R R] (x : G) (y z : R)
Goal: IsCoprime (x • y) z ↔ IsCoprime y z
-- isCoprime_group_smul_right
theorem isCoprime_group_smul_right : IsCoprime y (x • z) ↔ IsCoprime y z 
Arguments: {R : Type u_1} {G : Type u_2} [CommSemiring R] [Group G] [MulAction G R] [SMulCommClass G R R] [IsScalarTower G R R] (x : G) (y z : R)
Goal: IsCoprime y (x • z) ↔ IsCoprime y z
-- isCoprime_group_smul
theorem isCoprime_group_smul : IsCoprime (x • y) (x • z) ↔ IsCoprime y z 
Arguments: {R : Type u_1} {G : Type u_2} [CommSemiring R] [Group G] [MulAction G R] [SMulCommClass G R R] [IsScalarTower G R R] (x : G) (y z : R)
Goal: IsCoprime (x • y) (x • z) ↔ IsCoprime y z
-- isCoprime_mul_unit_left_left
theorem isCoprime_mul_unit_left_left : IsCoprime (x * y) z ↔ IsCoprime y z 
Arguments: {R : Type u_1} [CommSemiring R] {x : R} (hu : IsUnit x) (y z : R)
Goal: IsCoprime (x * y) z ↔ IsCoprime y z
-- isCoprime_mul_unit_left_right
theorem isCoprime_mul_unit_left_right : IsCoprime y (x * z) ↔ IsCoprime y z 
Arguments: {R : Type u_1} [CommSemiring R] {x : R} (hu : IsUnit x) (y z : R)
Goal: IsCoprime y (x * z) ↔ IsCoprime y z
-- isCoprime_mul_unit_left
theorem isCoprime_mul_unit_left : IsCoprime (x * y) (x * z) ↔ IsCoprime y z 
Arguments: {R : Type u_1} [CommSemiring R] {x : R} (hu : IsUnit x) (y z : R)
Goal: IsCoprime (x * y) (x * z) ↔ IsCoprime y z
-- isCoprime_mul_unit_right_left
theorem isCoprime_mul_unit_right_left : IsCoprime (y * x) z ↔ IsCoprime y z 
Arguments: {R : Type u_1} [CommSemiring R] {x : R} (hu : IsUnit x) (y z : R)
Goal: IsCoprime (y * x) z ↔ IsCoprime y z
-- isCoprime_mul_unit_right_right
theorem isCoprime_mul_unit_right_right : IsCoprime y (z * x) ↔ IsCoprime y z 
Arguments: {R : Type u_1} [CommSemiring R] {x : R} (hu : IsUnit x) (y z : R)
Goal: IsCoprime y (z * x) ↔ IsCoprime y z
-- isCoprime_mul_unit_right
theorem isCoprime_mul_unit_right : IsCoprime (y * x) (z * x) ↔ IsCoprime y z 
Arguments: {R : Type u_1} [CommSemiring R] {x : R} (hu : IsUnit x) (y z : R)
Goal: IsCoprime (y * x) (z * x) ↔ IsCoprime y z
-- IsCoprime.add_mul_left_left
theorem add_mul_left_left {x y : R} (h : IsCoprime x y) (z : R) : IsCoprime (x + y * z) y 
Arguments: {R : Type u} [CommRing R] {x y : R} (h : IsCoprime x y) (z : R)
Goal: IsCoprime (x + y * z) y
-- IsCoprime.add_mul_right_left
theorem add_mul_right_left {x y : R} (h : IsCoprime x y) (z : R) : IsCoprime (x + z * y) y 
Arguments: {R : Type u} [CommRing R] {x y : R} (h : IsCoprime x y) (z : R)
Goal: IsCoprime (x + z * y) y
-- IsCoprime.add_mul_left_right
theorem add_mul_left_right {x y : R} (h : IsCoprime x y) (z : R) : IsCoprime x (y + x * z) 
Arguments: {R : Type u} [CommRing R] {x y : R} (h : IsCoprime x y) (z : R)
Goal: IsCoprime x (y + x * z)
-- IsCoprime.add_mul_right_right
theorem add_mul_right_right {x y : R} (h : IsCoprime x y) (z : R) : IsCoprime x (y + z * x) 
Arguments: {R : Type u} [CommRing R] {x y : R} (h : IsCoprime x y) (z : R)
Goal: IsCoprime x (y + z * x)
-- IsCoprime.mul_add_left_left
theorem mul_add_left_left {x y : R} (h : IsCoprime x y) (z : R) : IsCoprime (y * z + x) y 
Arguments: {R : Type u} [CommRing R] {x y : R} (h : IsCoprime x y) (z : R)
Goal: IsCoprime (y * z + x) y
-- IsCoprime.mul_add_right_left
theorem mul_add_right_left {x y : R} (h : IsCoprime x y) (z : R) : IsCoprime (z * y + x) y 
Arguments: {R : Type u} [CommRing R] {x y : R} (h : IsCoprime x y) (z : R)
Goal: IsCoprime (z * y + x) y
-- IsCoprime.mul_add_left_right
theorem mul_add_left_right {x y : R} (h : IsCoprime x y) (z : R) : IsCoprime x (x * z + y) 
Arguments: {R : Type u} [CommRing R] {x y : R} (h : IsCoprime x y) (z : R)
Goal: IsCoprime x (x * z + y)
-- IsCoprime.mul_add_right_right
theorem mul_add_right_right {x y : R} (h : IsCoprime x y) (z : R) : IsCoprime x (z * x + y) 
Arguments: {R : Type u} [CommRing R] {x y : R} (h : IsCoprime x y) (z : R)
Goal: IsCoprime x (z * x + y)
-- IsCoprime.add_mul_left_left_iff
theorem add_mul_left_left_iff {x y z : R} : IsCoprime (x + y * z) y ↔ IsCoprime x y 
Arguments: {R : Type u} [CommRing R] {x y z : R}
Goal: IsCoprime (x + y * z) y ↔ IsCoprime x y
-- IsCoprime.add_mul_right_left_iff
theorem add_mul_right_left_iff {x y z : R} : IsCoprime (x + z * y) y ↔ IsCoprime x y 
Arguments: {R : Type u} [CommRing R] {x y z : R}
Goal: IsCoprime (x + z * y) y ↔ IsCoprime x y