ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Algebra.TensorProduct.congr_symm
theorem congr_symm (f : A ≃ₐ[S] B) (g : C ≃ₐ[R] D) : congr f.symm g.symm = (congr f g).symm 
Arguments: {R : Type uR} {S : Type uS} {A : Type uA} {B : Type uB} {C : Type uC} {D : Type uD} [CommSemiring R] [CommSemiring S] [Algebra R S] [Semiring A] [Algebra R A] [Algebra S A] [IsScalarTower R S A] [Semiring B] [Algebra R B] [Algebra S B] [IsScalarTower R S B] [Semiring C] [Algebra R C] [Semiring D] [Algebra R D] (f : A ≃ₐ[S] B) (g : C ≃ₐ[R] D)
Goal: Algebra.TensorProduct.congr f.symm g.symm = (Algebra.TensorProduct.congr f g).symm
-- Algebra.TensorProduct.lmul'_toLinearMap
theorem lmul'_toLinearMap : (lmul' R : _ →ₐ[R] S).toLinearMap = LinearMap.mul' R S 
Arguments: {R : Type uR} {S : Type uS} [CommSemiring R] [CommSemiring S] [Algebra R S]
Goal: (Algebra.TensorProduct.lmul' R).toLinearMap = LinearMap.mul' R S
-- Algebra.TensorProduct.lmul'_apply_tmul
@[simp]
theorem lmul'_apply_tmul (a b : S) : lmul' (S := S) R (a ⊗ₜ[R] b) = a * b 
Arguments: {R : Type uR} {S : Type uS} [CommSemiring R] [CommSemiring S] [Algebra R S] (a b : S)
Goal: (Algebra.TensorProduct.lmul' R) (a ⊗ₜ[R] b) = a * b
-- Algebra.TensorProduct.lmul'_comp_includeLeft
@[simp]
theorem lmul'_comp_includeLeft : (lmul' R : _ →ₐ[R] S).comp includeLeft = AlgHom.id R S 
Arguments: {R : Type uR} {S : Type uS} [CommSemiring R] [CommSemiring S] [Algebra R S]
Goal: (Algebra.TensorProduct.lmul' R).comp Algebra.TensorProduct.includeLeft = AlgHom.id R S
-- Algebra.TensorProduct.lmul'_comp_includeRight
@[simp]
theorem lmul'_comp_includeRight : (lmul' R : _ →ₐ[R] S).comp includeRight = AlgHom.id R S 
Arguments: {R : Type uR} {S : Type uS} [CommSemiring R] [CommSemiring S] [Algebra R S]
Goal: (Algebra.TensorProduct.lmul' R).comp Algebra.TensorProduct.includeRight = AlgHom.id R S
-- Algebra.TensorProduct.productMap_eq_comp_map
theorem productMap_eq_comp_map : productMap f g = (lmul' R).comp (TensorProduct.map f g) 
Arguments: {R : Type uR} {S : Type uS} {A : Type uA} {B : Type uB} [CommSemiring R] [Semiring A] [Semiring B] [CommSemiring S] [Algebra R A] [Algebra R B] [Algebra R S] (f : A →ₐ[R] S) (g : B →ₐ[R] S)
Goal: Algebra.TensorProduct.productMap f g = (Algebra.TensorProduct.lmul' R).comp (Algebra.TensorProduct.map f g)
-- Algebra.TensorProduct.productMap_apply_tmul
@[simp]
theorem productMap_apply_tmul (a : A) (b : B) : productMap f g (a ⊗ₜ b) = f a * g b 
Arguments: {R : Type uR} {S : Type uS} {A : Type uA} {B : Type uB} [CommSemiring R] [Semiring A] [Semiring B] [CommSemiring S] [Algebra R A] [Algebra R B] [Algebra R S] (f : A →ₐ[R] S) (g : B →ₐ[R] S) (a : A) (b : B)
Goal: (Algebra.TensorProduct.productMap f g) (a ⊗ₜ[R] b) = f a * g b
-- Algebra.TensorProduct.productMap_left_apply
theorem productMap_left_apply (a : A) : productMap f g (a ⊗ₜ 1) = f a 
Arguments: {R : Type uR} {S : Type uS} {A : Type uA} {B : Type uB} [CommSemiring R] [Semiring A] [Semiring B] [CommSemiring S] [Algebra R A] [Algebra R B] [Algebra R S] (f : A →ₐ[R] S) (g : B →ₐ[R] S) (a : A)
Goal: (Algebra.TensorProduct.productMap f g) (a ⊗ₜ[R] 1) = f a
-- Algebra.TensorProduct.productMap_left
@[simp]
theorem productMap_left : (productMap f g).comp includeLeft = f 
Arguments: {R : Type uR} {S : Type uS} {A : Type uA} {B : Type uB} [CommSemiring R] [Semiring A] [Semiring B] [CommSemiring S] [Algebra R A] [Algebra R B] [Algebra R S] (f : A →ₐ[R] S) (g : B →ₐ[R] S)
Goal: (Algebra.TensorProduct.productMap f g).comp Algebra.TensorProduct.includeLeft = f
-- Algebra.TensorProduct.productMap_right_apply
theorem productMap_right_apply (b : B) :
    productMap f g (1 ⊗ₜ b) = g b 
Arguments: {R : Type uR} {S : Type uS} {A : Type uA} {B : Type uB} [CommSemiring R] [Semiring A] [Semiring B] [CommSemiring S] [Algebra R A] [Algebra R B] [Algebra R S] (f : A →ₐ[R] S) (g : B →ₐ[R] S) (b : B)
Goal: (Algebra.TensorProduct.productMap f g) (1 ⊗ₜ[R] b) = g b
-- Algebra.TensorProduct.productMap_right
@[simp]
theorem productMap_right : (productMap f g).comp includeRight = g 
Arguments: {R : Type uR} {S : Type uS} {A : Type uA} {B : Type uB} [CommSemiring R] [Semiring A] [Semiring B] [CommSemiring S] [Algebra R A] [Algebra R B] [Algebra R S] (f : A →ₐ[R] S) (g : B →ₐ[R] S)
Goal: (Algebra.TensorProduct.productMap f g).comp Algebra.TensorProduct.includeRight = g
-- Algebra.TensorProduct.productMap_range
theorem productMap_range : (productMap f g).range = f.range ⊔ g.range 
Arguments: {R : Type uR} {S : Type uS} {A : Type uA} {B : Type uB} [CommSemiring R] [Semiring A] [Semiring B] [CommSemiring S] [Algebra R A] [Algebra R B] [Algebra R S] (f : A →ₐ[R] S) (g : B →ₐ[R] S)
Goal: (Algebra.TensorProduct.productMap f g).range = f.range ⊔ g.range
-- Algebra.TensorProduct.basisAux_tmul
theorem basisAux_tmul (a : A) (m : M) :
    basisAux A b (a ⊗ₜ m) = a • Finsupp.mapRange (algebraMap R A) (map_zero _) (b.repr m) 
Arguments: {R : Type uR} {A : Type uA} {M : Type uM} {ι : Type uι} [CommSemiring R] [Semiring A] [Algebra R A] [AddCommMonoid M] [Module R M] (b : Basis ι R M) (a : A) (m : M)
Goal: (Algebra.TensorProduct.basisAux A b) (a ⊗ₜ[R] m) = a • Finsupp.mapRange ⇑(algebraMap R A) ⋯ (b.repr m)
-- Algebra.TensorProduct.basisAux_map_smul
theorem basisAux_map_smul (a : A) (x : A ⊗[R] M) : basisAux A b (a • x) = a • basisAux A b x 
Arguments: {R : Type uR} {A : Type uA} {M : Type uM} {ι : Type uι} [CommSemiring R] [Semiring A] [Algebra R A] [AddCommMonoid M] [Module R M] (b : Basis ι R M) (a : A) (x : A ⊗[R] M)
Goal: (Algebra.TensorProduct.basisAux A b) (a • x) = a • (Algebra.TensorProduct.basisAux A b) x
-- Algebra.TensorProduct.basis_repr_tmul
@[simp]
theorem basis_repr_tmul (a : A) (m : M) :
    (basis A b).repr (a ⊗ₜ m) = a • Finsupp.mapRange (algebraMap R A) (map_zero _) (b.repr m) 
Arguments: {R : Type uR} {A : Type uA} {M : Type uM} {ι : Type uι} [CommSemiring R] [Semiring A] [Algebra R A] [AddCommMonoid M] [Module R M] (b : Basis ι R M) (a : A) (m : M)
Goal: (Algebra.TensorProduct.basis A b).repr (a ⊗ₜ[R] m) = a • Finsupp.mapRange ⇑(algebraMap R A) ⋯ (b.repr m)
-- Algebra.TensorProduct.basis_repr_symm_apply
theorem basis_repr_symm_apply (a : A) (i : ι) :
    (basis A b).repr.symm (Finsupp.single i a) = a ⊗ₜ b.repr.symm (Finsupp.single i 1) 
Arguments: {R : Type uR} {A : Type uA} {M : Type uM} {ι : Type uι} [CommSemiring R] [Semiring A] [Algebra R A] [AddCommMonoid M] [Module R M] (b : Basis ι R M) (a : A) (i : ι)
Goal: (Algebra.TensorProduct.basis A b).repr.symm (Finsupp.single i a) = a ⊗ₜ[R] b.repr.symm (Finsupp.single i 1)
-- Algebra.TensorProduct.basis_apply
@[simp]
theorem basis_apply (i : ι) : basis A b i = 1 ⊗ₜ b i 
Arguments: {R : Type uR} {A : Type uA} {M : Type uM} {ι : Type uι} [CommSemiring R] [Semiring A] [Algebra R A] [AddCommMonoid M] [Module R M] (b : Basis ι R M) (i : ι)
Goal: (Algebra.TensorProduct.basis A b) i = 1 ⊗ₜ[R] b i
-- Algebra.TensorProduct.basis_repr_symm_apply'
theorem basis_repr_symm_apply' (a : A) (i : ι) : a • basis A b i = a ⊗ₜ b i 
Arguments: {R : Type uR} {A : Type uA} {M : Type uM} {ι : Type uι} [CommSemiring R] [Semiring A] [Algebra R A] [AddCommMonoid M] [Module R M] (b : Basis ι R M) (a : A) (i : ι)
Goal: a • (Algebra.TensorProduct.basis A b) i = a ⊗ₜ[R] b i
-- Basis.baseChange_linearMap
lemma _root_.Basis.baseChange_linearMap (b : Basis ι R M) (b' : Basis ι' R N) (ij : ι × ι') :
    baseChange A (b'.linearMap b ij) = (basis A b').linearMap (basis A b) ij 
Arguments: {R : Type uR} {M : Type uM} {ι : Type uι} [CommSemiring R] [AddCommMonoid M] [Module R M] [Fintype ι] {ι' : Type u_1} {N : Type u_2} [Fintype ι'] [DecidableEq ι'] [AddCommMonoid N] [Module R N] (A : Type u_3) [CommSemiring A] [Algebra R A] (b : Basis ι R M) (b' : Basis ι' R N) (ij : ι × ι')
Goal: LinearMap.baseChange A ((b'.linearMap b) ij) =
  ((Algebra.TensorProduct.basis A b').linearMap (Algebra.TensorProduct.basis A b)) ij
-- Basis.baseChange_end
lemma _root_.Basis.baseChange_end (b : Basis ι R M) (ij : ι × ι) :
    baseChange A (b.end ij) = (basis A b).end ij 
Arguments: {R : Type uR} {M : Type uM} {ι : Type uι} [CommSemiring R] [AddCommMonoid M] [Module R M] [Fintype ι] (A : Type u_3) [CommSemiring A] [Algebra R A] [DecidableEq ι] (b : Basis ι R M) (ij : ι × ι)
Goal: LinearMap.baseChange A (b.end ij) = (Algebra.TensorProduct.basis A b).end ij
-- LinearMap.toMatrix_baseChange
@[simp]
lemma toMatrix_baseChange (f : M₁ →ₗ[R] M₂) (b₁ : Basis ι R M₁) (b₂ : Basis ι₂ R M₂) :
    toMatrix (basis A b₁) (basis A b₂) (f.baseChange A) =
    (toMatrix b₁ b₂ f).map (algebraMap R A) 
Arguments: {R : Type u_1} {M₁ : Type u_2} {M₂ : Type u_3} {ι : Type u_4} {ι₂ : Type u_5} (A : Type u_6) [Fintype ι] [Finite ι₂] [DecidableEq ι] [CommSemiring R] [CommSemiring A] [Algebra R A] [AddCommMonoid M₁] [Module R M₁] [AddCommMonoid M₂] [Module R M₂] (f : M₁ →ₗ[R] M₂) (b₁ : Basis ι R M₁) (b₂ : Basis ι₂ R M₂)
Goal: (LinearMap.toMatrix (Algebra.TensorProduct.basis A b₁) (Algebra.TensorProduct.basis A b₂)) (LinearMap.baseChange A f) =
  ((LinearMap.toMatrix b₁ b₂) f).map ⇑(algebraMap R A)
-- LinearMap.tensorProduct_apply
@[simps!]
noncomputable
def tensorProduct : A ⊗[R] (M →ₗ[R] N) →ₗ[A] (A ⊗[R] M) →ₗ[A] (A ⊗[R] N) 
Arguments: (R : Type u_1) (A : Type u_2) (M : Type u_3) (N : Type u_4) [CommRing R] [CommRing A] [Algebra R A] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N]
Goal: ∀ (a : A ⊗[R] (M →ₗ[R] N)),
  (LinearMap.tensorProduct R A M N) a =
    (TensorProduct.liftAux
        (↑R { toFun := fun a => a • LinearMap.baseChangeHom R A M N, map_add' := ⋯, map_smul' := ⋯ }))
      a
-- LinearMap.tensorProductEnd_apply
@[simps!]
noncomputable
def tensorProductEnd : A ⊗[R] (End R M) →ₐ[A] End A (A ⊗[R] M) 
Arguments: (R : Type u_1) (A : Type u_2) (M : Type u_3) [CommRing R] [CommRing A] [Algebra R A] [AddCommGroup M] [Module R M] (a : A ⊗[R] Module.End R M)
Goal: (LinearMap.tensorProductEnd R A M) a =
  (TensorProduct.liftAux (↑R { toFun := fun a => a • LinearMap.baseChangeHom R A M M, map_add' := ⋯, map_smul' := ⋯ }))
    a
-- Module.endTensorEndAlgHom_apply
theorem endTensorEndAlgHom_apply (f : End A M) (g : End R N) :
    endTensorEndAlgHom (R := R) (S := S) (A := A) (M := M) (N := N) (f ⊗ₜ[R] g)
      = AlgebraTensorModule.map f g 
Arguments: {R : Type u_1} {S : Type u_2} {A : Type u_3} {M : Type u_4} {N : Type u_5} [CommSemiring R] [CommSemiring S] [Semiring A] [AddCommMonoid M] [AddCommMonoid N] [Algebra R S] [Algebra S A] [Algebra R A] [Module R M] [Module S M] [Module A M] [Module R N] [IsScalarTower R A M] [IsScalarTower S A M] [IsScalarTower R S M] (f : Module.End A M) (g : Module.End R N)
Goal: Module.endTensorEndAlgHom (f ⊗ₜ[R] g) = TensorProduct.AlgebraTensorModule.map f g
-- Subalgebra.finite_sup
theorem Subalgebra.finite_sup {K L : Type*} [CommSemiring K] [CommSemiring L] [Algebra K L]
    (E1 E2 : Subalgebra K L) [Module.Finite K E1] [Module.Finite K E2] :
    Module.Finite K ↥(E1 ⊔ E2) 
Arguments: {K : Type u_1} {L : Type u_2} [CommSemiring K] [CommSemiring L] [Algebra K L] (E1 E2 : Subalgebra K L) [Module.Finite K ↥E1] [Module.Finite K ↥E2]
Goal: Module.Finite K ↥(E1 ⊔ E2)
-- Subalgebra.finiteDimensional_sup
@[deprecated Subalgebra.finite_sup (since := "2024-04-11")]
theorem Subalgebra.finiteDimensional_sup {K L : Type*} [Field K] [CommRing L] [Algebra K L]
    (E1 E2 : Subalgebra K L) [FiniteDimensional K E1] [FiniteDimensional K E2] :
    FiniteDimensional K (E1 ⊔ E2 : Subalgebra K L) 
Arguments: {K : Type u_1} {L : Type u_2} [Field K] [CommRing L] [Algebra K L] (E1 E2 : Subalgebra K L) [FiniteDimensional K ↥E1] [FiniteDimensional K ↥E2]
Goal: FiniteDimensional K ↥(E1 ⊔ E2)
-- TensorProduct.Algebra.moduleAux_apply
theorem moduleAux_apply (a : A) (b : B) (m : M) : moduleAux (a ⊗ₜ[R] b) m = a • b • m 
Arguments: {R : Type u_1} {A : Type u_2} {B : Type u_3} {M : Type u_4} [CommSemiring R] [AddCommMonoid M] [Module R M] [Semiring A] [Semiring B] [Module A M] [Module B M] [Algebra R A] [Algebra R B] [IsScalarTower R A M] [IsScalarTower R B M] (a : A) (b : B) (m : M)
Goal: (TensorProduct.Algebra.moduleAux (a ⊗ₜ[R] b)) m = a • b • m
-- TensorProduct.Algebra.smul_def
theorem smul_def (a : A) (b : B) (m : M) : a ⊗ₜ[R] b • m = a • b • m 
Arguments: {R : Type u_1} {A : Type u_2} {B : Type u_3} {M : Type u_4} [CommSemiring R] [AddCommMonoid M] [Module R M] [Semiring A] [Semiring B] [Module A M] [Module B M] [Algebra R A] [Algebra R B] [IsScalarTower R A M] [IsScalarTower R B M] [SMulCommClass A B M] (a : A) (b : B) (m : M)
Goal: a ⊗ₜ[R] b • m = a • b • m
-- RingHom.isIntegralElem_map
theorem RingHom.isIntegralElem_map {x : R} : f.IsIntegralElem (f x) 
Arguments: {R : Type u_1} {S : Type u_2} [CommRing R] [Ring S] (f : R →+* S) {x : R}
Goal: f.IsIntegralElem (f x)
-- isIntegral_algebraMap
theorem isIntegral_algebraMap {x : R} : IsIntegral R (algebraMap R A x) 
Arguments: {R : Type u_1} {A : Type u_3} [CommRing R] [Ring A] [Algebra R A] {x : R}
Goal: IsIntegral R ((algebraMap R A) x)
-- IsIntegral.map
theorem IsIntegral.map {B C F : Type*} [Ring B] [Ring C] [Algebra R B] [Algebra A B] [Algebra R C]
    [IsScalarTower R A B] [Algebra A C] [IsScalarTower R A C] {b : B}
    [FunLike F B C] [AlgHomClass F A B C] (f : F)
    (hb : IsIntegral R b) : IsIntegral R (f b) 
Arguments: {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] {B : Type u_5} {C : Type u_6} {F : Type u_7} [Ring B] [Ring C] [Algebra R B] [Algebra A B] [Algebra R C] [IsScalarTower R A B] [Algebra A C] [IsScalarTower R A C] {b : B} [FunLike F B C] [AlgHomClass F A B C] (f : F) (hb : IsIntegral R b)
Goal: IsIntegral R (f b)
-- isIntegral_algHom_iff
theorem isIntegral_algHom_iff {x : A} : IsIntegral R (f x) ↔ IsIntegral R x 
Arguments: {R : Type u_1} [CommRing R] {A : Type u_5} {B : Type u_6} [Ring A] [Ring B] [Algebra R A] [Algebra R B] (f : A →ₐ[R] B) (hf : Function.Injective ⇑f) {x : A}
Goal: IsIntegral R (f x) ↔ IsIntegral R x
-- Submodule.span_range_natDegree_eq_adjoin
theorem Submodule.span_range_natDegree_eq_adjoin {R A} [CommRing R] [Semiring A] [Algebra R A]
    {x : A} {f : R[X]} (hf : f.Monic) (hfx : aeval x f = 0) :
    span R (Finset.image (x ^ ·) (Finset.range (natDegree f))) =
      Subalgebra.toSubmodule (Algebra.adjoin R {x}) 
Arguments: {R : Type u_5} {A : Type u_6} [CommRing R] [Semiring A] [Algebra R A] {x : A} {f : R[X]} (hf : f.Monic) (hfx : (Polynomial.aeval x) f = 0)
Goal: Submodule.span R ↑(Finset.image (fun x_1 => x ^ x_1) (Finset.range f.natDegree)) =
  Subalgebra.toSubmodule (Algebra.adjoin R {x})
-- IsIntegral.fg_adjoin_singleton
theorem IsIntegral.fg_adjoin_singleton {x : B} (hx : IsIntegral R x) :
    (Algebra.adjoin R {x}).toSubmodule.FG 
Arguments: {R : Type u_1} {B : Type u_3} [CommRing R] [Ring B] [Algebra R B] {x : B} (hx : IsIntegral R x)
Goal: (Subalgebra.toSubmodule (Algebra.adjoin R {x})).FG
-- RingHom.isIntegralElem_zero
theorem RingHom.isIntegralElem_zero : f.IsIntegralElem 0 
Arguments: {R : Type u_1} {B : Type u_3} [CommRing R] [Ring B] (f : R →+* B)
Goal: f.IsIntegralElem 0
-- isIntegral_zero
theorem isIntegral_zero : IsIntegral R (0 : B) 
Arguments: {R : Type u_1} {B : Type u_3} [CommRing R] [Ring B] [Algebra R B]
Goal: IsIntegral R 0
-- RingHom.isIntegralElem_one
theorem RingHom.isIntegralElem_one : f.IsIntegralElem 1 
Arguments: {R : Type u_1} {B : Type u_3} [CommRing R] [Ring B] (f : R →+* B)
Goal: f.IsIntegralElem 1
-- isIntegral_one
theorem isIntegral_one : IsIntegral R (1 : B) 
Arguments: {R : Type u_1} {B : Type u_3} [CommRing R] [Ring B] [Algebra R B]
Goal: IsIntegral R 1
-- IsIntegral.of_pow
theorem IsIntegral.of_pow {x : B} {n : ℕ} (hn : 0 < n) (hx : IsIntegral R <| x ^ n) :
    IsIntegral R x 
Arguments: {R : Type u_1} {B : Type u_3} [CommRing R] [Ring B] [Algebra R B] {x : B} {n : ℕ} (hn : 0 < n) (hx : IsIntegral R (x ^ n))
Goal: IsIntegral R x
-- IsIntegralClosure.isIntegral_iff
class IsIntegralClosure (A R B : Type*) [CommRing R] [CommSemiring A] [CommRing B] [Algebra R B]
  [Algebra A B] : Prop where
  algebraMap_injective' : Function.Injective (algebraMap A B)
  isIntegral_iff : ∀ {x : B}, IsIntegral R x ↔ ∃ y, algebraMap A B y = x
Arguments: {A : Type u_1} {R : Type u_2} {B : Type u_3} [CommRing R] [CommSemiring A] [CommRing B] [Algebra R B] [Algebra A B] [self : IsIntegralClosure A R B] {x : B}
Goal: IsIntegral R x ↔ ∃ y, (algebraMap A B) y = x
-- IsIntegralClosure.algebraMap_injective'
class IsIntegralClosure (A R B : Type*) [CommRing R] [CommSemiring A] [CommRing B] [Algebra R B]
  [Algebra A B] : Prop where
  algebraMap_injective' : Function.Injective (algebraMap A B)
  isIntegral_iff : ∀ {x : B}, IsIntegral R x ↔ ∃ y, algebraMap A B y = x
Arguments: {A : Type u_1} (R : Type u_2) {B : Type u_3} [CommRing R] [CommSemiring A] [CommRing B] [Algebra R B] [Algebra A B] [self : IsIntegralClosure A R B]
Goal: Function.Injective ⇑(algebraMap A B)
-- HahnSeries.one_coeff
@[simp]
theorem one_coeff [Zero R] [One R] {a : Γ} :
    (1 : HahnSeries Γ R).coeff a = if a = 0 then 1 else 0 
Arguments: {Γ : Type u_1} {R : Type u_3} [Zero Γ] [PartialOrder Γ] [Zero R] [One R] {a : Γ}
Goal: HahnSeries.coeff 1 a = if a = 0 then 1 else 0
-- HahnSeries.single_zero_one
@[simp]
theorem single_zero_one [Zero R] [One R] : single (0 : Γ) (1 : R) = 1 
Arguments: {Γ : Type u_1} {R : Type u_3} [Zero Γ] [PartialOrder Γ] [Zero R] [One R]
Goal: (HahnSeries.single 0) 1 = 1
-- HahnSeries.support_one
@[simp]
theorem support_one [MulZeroOneClass R] [Nontrivial R] : support (1 : HahnSeries Γ R) = {0} 
Arguments: {Γ : Type u_1} {R : Type u_3} [Zero Γ] [PartialOrder Γ] [MulZeroOneClass R] [Nontrivial R]
Goal: HahnSeries.support 1 = {0}
-- HahnSeries.orderTop_one
@[simp]
theorem orderTop_one [MulZeroOneClass R] [Nontrivial R] : orderTop (1 : HahnSeries Γ R) = 0 
Arguments: {Γ : Type u_1} {R : Type u_3} [Zero Γ] [PartialOrder Γ] [MulZeroOneClass R] [Nontrivial R]
Goal: HahnSeries.orderTop 1 = 0
-- HahnSeries.order_one
@[simp]
theorem order_one [MulZeroOneClass R] : order (1 : HahnSeries Γ R) = 0 
Arguments: {Γ : Type u_1} {R : Type u_3} [Zero Γ] [PartialOrder Γ] [MulZeroOneClass R]
Goal: HahnSeries.order 1 = 0
-- HahnSeries.leadingCoeff_one
@[simp]
theorem leadingCoeff_one [MulZeroOneClass R] : (1 : HahnSeries Γ R).leadingCoeff = 1 
Arguments: {Γ : Type u_1} {R : Type u_3} [Zero Γ] [PartialOrder Γ] [MulZeroOneClass R]
Goal: HahnSeries.leadingCoeff 1 = 1
-- HahnModule.ext
@[ext]
theorem ext (x y : HahnModule Γ R V) (h : ((of R).symm x).coeff = ((of R).symm y).coeff) : x = y 
Arguments: {Γ : Type u_1} {R : Type u_3} {V : Type u_4} [PartialOrder Γ] [Zero V] [SMul R V] (x y : HahnModule Γ R V) (h : ((HahnModule.of R).symm x).coeff = ((HahnModule.of R).symm y).coeff)
Goal: x = y
-- HahnModule.ext_iff
theorem ext_iff (x y : HahnModule Γ R V) :
    ((of R).symm x).coeff = ((of R).symm y).coeff ↔ x = y 
Arguments: {Γ : Type u_1} {R : Type u_3} {V : Type u_4} [PartialOrder Γ] [Zero V] [SMul R V] (x y : HahnModule Γ R V)
Goal: ((HahnModule.of R).symm x).coeff = ((HahnModule.of R).symm y).coeff ↔ x = y
-- HahnModule.of_zero
@[simp] theorem of_zero : of R (0 : HahnSeries Γ V) = 0 
Arguments: {Γ : Type u_1} {R : Type u_3} {V : Type u_4} [PartialOrder Γ] [AddCommMonoid V] [SMul R V]
Goal: (HahnModule.of R) 0 = 0
-- HahnModule.of_add
@[simp] theorem of_add (x y : HahnSeries Γ V) : of R (x + y) = of R x + of R y 
Arguments: {Γ : Type u_1} {R : Type u_3} {V : Type u_4} [PartialOrder Γ] [AddCommMonoid V] [SMul R V] (x y : HahnSeries Γ V)
Goal: (HahnModule.of R) (x + y) = (HahnModule.of R) x + (HahnModule.of R) y
-- HahnModule.of_symm_zero
@[simp] theorem of_symm_zero : (of R).symm (0 : HahnModule Γ R V) = 0 
Arguments: {Γ : Type u_1} {R : Type u_3} {V : Type u_4} [PartialOrder Γ] [AddCommMonoid V] [SMul R V]
Goal: (HahnModule.of R).symm 0 = 0
-- HahnModule.of_symm_add
@[simp] theorem of_symm_add (x y : HahnModule Γ R V) :
  (of R).symm (x + y) = (of R).symm x + (of R).symm y 
Arguments: {Γ : Type u_1} {R : Type u_3} {V : Type u_4} [PartialOrder Γ] [AddCommMonoid V] [SMul R V] (x y : HahnModule Γ R V)
Goal: (HahnModule.of R).symm (x + y) = (HahnModule.of R).symm x + (HahnModule.of R).symm y
-- HahnModule.smul_coeff
theorem smul_coeff [Zero R] (x : HahnSeries Γ R) (y : HahnModule Γ' R V) (a : Γ') :
    ((of R).symm <| x • y).coeff a =
      ∑ ij ∈ VAddAntidiagonal x.isPWO_support ((of R).symm y).isPWO_support a,
        x.coeff ij.fst • ((of R).symm y).coeff ij.snd 
Arguments: {Γ : Type u_1} {Γ' : Type u_2} {R : Type u_3} {V : Type u_4} [PartialOrder Γ] [AddCommMonoid V] [SMul R V] [PartialOrder Γ'] [VAdd Γ Γ'] [IsOrderedCancelVAdd Γ Γ'] [Zero R] (x : HahnSeries Γ R) (y : HahnModule Γ' R V) (a : Γ')
Goal: ((HahnModule.of R).symm (x • y)).coeff a =
  ∑ ij ∈ Finset.VAddAntidiagonal ⋯ ⋯ a, x.coeff ij.1 • ((HahnModule.of R).symm y).coeff ij.2
-- HahnModule.of_smul
@[simp] theorem of_smul [SMulZeroClass R V] (r : R) (x : HahnSeries Γ V) :
  (of R) (r • x) = r • (of R) x 
Arguments: {Γ : Type u_1} {R : Type u_3} {V : Type u_4} [PartialOrder Γ] [AddCommMonoid V] [SMulZeroClass R V] (r : R) (x : HahnSeries Γ V)
Goal: (HahnModule.of R) (r • x) = r • (HahnModule.of R) x
-- HahnModule.of_symm_smul
@[simp] theorem of_symm_smul [PartialOrder Γ] [SMulZeroClass R V] (r : R) (x : HahnModule Γ R V) :
  (of R).symm (r • x) = r • (of R).symm x 
Arguments: {Γ : Type u_1} {R : Type u_3} {V : Type u_4} [AddCommMonoid V] [PartialOrder Γ] [SMulZeroClass R V] (r : R) (x : HahnModule Γ R V)
Goal: (HahnModule.of R).symm (r • x) = r • (HahnModule.of R).symm x
-- HahnModule.smul_coeff_right
theorem smul_coeff_right [SMulZeroClass R V] {x : HahnSeries Γ R} {y : HahnModule Γ' R V} {a : Γ'}
    {s : Set Γ'} (hs : s.IsPWO) (hys : ((of R).symm y).support ⊆ s) :
    ((of R).symm <| x • y).coeff a =
      ∑ ij ∈ VAddAntidiagonal x.isPWO_support hs a,
        x.coeff ij.fst • ((of R).symm y).coeff ij.snd 
Arguments: {Γ : Type u_1} {Γ' : Type u_2} {R : Type u_3} {V : Type u_4} [PartialOrder Γ] [PartialOrder Γ'] [VAdd Γ Γ'] [IsOrderedCancelVAdd Γ Γ'] [Zero R] [AddCommMonoid V] [SMulZeroClass R V] {x : HahnSeries Γ R} {y : HahnModule Γ' R V} {a : Γ'} {s : Set Γ'} (hs : s.IsPWO) (hys : ((HahnModule.of R).symm y).support ⊆ s)
Goal: ((HahnModule.of R).symm (x • y)).coeff a =
  ∑ ij ∈ Finset.VAddAntidiagonal ⋯ hs a, x.coeff ij.1 • ((HahnModule.of R).symm y).coeff ij.2
-- HahnModule.smul_coeff_left
theorem smul_coeff_left [SMulWithZero R V] {x : HahnSeries Γ R}
    {y : HahnModule Γ' R V} {a : Γ'} {s : Set Γ}
    (hs : s.IsPWO) (hxs : x.support ⊆ s) :
    ((of R).symm <| x • y).coeff a =
      ∑ ij ∈ VAddAntidiagonal hs ((of R).symm y).isPWO_support a,
        x.coeff ij.fst • ((of R).symm y).coeff ij.snd 
Arguments: {Γ : Type u_1} {Γ' : Type u_2} {R : Type u_3} {V : Type u_4} [PartialOrder Γ] [PartialOrder Γ'] [VAdd Γ Γ'] [IsOrderedCancelVAdd Γ Γ'] [Zero R] [AddCommMonoid V] [SMulWithZero R V] {x : HahnSeries Γ R} {y : HahnModule Γ' R V} {a : Γ'} {s : Set Γ} (hs : s.IsPWO) (hxs : x.support ⊆ s)
Goal: ((HahnModule.of R).symm (x • y)).coeff a =
  ∑ ij ∈ Finset.VAddAntidiagonal hs ⋯ a, x.coeff ij.1 • ((HahnModule.of R).symm y).coeff ij.2
-- HahnModule.smul_add
theorem smul_add [Zero R] [DistribSMul R V] (x : HahnSeries Γ R) (y z : HahnModule Γ' R V) :
    x • (y + z) = x • y + x • z 
Arguments: {Γ : Type u_1} {Γ' : Type u_2} {R : Type u_3} {V : Type u_4} [PartialOrder Γ] [PartialOrder Γ'] [VAdd Γ Γ'] [IsOrderedCancelVAdd Γ Γ'] [AddCommMonoid V] [Zero R] [DistribSMul R V] (x : HahnSeries Γ R) (y z : HahnModule Γ' R V)
Goal: x • (y + z) = x • y + x • z
-- HahnModule.add_smul
theorem add_smul [AddCommMonoid R] [SMulWithZero R V] {x y : HahnSeries Γ R}
    {z : HahnModule Γ' R V} (h : ∀ (r s : R) (u : V), (r + s) • u = r • u + s • u) :
    (x + y) • z = x • z + y • z 
Arguments: {Γ : Type u_1} {Γ' : Type u_2} {R : Type u_3} {V : Type u_4} [PartialOrder Γ] [PartialOrder Γ'] [VAdd Γ Γ'] [IsOrderedCancelVAdd Γ Γ'] [AddCommMonoid V] [AddCommMonoid R] [SMulWithZero R V] {x y : HahnSeries Γ R} {z : HahnModule Γ' R V} (h : ∀ (r s : R) (u : V), (r + s) • u = r • u + s • u)
Goal: (x + y) • z = x • z + y • z