ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- LieAlgebra.IsKilling.iInf_ker_weight_eq_bot
@[simp]
lemma iInf_ker_weight_eq_bot :
    ⨅ α : Weight K H L, α.ker = ⊥ 
Arguments: (K : Type u_2) (L : Type u_3) [LieRing L] [Field K] [LieAlgebra K L] [FiniteDimensional K L] (H : LieSubalgebra K L) [H.IsCartanSubalgebra] [LieModule.IsTriangularizable K (↥H) L] [LieAlgebra.IsKilling K L]
Goal: ⨅ α, LieModule.Weight.ker = ⊥
-- LieAlgebra.IsKilling.corootSpace_zero_eq_bot
@[simp]
lemma corootSpace_zero_eq_bot :
    corootSpace (0 : H → K) = ⊥ 
Arguments: (K : Type u_2) (L : Type u_3) [LieRing L] [Field K] [LieAlgebra K L] [FiniteDimensional K L] (H : LieSubalgebra K L) [H.IsCartanSubalgebra] [LieAlgebra.IsKilling K L]
Goal: LieAlgebra.corootSpace 0 = ⊥
-- LieAlgebra.IsKilling.isSemisimple_ad_of_mem_isCartanSubalgebra
lemma isSemisimple_ad_of_mem_isCartanSubalgebra {x : L} (hx : x ∈ H) :
    (ad K L x).IsSemisimple 
Arguments: {K : Type u_2} {L : Type u_3} [LieRing L] [Field K] [LieAlgebra K L] [FiniteDimensional K L] {H : LieSubalgebra K L} [H.IsCartanSubalgebra] [LieModule.IsTriangularizable K (↥H) L] [LieAlgebra.IsKilling K L] [PerfectField K] {x : L} (hx : x ∈ H)
Goal: ((LieAlgebra.ad K L) x).IsSemisimple
-- LieAlgebra.IsKilling.lie_eq_smul_of_mem_rootSpace
lemma lie_eq_smul_of_mem_rootSpace {α : H → K} {x : L} (hx : x ∈ rootSpace H α) (h : H) :
    ⁅h, x⁆ = α h • x 
Arguments: {K : Type u_2} {L : Type u_3} [LieRing L] [Field K] [LieAlgebra K L] [FiniteDimensional K L] {H : LieSubalgebra K L} [H.IsCartanSubalgebra] [LieModule.IsTriangularizable K (↥H) L] [LieAlgebra.IsKilling K L] [PerfectField K] {α : ↥H → K} {x : L} (hx : x ∈ LieAlgebra.rootSpace H α) (h : ↥H)
Goal: ⁅h, x⁆ = α h • x
-- LieAlgebra.IsKilling.lie_eq_killingForm_smul_of_mem_rootSpace_of_mem_rootSpace_neg
lemma lie_eq_killingForm_smul_of_mem_rootSpace_of_mem_rootSpace_neg
    {α : Weight K H L} {e f : L} (heα : e ∈ rootSpace H α) (hfα : f ∈ rootSpace H (-α)) :
    ⁅e, f⁆ = killingForm K L e f • (cartanEquivDual H).symm α 
Arguments: {K : Type u_2} {L : Type u_3} [LieRing L] [Field K] [LieAlgebra K L] [FiniteDimensional K L] {H : LieSubalgebra K L} [H.IsCartanSubalgebra] [LieModule.IsTriangularizable K (↥H) L] [LieAlgebra.IsKilling K L] [PerfectField K] {α : LieModule.Weight K (↥H) L} {e f : L} (heα : e ∈ LieAlgebra.rootSpace H ⇑α) (hfα : f ∈ LieAlgebra.rootSpace H (-⇑α))
Goal: ⁅e, f⁆ =
  ((killingForm K L) e) f • ↑((LieAlgebra.IsKilling.cartanEquivDual H).symm (LieModule.Weight.toLinear K (↥H) L α))
-- LieAlgebra.IsKilling.coe_corootSpace_eq_span_singleton'
lemma coe_corootSpace_eq_span_singleton' (α : Weight K H L) :
    (corootSpace α).toSubmodule = K ∙ (cartanEquivDual H).symm α 
Arguments: {K : Type u_2} {L : Type u_3} [LieRing L] [Field K] [LieAlgebra K L] [FiniteDimensional K L] {H : LieSubalgebra K L} [H.IsCartanSubalgebra] [LieModule.IsTriangularizable K (↥H) L] [LieAlgebra.IsKilling K L] [PerfectField K] (α : LieModule.Weight K (↥H) L)
Goal: ↑(LieAlgebra.corootSpace ⇑α) =
  Submodule.span K {(LieAlgebra.IsKilling.cartanEquivDual H).symm (LieModule.Weight.toLinear K (↥H) L α)}
-- LieAlgebra.IsKilling.eq_zero_of_apply_eq_zero_of_mem_corootSpace
lemma eq_zero_of_apply_eq_zero_of_mem_corootSpace
    (x : H) (α : H → K) (hαx : α x = 0) (hx : x ∈ corootSpace α) :
    x = 0 
Arguments: {K : Type u_2} {L : Type u_3} [LieRing L] [Field K] [LieAlgebra K L] [FiniteDimensional K L] {H : LieSubalgebra K L} [H.IsCartanSubalgebra] [LieModule.IsTriangularizable K (↥H) L] [LieAlgebra.IsKilling K L] [CharZero K] (x : ↥H) (α : ↥H → K) (hαx : α x = 0) (hx : x ∈ LieAlgebra.corootSpace α)
Goal: x = 0
-- LieAlgebra.IsKilling.disjoint_ker_weight_corootSpace
lemma disjoint_ker_weight_corootSpace (α : Weight K H L) :
    Disjoint α.ker (corootSpace α) 
Arguments: {K : Type u_2} {L : Type u_3} [LieRing L] [Field K] [LieAlgebra K L] [FiniteDimensional K L] {H : LieSubalgebra K L} [H.IsCartanSubalgebra] [LieModule.IsTriangularizable K (↥H) L] [LieAlgebra.IsKilling K L] [CharZero K] (α : LieModule.Weight K (↥H) L)
Goal: Disjoint LieModule.Weight.ker (lieIdealSubalgebra K (↥H) (LieAlgebra.corootSpace ⇑α)).toSubmodule
-- LieAlgebra.IsKilling.root_apply_cartanEquivDual_symm_ne_zero
lemma root_apply_cartanEquivDual_symm_ne_zero {α : Weight K H L} (hα : α.IsNonZero) :
    α ((cartanEquivDual H).symm α) ≠ 0 
Arguments: {K : Type u_2} {L : Type u_3} [LieRing L] [Field K] [LieAlgebra K L] [FiniteDimensional K L] {H : LieSubalgebra K L} [H.IsCartanSubalgebra] [LieModule.IsTriangularizable K (↥H) L] [LieAlgebra.IsKilling K L] [CharZero K] {α : LieModule.Weight K (↥H) L} (hα : α.IsNonZero)
Goal: α ((LieAlgebra.IsKilling.cartanEquivDual H).symm (LieModule.Weight.toLinear K (↥H) L α)) ≠ 0
-- LieAlgebra.IsKilling.root_apply_coroot
lemma root_apply_coroot {α : Weight K H L} (hα : α.IsNonZero) :
    α (coroot α) = 2 
Arguments: {K : Type u_2} {L : Type u_3} [LieRing L] [Field K] [LieAlgebra K L] [FiniteDimensional K L] {H : LieSubalgebra K L} [H.IsCartanSubalgebra] [LieModule.IsTriangularizable K (↥H) L] [LieAlgebra.IsKilling K L] [CharZero K] {α : LieModule.Weight K (↥H) L} (hα : α.IsNonZero)
Goal: α (LieAlgebra.IsKilling.coroot α) = 2
-- LieAlgebra.IsKilling.traceForm_coroot
lemma traceForm_coroot (α : Weight K H L) (x : H) :
    traceForm K H L (coroot α) x = 2 • (α <| (cartanEquivDual H).symm α)⁻¹ • α x 
Arguments: {K : Type u_2} {L : Type u_3} [LieRing L] [Field K] [LieAlgebra K L] [FiniteDimensional K L] {H : LieSubalgebra K L} [H.IsCartanSubalgebra] [LieAlgebra.IsKilling K L] [CharZero K] (α : LieModule.Weight K (↥H) L) (x : ↥H)
Goal: ((LieModule.traceForm K (↥H) L) (LieAlgebra.IsKilling.coroot α)) x =
  2 • (α ((LieAlgebra.IsKilling.cartanEquivDual H).symm (LieModule.Weight.toLinear K (↥H) L α)))⁻¹ • α x
-- LieAlgebra.IsKilling.coroot_eq_zero_iff
@[simp] lemma coroot_eq_zero_iff {α : Weight K H L} :
    coroot α = 0 ↔ α.IsZero 
Arguments: {K : Type u_2} {L : Type u_3} [LieRing L] [Field K] [LieAlgebra K L] [FiniteDimensional K L] {H : LieSubalgebra K L} [H.IsCartanSubalgebra] [LieModule.IsTriangularizable K (↥H) L] [LieAlgebra.IsKilling K L] [CharZero K] {α : LieModule.Weight K (↥H) L}
Goal: LieAlgebra.IsKilling.coroot α = 0 ↔ α.IsZero
-- LieAlgebra.IsKilling.coroot_zero
@[simp]
lemma coroot_zero [Nontrivial L] : coroot (0 : Weight K H L) = 0 
Arguments: {K : Type u_2} {L : Type u_3} [LieRing L] [Field K] [LieAlgebra K L] [FiniteDimensional K L] {H : LieSubalgebra K L} [H.IsCartanSubalgebra] [LieModule.IsTriangularizable K (↥H) L] [LieAlgebra.IsKilling K L] [CharZero K] [Nontrivial L]
Goal: LieAlgebra.IsKilling.coroot 0 = 0
-- LieAlgebra.IsKilling.coe_corootSpace_eq_span_singleton
lemma coe_corootSpace_eq_span_singleton (α : Weight K H L) :
    (corootSpace α).toSubmodule = K ∙ coroot α 
Arguments: {K : Type u_2} {L : Type u_3} [LieRing L] [Field K] [LieAlgebra K L] [FiniteDimensional K L] {H : LieSubalgebra K L} [H.IsCartanSubalgebra] [LieModule.IsTriangularizable K (↥H) L] [LieAlgebra.IsKilling K L] [CharZero K] (α : LieModule.Weight K (↥H) L)
Goal: ↑(LieAlgebra.corootSpace ⇑α) = Submodule.span K {LieAlgebra.IsKilling.coroot α}
-- LieAlgebra.IsKilling.corootSpace_eq_bot_iff
@[simp]
lemma corootSpace_eq_bot_iff {α : Weight K H L} :
    corootSpace α = ⊥ ↔ α.IsZero 
Arguments: {K : Type u_2} {L : Type u_3} [LieRing L] [Field K] [LieAlgebra K L] [FiniteDimensional K L] {H : LieSubalgebra K L} [H.IsCartanSubalgebra] [LieModule.IsTriangularizable K (↥H) L] [LieAlgebra.IsKilling K L] [CharZero K] {α : LieModule.Weight K (↥H) L}
Goal: LieAlgebra.corootSpace ⇑α = ⊥ ↔ α.IsZero
-- LieAlgebra.IsKilling.isCompl_ker_weight_span_coroot
lemma isCompl_ker_weight_span_coroot (α : Weight K H L) :
    IsCompl α.ker (K ∙ coroot α) 
Arguments: {K : Type u_2} {L : Type u_3} [LieRing L] [Field K] [LieAlgebra K L] [FiniteDimensional K L] {H : LieSubalgebra K L} [H.IsCartanSubalgebra] [LieModule.IsTriangularizable K (↥H) L] [LieAlgebra.IsKilling K L] [CharZero K] (α : LieModule.Weight K (↥H) L)
Goal: IsCompl LieModule.Weight.ker (Submodule.span K {LieAlgebra.IsKilling.coroot α})
-- LieAlgebra.IsKilling.traceForm_eq_zero_of_mem_ker_of_mem_span_coroot
lemma traceForm_eq_zero_of_mem_ker_of_mem_span_coroot {α : Weight K H L} {x y : H}
    (hx : x ∈ α.ker) (hy : y ∈ K ∙ coroot α) :
    traceForm K H L x y = 0 
Arguments: {K : Type u_2} {L : Type u_3} [LieRing L] [Field K] [LieAlgebra K L] [FiniteDimensional K L] {H : LieSubalgebra K L} [H.IsCartanSubalgebra] [LieModule.IsTriangularizable K (↥H) L] [LieAlgebra.IsKilling K L] [CharZero K] {α : LieModule.Weight K (↥H) L} {x y : ↥H} (hx : x ∈ LieModule.Weight.ker) (hy : y ∈ Submodule.span K {LieAlgebra.IsKilling.coroot α})
Goal: ((LieModule.traceForm K (↥H) L) x) y = 0
-- LieAlgebra.IsKilling.orthogonal_span_coroot_eq_ker
@[simp] lemma orthogonal_span_coroot_eq_ker (α : Weight K H L) :
    (traceForm K H L).orthogonal (K ∙ coroot α) = α.ker 
Arguments: {K : Type u_2} {L : Type u_3} [LieRing L] [Field K] [LieAlgebra K L] [FiniteDimensional K L] {H : LieSubalgebra K L} [H.IsCartanSubalgebra] [LieModule.IsTriangularizable K (↥H) L] [LieAlgebra.IsKilling K L] [CharZero K] (α : LieModule.Weight K (↥H) L)
Goal: (LieModule.traceForm K (↥H) L).orthogonal (Submodule.span K {LieAlgebra.IsKilling.coroot α}) = LieModule.Weight.ker
-- LieAlgebra.IsKilling.coroot_eq_iff
@[simp] lemma coroot_eq_iff (α β : Weight K H L) :
    coroot α = coroot β ↔ α = β 
Arguments: {K : Type u_2} {L : Type u_3} [LieRing L] [Field K] [LieAlgebra K L] [FiniteDimensional K L] {H : LieSubalgebra K L} [H.IsCartanSubalgebra] [LieModule.IsTriangularizable K (↥H) L] [LieAlgebra.IsKilling K L] [CharZero K] (α β : LieModule.Weight K (↥H) L)
Goal: LieAlgebra.IsKilling.coroot α = LieAlgebra.IsKilling.coroot β ↔ α = β
-- LieAlgebra.IsKilling.exists_isSl2Triple_of_weight_isNonZero
lemma exists_isSl2Triple_of_weight_isNonZero {α : Weight K H L} (hα : α.IsNonZero) :
    ∃ h e f : L, IsSl2Triple h e f ∧ e ∈ rootSpace H α ∧ f ∈ rootSpace H (- α) 
Arguments: {K : Type u_2} {L : Type u_3} [LieRing L] [Field K] [LieAlgebra K L] [FiniteDimensional K L] {H : LieSubalgebra K L} [H.IsCartanSubalgebra] [LieModule.IsTriangularizable K (↥H) L] [LieAlgebra.IsKilling K L] [CharZero K] {α : LieModule.Weight K (↥H) L} (hα : α.IsNonZero)
Goal: ∃ h e f, IsSl2Triple h e f ∧ e ∈ LieAlgebra.rootSpace H ⇑α ∧ f ∈ LieAlgebra.rootSpace H (-⇑α)
-- IsSl2Triple.h_eq_coroot
lemma _root_.IsSl2Triple.h_eq_coroot {α : Weight K H L} (hα : α.IsNonZero)
    {h e f : L} (ht : IsSl2Triple h e f) (heα : e ∈ rootSpace H α) (hfα : f ∈ rootSpace H (- α)) :
    h = coroot α 
Arguments: {K : Type u_2} {L : Type u_3} [LieRing L] [Field K] [LieAlgebra K L] [FiniteDimensional K L] {H : LieSubalgebra K L} [H.IsCartanSubalgebra] [LieModule.IsTriangularizable K (↥H) L] [LieAlgebra.IsKilling K L] [CharZero K] {α : LieModule.Weight K (↥H) L} (hα : α.IsNonZero) {h e f : L} (ht : IsSl2Triple h e f) (heα : e ∈ LieAlgebra.rootSpace H ⇑α) (hfα : f ∈ LieAlgebra.rootSpace H (-⇑α))
Goal: h = ↑(LieAlgebra.IsKilling.coroot α)
-- LieAlgebra.IsKilling.finrank_rootSpace_eq_one
lemma finrank_rootSpace_eq_one (α : Weight K H L) (hα : α.IsNonZero) :
    finrank K (rootSpace H α) = 1 
Arguments: {K : Type u_2} {L : Type u_3} [LieRing L] [Field K] [LieAlgebra K L] [FiniteDimensional K L] {H : LieSubalgebra K L} [H.IsCartanSubalgebra] [LieModule.IsTriangularizable K (↥H) L] [LieAlgebra.IsKilling K L] [CharZero K] (α : LieModule.Weight K (↥H) L) (hα : α.IsNonZero)
Goal: FiniteDimensional.finrank K ↥↑(LieAlgebra.rootSpace H ⇑α) = 1
-- LieModule.Weight.coe_neg
@[simp] lemma coe_neg : ((-α : Weight K H L) : H → K) = -α 
Arguments: {K : Type u_2} {L : Type u_3} [LieRing L] [Field K] [LieAlgebra K L] [FiniteDimensional K L] [LieAlgebra.IsKilling K L] {H : LieSubalgebra K L} [H.IsCartanSubalgebra] [LieModule.IsTriangularizable K (↥H) L] {α : LieModule.Weight K (↥H) L}
Goal: ⇑(-α) = -⇑α
-- LieModule.Weight.IsZero.neg
lemma IsZero.neg (h : α.IsZero) : (-α).IsZero 
Arguments: {K : Type u_2} {L : Type u_3} [LieRing L] [Field K] [LieAlgebra K L] [FiniteDimensional K L] [LieAlgebra.IsKilling K L] {H : LieSubalgebra K L} [H.IsCartanSubalgebra] [LieModule.IsTriangularizable K (↥H) L] {α : LieModule.Weight K (↥H) L} (h : α.IsZero)
Goal: (-α).IsZero
-- LieModule.Weight.isZero_neg
@[simp] lemma isZero_neg : (-α).IsZero ↔ α.IsZero 
Arguments: {K : Type u_2} {L : Type u_3} [LieRing L] [Field K] [LieAlgebra K L] [FiniteDimensional K L] [LieAlgebra.IsKilling K L] {H : LieSubalgebra K L} [H.IsCartanSubalgebra] [LieModule.IsTriangularizable K (↥H) L] {α : LieModule.Weight K (↥H) L}
Goal: (-α).IsZero ↔ α.IsZero
-- LieModule.Weight.IsNonZero.neg
lemma IsNonZero.neg (h : α.IsNonZero) : (-α).IsNonZero 
Arguments: {K : Type u_2} {L : Type u_3} [LieRing L] [Field K] [LieAlgebra K L] [FiniteDimensional K L] [LieAlgebra.IsKilling K L] {H : LieSubalgebra K L} [H.IsCartanSubalgebra] [LieModule.IsTriangularizable K (↥H) L] {α : LieModule.Weight K (↥H) L} (h : α.IsNonZero)
Goal: (-α).IsNonZero
-- LieModule.Weight.isNonZero_neg
@[simp] lemma isNonZero_neg {α : Weight K H L} : (-α).IsNonZero ↔ α.IsNonZero 
Arguments: {K : Type u_2} {L : Type u_3} [LieRing L] [Field K] [LieAlgebra K L] [FiniteDimensional K L] [LieAlgebra.IsKilling K L] {H : LieSubalgebra K L} [H.IsCartanSubalgebra] [LieModule.IsTriangularizable K (↥H) L] {α : LieModule.Weight K (↥H) L}
Goal: (-α).IsNonZero ↔ α.IsNonZero
-- LieModule.Weight.toLinear_neg
@[simp] lemma toLinear_neg {α : Weight K H L} : (-α).toLinear = -α.toLinear 
Arguments: {K : Type u_2} {L : Type u_3} [LieRing L] [Field K] [LieAlgebra K L] [FiniteDimensional K L] [LieAlgebra.IsKilling K L] {H : LieSubalgebra K L} [H.IsCartanSubalgebra] [LieModule.IsTriangularizable K (↥H) L] {α : LieModule.Weight K (↥H) L}
Goal: LieModule.Weight.toLinear K (↥H) L (-α) = -LieModule.Weight.toLinear K (↥H) L α
-- LieAlgebra.IsKilling.coroot_neg
@[simp]
lemma _root_.LieAlgebra.IsKilling.coroot_neg (α : Weight K H L) : coroot (-α) = -coroot α 
Arguments: {K : Type u_2} {L : Type u_3} [LieRing L] [Field K] [LieAlgebra K L] [FiniteDimensional K L] [LieAlgebra.IsKilling K L] {H : LieSubalgebra K L} [H.IsCartanSubalgebra] [LieModule.IsTriangularizable K (↥H) L] [CharZero K] (α : LieModule.Weight K (↥H) L)
Goal: LieAlgebra.IsKilling.coroot (-α) = -LieAlgebra.IsKilling.coroot α
-- ModuleCat.ker_eq_bot_of_mono
theorem ker_eq_bot_of_mono [Mono f] : LinearMap.ker f = ⊥ 
Arguments: {R : Type u} [Ring R] {X Y : ModuleCat R} (f : X ⟶ Y) [CategoryTheory.Mono f]
Goal: LinearMap.ker f = ⊥
-- ModuleCat.range_eq_top_of_epi
theorem range_eq_top_of_epi [Epi f] : LinearMap.range f = ⊤ 
Arguments: {R : Type u} [Ring R] {X Y : ModuleCat R} (f : X ⟶ Y) [CategoryTheory.Epi f]
Goal: LinearMap.range f = ⊤
-- ModuleCat.mono_iff_ker_eq_bot
theorem mono_iff_ker_eq_bot : Mono f ↔ LinearMap.ker f = ⊥ 
Arguments: {R : Type u} [Ring R] {X Y : ModuleCat R} (f : X ⟶ Y)
Goal: CategoryTheory.Mono f ↔ LinearMap.ker f = ⊥
-- ModuleCat.mono_iff_injective
theorem mono_iff_injective : Mono f ↔ Function.Injective f 
Arguments: {R : Type u} [Ring R] {X Y : ModuleCat R} (f : X ⟶ Y)
Goal: CategoryTheory.Mono f ↔ Function.Injective ⇑f
-- ModuleCat.epi_iff_range_eq_top
theorem epi_iff_range_eq_top : Epi f ↔ LinearMap.range f = ⊤ 
Arguments: {R : Type u} [Ring R] {X Y : ModuleCat R} (f : X ⟶ Y)
Goal: CategoryTheory.Epi f ↔ LinearMap.range f = ⊤
-- ModuleCat.epi_iff_surjective
theorem epi_iff_surjective : Epi f ↔ Function.Surjective f 
Arguments: {R : Type u} [Ring R] {X Y : ModuleCat R} (f : X ⟶ Y)
Goal: CategoryTheory.Epi f ↔ Function.Surjective ⇑f
-- AddMagmaCat.coe_of
@[to_additive (attr := simp)]
theorem coe_of (R : Type u) [Mul R] : (MagmaCat.of R : Type u) = R 
Arguments: (R : Type u) [Add R]
Goal: ↑(AddMagmaCat.of R) = R
-- MagmaCat.coe_of
@[to_additive (attr := simp)]
theorem coe_of (R : Type u) [Mul R] : (MagmaCat.of R : Type u) = R 
Arguments: (R : Type u) [Mul R]
Goal: ↑(MagmaCat.of R) = R
-- MagmaCat.mulEquiv_coe_eq
@[to_additive (attr := simp)]
lemma mulEquiv_coe_eq {X Y : Type _} [Mul X] [Mul Y] (e : X ≃* Y) :
    (@DFunLike.coe (MagmaCat.of X ⟶ MagmaCat.of Y) _ (fun _ => (forget MagmaCat).obj _)
      ConcreteCategory.instFunLike (e : X →ₙ* Y) : X → Y) = ↑e 
Arguments: {X Y : Type u_1} [Mul X] [Mul Y] (e : X ≃* Y)
Goal: ⇑↑e = ⇑e
-- AddMagmaCat.addEquiv_coe_eq
@[to_additive (attr := simp)]
lemma mulEquiv_coe_eq {X Y : Type _} [Mul X] [Mul Y] (e : X ≃* Y) :
    (@DFunLike.coe (MagmaCat.of X ⟶ MagmaCat.of Y) _ (fun _ => (forget MagmaCat).obj _)
      ConcreteCategory.instFunLike (e : X →ₙ* Y) : X → Y) = ↑e 
Arguments: {X Y : Type u_1} [Add X] [Add Y] (e : X ≃+ Y)
Goal: ⇑↑e = ⇑e
-- AddMagmaCat.ofHom_apply
@[to_additive] -- Porting note: simp removed, simpNF says LHS simplifies to itself
theorem ofHom_apply {X Y : Type u} [Mul X] [Mul Y] (f : X →ₙ* Y) (x : X) : ofHom f x = f x 
Arguments: {X Y : Type u} [Add X] [Add Y] (f : AddHom X Y) (x : X)
Goal: (AddMagmaCat.ofHom f) x = f x
-- MagmaCat.ofHom_apply
@[to_additive] -- Porting note: simp removed, simpNF says LHS simplifies to itself
theorem ofHom_apply {X Y : Type u} [Mul X] [Mul Y] (f : X →ₙ* Y) (x : X) : ofHom f x = f x 
Arguments: {X Y : Type u} [Mul X] [Mul Y] (f : X →ₙ* Y) (x : X)
Goal: (MagmaCat.ofHom f) x = f x
-- AddSemigrp.coe_of
@[to_additive (attr := simp)]
theorem coe_of (R : Type u) [Semigroup R] : (Semigrp.of R : Type u) = R 
Arguments: (R : Type u) [AddSemigroup R]
Goal: ↑(AddSemigrp.of R) = R
-- Semigrp.coe_of
@[to_additive (attr := simp)]
theorem coe_of (R : Type u) [Semigroup R] : (Semigrp.of R : Type u) = R 
Arguments: (R : Type u) [Semigroup R]
Goal: ↑(Semigrp.of R) = R
-- Semigrp.mulEquiv_coe_eq
@[to_additive (attr := simp)]
lemma mulEquiv_coe_eq {X Y : Type _} [Semigroup X] [Semigroup Y] (e : X ≃* Y) :
    (@DFunLike.coe (Semigrp.of X ⟶ Semigrp.of Y) _ (fun _ => (forget Semigrp).obj _)
      ConcreteCategory.instFunLike (e : X →ₙ* Y) : X → Y) = ↑e 
Arguments: {X Y : Type u_1} [Semigroup X] [Semigroup Y] (e : X ≃* Y)
Goal: ⇑↑e = ⇑e
-- AddSemigrp.addEquiv_coe_eq
@[to_additive (attr := simp)]
lemma mulEquiv_coe_eq {X Y : Type _} [Semigroup X] [Semigroup Y] (e : X ≃* Y) :
    (@DFunLike.coe (Semigrp.of X ⟶ Semigrp.of Y) _ (fun _ => (forget Semigrp).obj _)
      ConcreteCategory.instFunLike (e : X →ₙ* Y) : X → Y) = ↑e 
Arguments: {X Y : Type u_1} [AddSemigroup X] [AddSemigroup Y] (e : X ≃+ Y)
Goal: ⇑↑e = ⇑e
-- Semigrp.ofHom_apply
@[to_additive] -- Porting note: simp removed, simpNF says LHS simplifies to itself
theorem ofHom_apply {X Y : Type u} [Semigroup X] [Semigroup Y] (f : X →ₙ* Y) (x : X) :
    ofHom f x = f x 
Arguments: {X Y : Type u} [Semigroup X] [Semigroup Y] (f : X →ₙ* Y) (x : X)
Goal: (Semigrp.ofHom f) x = f x
-- AddSemigrp.ofHom_apply
@[to_additive] -- Porting note: simp removed, simpNF says LHS simplifies to itself
theorem ofHom_apply {X Y : Type u} [Semigroup X] [Semigroup Y] (f : X →ₙ* Y) (x : X) :
    ofHom f x = f x 
Arguments: {X Y : Type u} [AddSemigroup X] [AddSemigroup Y] (f : AddHom X Y) (x : X)
Goal: (AddSemigrp.ofHom f) x = f x
-- MulEquiv.toMagmaCatIso_inv_apply
@[to_additive (attr := simps)
      "Build an isomorphism in the category `AddMagmaCat` from an `AddEquiv` between `Add`s."]
def MulEquiv.toMagmaCatIso (e : X ≃* Y) : MagmaCat.of X ≅ MagmaCat.of Y 
Arguments: {X Y : Type u} [Mul X] [Mul Y] (e : X ≃* Y)
Goal: ∀ (a : Y), e.toMagmaCatIso.inv a = e.symm.toFun a
-- AddEquiv.toAddMagmaCatIso_hom_apply
@[to_additive (attr := simps)
      "Build an isomorphism in the category `AddMagmaCat` from an `AddEquiv` between `Add`s."]
def MulEquiv.toMagmaCatIso (e : X ≃* Y) : MagmaCat.of X ≅ MagmaCat.of Y 
Arguments: {X Y : Type u} [Add X] [Add Y] (e : X ≃+ Y)
Goal: ∀ (a : X), e.toAddMagmaCatIso.hom a = e.toFun a
-- MulEquiv.toMagmaCatIso_hom_apply
@[to_additive (attr := simps)
      "Build an isomorphism in the category `AddMagmaCat` from an `AddEquiv` between `Add`s."]
def MulEquiv.toMagmaCatIso (e : X ≃* Y) : MagmaCat.of X ≅ MagmaCat.of Y 
Arguments: {X Y : Type u} [Mul X] [Mul Y] (e : X ≃* Y)
Goal: ∀ (a : X), e.toMagmaCatIso.hom a = e.toFun a
-- AddEquiv.toAddMagmaCatIso_inv_apply
@[to_additive (attr := simps)
      "Build an isomorphism in the category `AddMagmaCat` from an `AddEquiv` between `Add`s."]
def MulEquiv.toMagmaCatIso (e : X ≃* Y) : MagmaCat.of X ≅ MagmaCat.of Y 
Arguments: {X Y : Type u} [Add X] [Add Y] (e : X ≃+ Y)
Goal: ∀ (a : Y), e.toAddMagmaCatIso.inv a = e.symm.toFun a
-- AddEquiv.toAddSemigrpIso_inv_apply
@[to_additive (attr := simps)
  "Build an isomorphism in the category
  `AddSemigroup` from an `AddEquiv` between `AddSemigroup`s."]
def MulEquiv.toSemigrpIso (e : X ≃* Y) : Semigrp.of X ≅ Semigrp.of Y 
Arguments: {X Y : Type u} [AddSemigroup X] [AddSemigroup Y] (e : X ≃+ Y)
Goal: ∀ (a : Y), e.toAddSemigrpIso.inv a = e.symm.toFun a
-- MulEquiv.toSemigrpIso_inv_apply
@[to_additive (attr := simps)
  "Build an isomorphism in the category
  `AddSemigroup` from an `AddEquiv` between `AddSemigroup`s."]
def MulEquiv.toSemigrpIso (e : X ≃* Y) : Semigrp.of X ≅ Semigrp.of Y 
Arguments: {X Y : Type u} [Semigroup X] [Semigroup Y] (e : X ≃* Y)
Goal: ∀ (a : Y), e.toSemigrpIso.inv a = e.symm.toFun a
-- AddEquiv.toAddSemigrpIso_hom_apply
@[to_additive (attr := simps)
  "Build an isomorphism in the category
  `AddSemigroup` from an `AddEquiv` between `AddSemigroup`s."]
def MulEquiv.toSemigrpIso (e : X ≃* Y) : Semigrp.of X ≅ Semigrp.of Y 
Arguments: {X Y : Type u} [AddSemigroup X] [AddSemigroup Y] (e : X ≃+ Y)
Goal: ∀ (a : X), e.toAddSemigrpIso.hom a = e.toFun a
-- MulEquiv.toSemigrpIso_hom_apply
@[to_additive (attr := simps)
  "Build an isomorphism in the category
  `AddSemigroup` from an `AddEquiv` between `AddSemigroup`s."]
def MulEquiv.toSemigrpIso (e : X ≃* Y) : Semigrp.of X ≅ Semigrp.of Y 
Arguments: {X Y : Type u} [Semigroup X] [Semigroup Y] (e : X ≃* Y)
Goal: ∀ (a : X), e.toSemigrpIso.hom a = e.toFun a
-- HopfAlgebraCat.of_str
@[simps]
def of (X : Type v) [Ring X] [HopfAlgebra R X] :
    HopfAlgebraCat R 
Arguments: (R : Type u) [CommRing R] (X : Type v) [Ring X] [HopfAlgebra R X]
Goal: (HopfAlgebraCat.of R X).str = inferInstance
-- HopfAlgebraCat.of_instHopfAlgebra
@[simps]
def of (X : Type v) [Ring X] [HopfAlgebra R X] :
    HopfAlgebraCat R 
Arguments: (R : Type u) [CommRing R] (X : Type v) [Ring X] [HopfAlgebra R X]
Goal: (HopfAlgebraCat.of R X).instHopfAlgebra = inferInstance
-- HopfAlgebraCat.of_α
@[simps]
def of (X : Type v) [Ring X] [HopfAlgebra R X] :
    HopfAlgebraCat R 
Arguments: (R : Type u) [CommRing R] (X : Type v) [Ring X] [HopfAlgebra R X]
Goal: ↑(HopfAlgebraCat.of R X).toBundled = X
-- HopfAlgebraCat.of_comul
@[simp]
lemma of_comul {X : Type v} [Ring X] [HopfAlgebra R X] :
    Coalgebra.comul (A := of R X) = Coalgebra.comul (R := R) (A := X) 
Arguments: {R : Type u} [CommRing R] {X : Type v} [Ring X] [HopfAlgebra R X]
Goal: Coalgebra.comul = Coalgebra.comul
-- HopfAlgebraCat.of_counit
@[simp]
lemma of_counit {X : Type v} [Ring X] [HopfAlgebra R X] :
    Coalgebra.counit (A := of R X) = Coalgebra.counit (R := R) (A := X) 
Arguments: {R : Type u} [CommRing R] {X : Type v} [Ring X] [HopfAlgebra R X]
Goal: Coalgebra.counit = Coalgebra.counit
-- HopfAlgebraCat.Hom.toBialgHom_injective
lemma Hom.toBialgHom_injective (V W : HopfAlgebraCat.{v} R) :
    Function.Injective (Hom.toBialgHom : Hom V W → _) 
Arguments: {R : Type u} [CommRing R] (V W : HopfAlgebraCat R)
Goal: Function.Injective HopfAlgebraCat.Hom.toBialgHom
-- HopfAlgebraCat.hom_ext
@[ext]
lemma hom_ext {X Y : HopfAlgebraCat.{v} R} (f g : X ⟶ Y) (h : f.toBialgHom = g.toBialgHom) :
    f = g 
Arguments: {R : Type u} [CommRing R] {X Y : HopfAlgebraCat R} (f g : X ⟶ Y) (h : f.toBialgHom = g.toBialgHom)
Goal: f = g
-- HopfAlgebraCat.toBialgHom_comp
@[simp] theorem toBialgHom_comp {X Y Z : HopfAlgebraCat.{v} R} (f : X ⟶ Y) (g : Y ⟶ Z) :
    (f ≫ g).toBialgHom = g.toBialgHom.comp f.toBialgHom 
Arguments: {R : Type u} [CommRing R] {X Y Z : HopfAlgebraCat R} (f : X ⟶ Y) (g : Y ⟶ Z)
Goal: (f ≫ g).toBialgHom = g.toBialgHom.comp f.toBialgHom
-- HopfAlgebraCat.toBialgHom_id
@[simp] theorem toBialgHom_id {M : HopfAlgebraCat.{v} R} :
    Hom.toBialgHom (𝟙 M) = BialgHom.id _ _ 
Arguments: {R : Type u} [CommRing R] {M : HopfAlgebraCat R}
Goal: (𝟙 M).toBialgHom = BialgHom.id R ↑M.toBundled