ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Matrix.IsHermitian.inv
theorem IsHermitian.inv [Fintype m] [DecidableEq m] {A : Matrix m m α} (hA : A.IsHermitian) :
    A⁻¹.IsHermitian 
Arguments: {α : Type u_1} {m : Type u_3} [CommRing α] [StarRing α] [Fintype m] [DecidableEq m] {A : Matrix m m α} (hA : A.IsHermitian)
Goal: A⁻¹.IsHermitian
-- Matrix.isHermitian_inv
@[simp]
theorem isHermitian_inv [Fintype m] [DecidableEq m] (A : Matrix m m α) [Invertible A] :
    A⁻¹.IsHermitian ↔ A.IsHermitian 
Arguments: {α : Type u_1} {m : Type u_3} [CommRing α] [StarRing α] [Fintype m] [DecidableEq m] (A : Matrix m m α) [Invertible A]
Goal: A⁻¹.IsHermitian ↔ A.IsHermitian
-- Matrix.IsHermitian.adjugate
theorem IsHermitian.adjugate [Fintype m] [DecidableEq m] {A : Matrix m m α} (hA : A.IsHermitian) :
    A.adjugate.IsHermitian 
Arguments: {α : Type u_1} {m : Type u_3} [CommRing α] [StarRing α] [Fintype m] [DecidableEq m] {A : Matrix m m α} (hA : A.IsHermitian)
Goal: A.adjugate.IsHermitian
-- Matrix.IsHermitian.zpow
theorem IsHermitian.zpow [Fintype m] [DecidableEq m] {A : Matrix m m α} (h : A.IsHermitian)
    (k : ℤ) :
    (A ^ k).IsHermitian 
Arguments: {α : Type u_1} {m : Type u_3} [CommRing α] [StarRing α] [Fintype m] [DecidableEq m] {A : Matrix m m α} (h : A.IsHermitian) (k : ℤ)
Goal: (A ^ k).IsHermitian
-- Matrix.IsHermitian.coe_re_apply_self
theorem IsHermitian.coe_re_apply_self {A : Matrix n n α} (h : A.IsHermitian) (i : n) :
    (re (A i i) : α) = A i i 
Arguments: {α : Type u_1} {n : Type u_4} [RCLike α] {A : Matrix n n α} (h : A.IsHermitian) (i : n)
Goal: ↑(RCLike.re (A i i)) = A i i
-- Matrix.IsHermitian.coe_re_diag
theorem IsHermitian.coe_re_diag {A : Matrix n n α} (h : A.IsHermitian) :
    (fun i => (re (A.diag i) : α)) = A.diag 
Arguments: {α : Type u_1} {n : Type u_4} [RCLike α] {A : Matrix n n α} (h : A.IsHermitian)
Goal: (fun i => ↑(RCLike.re (A.diag i))) = A.diag
-- Matrix.isHermitian_iff_isSymmetric
theorem isHermitian_iff_isSymmetric [Fintype n] [DecidableEq n] {A : Matrix n n α} :
    IsHermitian A ↔ A.toEuclideanLin.IsSymmetric 
Arguments: {α : Type u_1} {n : Type u_4} [RCLike α] [Fintype n] [DecidableEq n] {A : Matrix n n α}
Goal: A.IsHermitian ↔ (Matrix.toEuclideanLin A).IsSymmetric
-- Finsupp.smul_sum
theorem smul_sum [Zero β] [AddCommMonoid M] [DistribSMul R M] {v : α →₀ β} {c : R} {h : α → β → M} :
    c • v.sum h = v.sum fun a b => c • h a b 
Arguments: {α : Type u_1} {β : Type u_2} {R : Type u_3} {M : Type u_4} [Zero β] [AddCommMonoid M] [DistribSMul R M] {v : α →₀ β} {c : R} {h : α → β → M}
Goal: c • v.sum h = v.sum fun a b => c • h a b
-- Finsupp.sum_smul_index_linearMap'
@[simp]
theorem sum_smul_index_linearMap' [Semiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid M₂]
    [Module R M₂] {v : α →₀ M} {c : R} {h : α → M →ₗ[R] M₂} :
    ((c • v).sum fun a => h a) = c • v.sum fun a => h a 
Arguments: {α : Type u_1} {R : Type u_3} {M : Type u_4} {M₂ : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid M₂] [Module R M₂] {v : α →₀ M} {c : R} {h : α → M →ₗ[R] M₂}
Goal: ((c • v).sum fun a => ⇑(h a)) = c • v.sum fun a => ⇑(h a)
-- Finsupp.linearEquivFunOnFinite_apply
@[simps apply]
noncomputable def linearEquivFunOnFinite : (α →₀ M) ≃ₗ[R] α → M 
Arguments: (R : Type u_1) (M : Type u_3) (α : Type u_4) [Finite α] [AddCommMonoid M] [Semiring R] [Module R M]
Goal: ∀ (a : α →₀ M) (a_1 : α), (Finsupp.linearEquivFunOnFinite R M α) a a_1 = a a_1
-- Finsupp.linearEquivFunOnFinite_single
@[simp]
theorem linearEquivFunOnFinite_single [DecidableEq α] (x : α) (m : M) :
    (linearEquivFunOnFinite R M α) (single x m) = Pi.single x m 
Arguments: (R : Type u_1) (M : Type u_3) (α : Type u_4) [Finite α] [AddCommMonoid M] [Semiring R] [Module R M] [DecidableEq α] (x : α) (m : M)
Goal: (Finsupp.linearEquivFunOnFinite R M α) (Finsupp.single x m) = Pi.single x m
-- Finsupp.linearEquivFunOnFinite_symm_single
@[simp]
theorem linearEquivFunOnFinite_symm_single [DecidableEq α] (x : α) (m : M) :
    (linearEquivFunOnFinite R M α).symm (Pi.single x m) = single x m 
Arguments: (R : Type u_1) (M : Type u_3) (α : Type u_4) [Finite α] [AddCommMonoid M] [Semiring R] [Module R M] [DecidableEq α] (x : α) (m : M)
Goal: (Finsupp.linearEquivFunOnFinite R M α).symm (Pi.single x m) = Finsupp.single x m
-- Finsupp.linearEquivFunOnFinite_symm_coe
@[simp]
theorem linearEquivFunOnFinite_symm_coe (f : α →₀ M) : (linearEquivFunOnFinite R M α).symm f = f 
Arguments: (R : Type u_1) (M : Type u_3) (α : Type u_4) [Finite α] [AddCommMonoid M] [Semiring R] [Module R M] (f : α →₀ M)
Goal: (Finsupp.linearEquivFunOnFinite R M α).symm ⇑f = f
-- Finsupp.LinearEquiv.finsuppUnique_apply
@[simp]
theorem LinearEquiv.finsuppUnique_apply (f : α →₀ M) :
    LinearEquiv.finsuppUnique R M α f = f default 
Arguments: {R : Type u_1} {M : Type u_3} [AddCommMonoid M] [Semiring R] [Module R M] (α : Type u_4) [Unique α] (f : α →₀ M)
Goal: (Finsupp.LinearEquiv.finsuppUnique R M α) f = f default
-- Finsupp.LinearEquiv.finsuppUnique_symm_apply
@[simp]
theorem LinearEquiv.finsuppUnique_symm_apply [Unique α] (m : M) :
    (LinearEquiv.finsuppUnique R M α).symm m = Finsupp.single default m 
Arguments: {R : Type u_1} {M : Type u_3} [AddCommMonoid M] [Semiring R] [Module R M] {α : Type u_4} [Unique α] (m : M)
Goal: (Finsupp.LinearEquiv.finsuppUnique R M α).symm m = Finsupp.single default m
-- Finsupp.lhom_ext
theorem lhom_ext ⦃φ ψ : (α →₀ M) →ₗ[R] N⦄ (h : ∀ a b, φ (single a b) = ψ (single a b)) : φ = ψ 
Arguments: {α : Type u_1} {M : Type u_2} {N : Type u_3} {R : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid N] [Module R N] ⦃φ ψ : (α →₀ M) →ₗ[R] N⦄ (h : ∀ (a : α) (b : M), φ (Finsupp.single a b) = ψ (Finsupp.single a b))
Goal: φ = ψ
-- Finsupp.lhom_ext'
-- Porting note: The priority should be higher than `LinearMap.ext`.
@[ext high]
theorem lhom_ext' ⦃φ ψ : (α →₀ M) →ₗ[R] N⦄ (h : ∀ a, φ.comp (lsingle a) = ψ.comp (lsingle a)) :
    φ = ψ 
Arguments: {α : Type u_1} {M : Type u_2} {N : Type u_3} {R : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid N] [Module R N] ⦃φ ψ : (α →₀ M) →ₗ[R] N⦄ (h : ∀ (a : α), φ ∘ₗ Finsupp.lsingle a = ψ ∘ₗ Finsupp.lsingle a)
Goal: φ = ψ
-- Finsupp.lcoeFun_apply
@[simps]
def lcoeFun : (α →₀ M) →ₗ[R] α → M 
Arguments: {α : Type u_1} {M : Type u_2} {R : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M]
Goal: ∀ (a : α →₀ M) (a_1 : α), Finsupp.lcoeFun a a_1 = a a_1
-- Finsupp.lsubtypeDomain_apply
theorem lsubtypeDomain_apply (f : α →₀ M) :
    (lsubtypeDomain s : (α →₀ M) →ₗ[R] s →₀ M) f = subtypeDomain (fun x => x ∈ s) f 
Arguments: {α : Type u_1} {M : Type u_2} {R : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M] (s : Set α) (f : α →₀ M)
Goal: (Finsupp.lsubtypeDomain s) f = Finsupp.subtypeDomain (fun x => x ∈ s) f
-- Finsupp.lsingle_apply
@[simp]
theorem lsingle_apply (a : α) (b : M) : (lsingle a : M →ₗ[R] α →₀ M) b = single a b 
Arguments: {α : Type u_1} {M : Type u_2} {R : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M] (a : α) (b : M)
Goal: (Finsupp.lsingle a) b = Finsupp.single a b
-- Finsupp.lapply_apply
@[simp]
theorem lapply_apply (a : α) (f : α →₀ M) : (lapply a : (α →₀ M) →ₗ[R] M) f = f a 
Arguments: {α : Type u_1} {M : Type u_2} {R : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M] (a : α) (f : α →₀ M)
Goal: (Finsupp.lapply a) f = f a
-- Finsupp.lapply_comp_lsingle_same
@[simp]
theorem lapply_comp_lsingle_same (a : α) : lapply a ∘ₗ lsingle a = (.id : M →ₗ[R] M) 
Arguments: {α : Type u_1} {M : Type u_2} {R : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M] (a : α)
Goal: Finsupp.lapply a ∘ₗ Finsupp.lsingle a = LinearMap.id
-- Finsupp.lapply_comp_lsingle_of_ne
@[simp]
theorem lapply_comp_lsingle_of_ne (a a' : α) (h : a ≠ a') :
    lapply a ∘ₗ lsingle a' = (0 : M →ₗ[R] M) 
Arguments: {α : Type u_1} {M : Type u_2} {R : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M] (a a' : α) (h : a ≠ a')
Goal: Finsupp.lapply a ∘ₗ Finsupp.lsingle a' = 0
-- Finsupp.ker_lsingle
@[simp]
theorem ker_lsingle (a : α) : ker (lsingle a : M →ₗ[R] α →₀ M) = ⊥ 
Arguments: {α : Type u_1} {M : Type u_2} {R : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M] (a : α)
Goal: LinearMap.ker (Finsupp.lsingle a) = ⊥
-- Finsupp.lsingle_range_le_ker_lapply
theorem lsingle_range_le_ker_lapply (s t : Set α) (h : Disjoint s t) :
    ⨆ a ∈ s, LinearMap.range (lsingle a : M →ₗ[R] α →₀ M) ≤
      ⨅ a ∈ t, ker (lapply a : (α →₀ M) →ₗ[R] M) 
Arguments: {α : Type u_1} {M : Type u_2} {R : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M] (s t : Set α) (h : Disjoint s t)
Goal: ⨆ a ∈ s, LinearMap.range (Finsupp.lsingle a) ≤ ⨅ a ∈ t, LinearMap.ker (Finsupp.lapply a)
-- Finsupp.iInf_ker_lapply_le_bot
theorem iInf_ker_lapply_le_bot : ⨅ a, ker (lapply a : (α →₀ M) →ₗ[R] M) ≤ ⊥ 
Arguments: {α : Type u_1} {M : Type u_2} {R : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M]
Goal: ⨅ a, LinearMap.ker (Finsupp.lapply a) ≤ ⊥
-- Finsupp.iSup_lsingle_range
theorem iSup_lsingle_range : ⨆ a, LinearMap.range (lsingle a : M →ₗ[R] α →₀ M) = ⊤ 
Arguments: {α : Type u_1} {M : Type u_2} {R : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M]
Goal: ⨆ a, LinearMap.range (Finsupp.lsingle a) = ⊤
-- Finsupp.disjoint_lsingle_lsingle
theorem disjoint_lsingle_lsingle (s t : Set α) (hs : Disjoint s t) :
    Disjoint (⨆ a ∈ s, LinearMap.range (lsingle a : M →ₗ[R] α →₀ M))
      (⨆ a ∈ t, LinearMap.range (lsingle a : M →ₗ[R] α →₀ M)) 
Arguments: {α : Type u_1} {M : Type u_2} {R : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M] (s t : Set α) (hs : Disjoint s t)
Goal: Disjoint (⨆ a ∈ s, LinearMap.range (Finsupp.lsingle a)) (⨆ a ∈ t, LinearMap.range (Finsupp.lsingle a))
-- Finsupp.span_single_image
theorem span_single_image (s : Set M) (a : α) :
    Submodule.span R (single a '' s) = (Submodule.span R s).map (lsingle a : M →ₗ[R] α →₀ M) 
Arguments: {α : Type u_1} {M : Type u_2} {R : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M] (s : Set M) (a : α)
Goal: Submodule.span R (Finsupp.single a '' s) = Submodule.map (Finsupp.lsingle a) (Submodule.span R s)
-- Finsupp.mem_supported
theorem mem_supported {s : Set α} (p : α →₀ M) : p ∈ supported M R s ↔ ↑p.support ⊆ s 
Arguments: {α : Type u_1} {M : Type u_2} (R : Type u_5) [Semiring R] [AddCommMonoid M] [Module R M] {s : Set α} (p : α →₀ M)
Goal: p ∈ Finsupp.supported M R s ↔ ↑p.support ⊆ s
-- Finsupp.mem_supported'
theorem mem_supported' {s : Set α} (p : α →₀ M) :
    p ∈ supported M R s ↔ ∀ x ∉ s, p x = 0 
Arguments: {α : Type u_1} {M : Type u_2} (R : Type u_5) [Semiring R] [AddCommMonoid M] [Module R M] {s : Set α} (p : α →₀ M)
Goal: p ∈ Finsupp.supported M R s ↔ ∀ x ∉ s, p x = 0
-- Finsupp.mem_supported_support
theorem mem_supported_support (p : α →₀ M) : p ∈ Finsupp.supported M R (p.support : Set α) 
Arguments: {α : Type u_1} {M : Type u_2} (R : Type u_5) [Semiring R] [AddCommMonoid M] [Module R M] (p : α →₀ M)
Goal: p ∈ Finsupp.supported M R ↑p.support
-- Finsupp.single_mem_supported
theorem single_mem_supported {s : Set α} {a : α} (b : M) (h : a ∈ s) :
    single a b ∈ supported M R s 
Arguments: {α : Type u_1} {M : Type u_2} (R : Type u_5) [Semiring R] [AddCommMonoid M] [Module R M] {s : Set α} {a : α} (b : M) (h : a ∈ s)
Goal: Finsupp.single a b ∈ Finsupp.supported M R s
-- Finsupp.supported_eq_span_single
theorem supported_eq_span_single (s : Set α) :
    supported R R s = span R ((fun i => single i 1) '' s) 
Arguments: {α : Type u_1} (R : Type u_5) [Semiring R] (s : Set α)
Goal: Finsupp.supported R R s = Submodule.span R ((fun i => Finsupp.single i 1) '' s)
-- Finsupp.restrictDom_apply
@[simp]
theorem restrictDom_apply (s : Set α) (l : α →₀ M) [DecidablePred (· ∈ s)] :
    (restrictDom M R s l : α →₀ M) = Finsupp.filter (· ∈ s) l 
Arguments: {α : Type u_1} {M : Type u_2} {R : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M] (s : Set α) (l : α →₀ M) [DecidablePred fun x => x ∈ s]
Goal: ↑((Finsupp.restrictDom M R s) l) = Finsupp.filter (fun x => x ∈ s) l
-- Finsupp.restrictDom_comp_subtype
theorem restrictDom_comp_subtype (s : Set α) [DecidablePred (· ∈ s)] :
    (restrictDom M R s).comp (Submodule.subtype _) = LinearMap.id 
Arguments: {α : Type u_1} {M : Type u_2} {R : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M] (s : Set α) [DecidablePred fun x => x ∈ s]
Goal: Finsupp.restrictDom M R s ∘ₗ (Finsupp.supported M R s).subtype = LinearMap.id
-- Finsupp.range_restrictDom
theorem range_restrictDom (s : Set α) [DecidablePred (· ∈ s)] :
    LinearMap.range (restrictDom M R s) = ⊤ 
Arguments: {α : Type u_1} {M : Type u_2} {R : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M] (s : Set α) [DecidablePred fun x => x ∈ s]
Goal: LinearMap.range (Finsupp.restrictDom M R s) = ⊤
-- Finsupp.supported_mono
theorem supported_mono {s t : Set α} (st : s ⊆ t) : supported M R s ≤ supported M R t 
Arguments: {α : Type u_1} {M : Type u_2} {R : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M] {s t : Set α} (st : s ⊆ t)
Goal: Finsupp.supported M R s ≤ Finsupp.supported M R t
-- Finsupp.supported_empty
@[simp]
theorem supported_empty : supported M R (∅ : Set α) = ⊥ 
Arguments: {α : Type u_1} {M : Type u_2} {R : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M]
Goal: Finsupp.supported M R ∅ = ⊥
-- Finsupp.supported_univ
@[simp]
theorem supported_univ : supported M R (Set.univ : Set α) = ⊤ 
Arguments: {α : Type u_1} {M : Type u_2} {R : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M]
Goal: Finsupp.supported M R Set.univ = ⊤
-- Finsupp.supported_iUnion
theorem supported_iUnion {δ : Type*} (s : δ → Set α) :
    supported M R (⋃ i, s i) = ⨆ i, supported M R (s i) 
Arguments: {α : Type u_1} {M : Type u_2} {R : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M] {δ : Type u_7} (s : δ → Set α)
Goal: Finsupp.supported M R (⋃ i, s i) = ⨆ i, Finsupp.supported M R (s i)
-- Finsupp.supported_union
theorem supported_union (s t : Set α) :
    supported M R (s ∪ t) = supported M R s ⊔ supported M R t 
Arguments: {α : Type u_1} {M : Type u_2} {R : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M] (s t : Set α)
Goal: Finsupp.supported M R (s ∪ t) = Finsupp.supported M R s ⊔ Finsupp.supported M R t
-- Finsupp.supported_iInter
theorem supported_iInter {ι : Type*} (s : ι → Set α) :
    supported M R (⋂ i, s i) = ⨅ i, supported M R (s i) 
Arguments: {α : Type u_1} {M : Type u_2} {R : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M] {ι : Type u_7} (s : ι → Set α)
Goal: Finsupp.supported M R (⋂ i, s i) = ⨅ i, Finsupp.supported M R (s i)
-- Finsupp.supported_inter
theorem supported_inter (s t : Set α) :
    supported M R (s ∩ t) = supported M R s ⊓ supported M R t 
Arguments: {α : Type u_1} {M : Type u_2} {R : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M] (s t : Set α)
Goal: Finsupp.supported M R (s ∩ t) = Finsupp.supported M R s ⊓ Finsupp.supported M R t
-- Finsupp.disjoint_supported_supported
theorem disjoint_supported_supported {s t : Set α} (h : Disjoint s t) :
    Disjoint (supported M R s) (supported M R t) 
Arguments: {α : Type u_1} {M : Type u_2} {R : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M] {s t : Set α} (h : Disjoint s t)
Goal: Disjoint (Finsupp.supported M R s) (Finsupp.supported M R t)
-- Finsupp.disjoint_supported_supported_iff
theorem disjoint_supported_supported_iff [Nontrivial M] {s t : Set α} :
    Disjoint (supported M R s) (supported M R t) ↔ Disjoint s t 
Arguments: {α : Type u_1} {M : Type u_2} {R : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M] [Nontrivial M] {s t : Set α}
Goal: Disjoint (Finsupp.supported M R s) (Finsupp.supported M R t) ↔ Disjoint s t
-- Finsupp.coe_lsum
@[simp]
theorem coe_lsum (f : α → M →ₗ[R] N) : (lsum S f : (α →₀ M) → N) = fun d => d.sum fun i => f i 
Arguments: {α : Type u_1} {M : Type u_2} {N : Type u_3} {R : Type u_5} (S : Type u_6) [Semiring R] [Semiring S] [AddCommMonoid M] [Module R M] [AddCommMonoid N] [Module R N] [Module S N] [SMulCommClass R S N] (f : α → M →ₗ[R] N)
Goal: ⇑((Finsupp.lsum S) f) = fun d => d.sum fun i => ⇑(f i)
-- Finsupp.lsum_apply
theorem lsum_apply (f : α → M →ₗ[R] N) (l : α →₀ M) : Finsupp.lsum S f l = l.sum fun b => f b 
Arguments: {α : Type u_1} {M : Type u_2} {N : Type u_3} {R : Type u_5} (S : Type u_6) [Semiring R] [Semiring S] [AddCommMonoid M] [Module R M] [AddCommMonoid N] [Module R N] [Module S N] [SMulCommClass R S N] (f : α → M →ₗ[R] N) (l : α →₀ M)
Goal: ((Finsupp.lsum S) f) l = l.sum fun b => ⇑(f b)
-- Finsupp.lsum_single
theorem lsum_single (f : α → M →ₗ[R] N) (i : α) (m : M) :
    Finsupp.lsum S f (Finsupp.single i m) = f i m 
Arguments: {α : Type u_1} {M : Type u_2} {N : Type u_3} {R : Type u_5} (S : Type u_6) [Semiring R] [Semiring S] [AddCommMonoid M] [Module R M] [AddCommMonoid N] [Module R N] [Module S N] [SMulCommClass R S N] (f : α → M →ₗ[R] N) (i : α) (m : M)
Goal: ((Finsupp.lsum S) f) (Finsupp.single i m) = (f i) m
-- Finsupp.lsum_comp_lsingle
@[simp] theorem lsum_comp_lsingle (f : α → M →ₗ[R] N) (i : α) :
    Finsupp.lsum S f ∘ₗ lsingle i = f i 
Arguments: {α : Type u_1} {M : Type u_2} {N : Type u_3} {R : Type u_5} (S : Type u_6) [Semiring R] [Semiring S] [AddCommMonoid M] [Module R M] [AddCommMonoid N] [Module R N] [Module S N] [SMulCommClass R S N] (f : α → M →ₗ[R] N) (i : α)
Goal: (Finsupp.lsum S) f ∘ₗ Finsupp.lsingle i = f i
-- Finsupp.lsum_symm_apply
theorem lsum_symm_apply (f : (α →₀ M) →ₗ[R] N) (x : α) : (lsum S).symm f x = f.comp (lsingle x) 
Arguments: {α : Type u_1} {M : Type u_2} {N : Type u_3} {R : Type u_5} (S : Type u_6) [Semiring R] [Semiring S] [AddCommMonoid M] [Module R M] [AddCommMonoid N] [Module R N] [Module S N] [SMulCommClass R S N] (f : (α →₀ M) →ₗ[R] N) (x : α)
Goal: (Finsupp.lsum S).symm f x = f ∘ₗ Finsupp.lsingle x
-- Finsupp.lift_symm_apply
@[simp]
theorem lift_symm_apply (f) (x) : ((lift M R X).symm f) x = f (single x 1) 
Arguments: (M : Type u_2) (R : Type u_5) [Semiring R] [AddCommMonoid M] [Module R M] (X : Type u_7) (f : (X →₀ R) →ₗ[R] M) (x : X)
Goal: (Finsupp.lift M R X).symm f x = f (Finsupp.single x 1)
-- Finsupp.lift_apply
@[simp]
theorem lift_apply (f) (g) : ((lift M R X) f) g = g.sum fun x r => r • f x 
Arguments: (M : Type u_2) (R : Type u_5) [Semiring R] [AddCommMonoid M] [Module R M] (X : Type u_7) (f : X → M) (g : X →₀ R)
Goal: ((Finsupp.lift M R X) f) g = g.sum fun x r => r • f x
-- Finsupp.llift_apply
@[simp]
theorem llift_apply (f : X → M) (x : X →₀ R) : llift M R S X f x = lift M R X f x 
Arguments: (M : Type u_2) (R : Type u_5) (S : Type u_6) [Semiring R] [Semiring S] [AddCommMonoid M] [Module R M] (X : Type u_7) [Module S M] [SMulCommClass R S M] (f : X → M) (x : X →₀ R)
Goal: ((Finsupp.llift M R S X) f) x = ((Finsupp.lift M R X) f) x
-- Finsupp.llift_symm_apply
@[simp]
theorem llift_symm_apply (f : (X →₀ R) →ₗ[R] M) (x : X) :
    (llift M R S X).symm f x = f (single x 1) 
Arguments: (M : Type u_2) (R : Type u_5) (S : Type u_6) [Semiring R] [Semiring S] [AddCommMonoid M] [Module R M] (X : Type u_7) [Module S M] [SMulCommClass R S M] (f : (X →₀ R) →ₗ[R] M) (x : X)
Goal: (Finsupp.llift M R S X).symm f x = f (Finsupp.single x 1)
-- Finsupp.lmapDomain_apply
@[simp]
theorem lmapDomain_apply (f : α → α') (l : α →₀ M) :
    (lmapDomain M R f : (α →₀ M) →ₗ[R] α' →₀ M) l = mapDomain f l 
Arguments: {α : Type u_1} (M : Type u_2) (R : Type u_5) [Semiring R] [AddCommMonoid M] [Module R M] {α' : Type u_7} (f : α → α') (l : α →₀ M)
Goal: (Finsupp.lmapDomain M R f) l = Finsupp.mapDomain f l
-- Finsupp.lmapDomain_id
@[simp]
theorem lmapDomain_id : (lmapDomain M R _root_.id : (α →₀ M) →ₗ[R] α →₀ M) = LinearMap.id 
Arguments: {α : Type u_1} (M : Type u_2) (R : Type u_5) [Semiring R] [AddCommMonoid M] [Module R M]
Goal: Finsupp.lmapDomain M R id = LinearMap.id
-- Finsupp.lmapDomain_comp
theorem lmapDomain_comp (f : α → α') (g : α' → α'') :
    lmapDomain M R (g ∘ f) = (lmapDomain M R g).comp (lmapDomain M R f) 
Arguments: {α : Type u_1} (M : Type u_2) (R : Type u_5) [Semiring R] [AddCommMonoid M] [Module R M] {α' : Type u_7} {α'' : Type u_8} (f : α → α') (g : α' → α'')
Goal: Finsupp.lmapDomain M R (g ∘ f) = Finsupp.lmapDomain M R g ∘ₗ Finsupp.lmapDomain M R f
-- Finsupp.supported_comap_lmapDomain
theorem supported_comap_lmapDomain (f : α → α') (s : Set α') :
    supported M R (f ⁻¹' s) ≤ (supported M R s).comap (lmapDomain M R f) 
Arguments: {α : Type u_1} (M : Type u_2) (R : Type u_5) [Semiring R] [AddCommMonoid M] [Module R M] {α' : Type u_7} (f : α → α') (s : Set α')
Goal: Finsupp.supported M R (f ⁻¹' s) ≤ Submodule.comap (Finsupp.lmapDomain M R f) (Finsupp.supported M R s)
-- Finsupp.lmapDomain_supported
theorem lmapDomain_supported (f : α → α') (s : Set α) :
    (supported M R s).map (lmapDomain M R f) = supported M R (f '' s) 
Arguments: {α : Type u_1} (M : Type u_2) (R : Type u_5) [Semiring R] [AddCommMonoid M] [Module R M] {α' : Type u_7} (f : α → α') (s : Set α)
Goal: Submodule.map (Finsupp.lmapDomain M R f) (Finsupp.supported M R s) = Finsupp.supported M R (f '' s)
-- Finsupp.lmapDomain_disjoint_ker
theorem lmapDomain_disjoint_ker (f : α → α') {s : Set α}
    (H : ∀ a ∈ s, ∀ b ∈ s, f a = f b → a = b) :
    Disjoint (supported M R s) (ker (lmapDomain M R f)) 
Arguments: {α : Type u_1} (M : Type u_2) (R : Type u_5) [Semiring R] [AddCommMonoid M] [Module R M] {α' : Type u_7} (f : α → α') {s : Set α} (H : ∀ a ∈ s, ∀ b ∈ s, f a = f b → a = b)
Goal: Disjoint (Finsupp.supported M R s) (LinearMap.ker (Finsupp.lmapDomain M R f))
-- Finsupp.total_apply
theorem total_apply (l : α →₀ R) : Finsupp.total α M R v l = l.sum fun i a => a • v i 
Arguments: {α : Type u_1} {M : Type u_2} (R : Type u_5) [Semiring R] [AddCommMonoid M] [Module R M] {v : α → M} (l : α →₀ R)
Goal: (Finsupp.total α M R v) l = l.sum fun i a => a • v i
-- Finsupp.total_apply_of_mem_supported
theorem total_apply_of_mem_supported {l : α →₀ R} {s : Finset α}
    (hs : l ∈ supported R R (↑s : Set α)) : Finsupp.total α M R v l = s.sum fun i => l i • v i 
Arguments: {α : Type u_1} {M : Type u_2} (R : Type u_5) [Semiring R] [AddCommMonoid M] [Module R M] {v : α → M} {l : α →₀ R} {s : Finset α} (hs : l ∈ Finsupp.supported R R ↑s)
Goal: (Finsupp.total α M R v) l = ∑ i ∈ s, l i • v i
-- Finsupp.total_single
@[simp]
theorem total_single (c : R) (a : α) : Finsupp.total α M R v (single a c) = c • v a 
Arguments: {α : Type u_1} {M : Type u_2} (R : Type u_5) [Semiring R] [AddCommMonoid M] [Module R M] {v : α → M} (c : R) (a : α)
Goal: (Finsupp.total α M R v) (Finsupp.single a c) = c • v a
-- Finsupp.total_zero_apply
theorem total_zero_apply (x : α →₀ R) : (Finsupp.total α M R 0) x = 0 
Arguments: {α : Type u_1} {M : Type u_2} (R : Type u_5) [Semiring R] [AddCommMonoid M] [Module R M] (x : α →₀ R)
Goal: (Finsupp.total α M R 0) x = 0
-- Finsupp.total_zero
@[simp]
theorem total_zero : Finsupp.total α M R 0 = 0 
Arguments: (α : Type u_1) (M : Type u_2) (R : Type u_5) [Semiring R] [AddCommMonoid M] [Module R M]
Goal: Finsupp.total α M R 0 = 0
-- Finsupp.apply_total
theorem apply_total (f : M →ₗ[R] M') (v) (l : α →₀ R) :
    f (Finsupp.total α M R v l) = Finsupp.total α M' R (f ∘ v) l 
Arguments: {α : Type u_1} {M : Type u_2} (R : Type u_5) [Semiring R] [AddCommMonoid M] [Module R M] {M' : Type u_8} [AddCommMonoid M'] [Module R M'] (f : M →ₗ[R] M') (v : α → M) (l : α →₀ R)
Goal: f ((Finsupp.total α M R v) l) = (Finsupp.total α M' R (⇑f ∘ v)) l
-- Finsupp.apply_total_id
theorem apply_total_id (f : M →ₗ[R] M') (l : M →₀ R) :
    f (Finsupp.total M M R _root_.id l) = Finsupp.total M M' R f l 
Arguments: {M : Type u_2} (R : Type u_5) [Semiring R] [AddCommMonoid M] [Module R M] {M' : Type u_8} [AddCommMonoid M'] [Module R M'] (f : M →ₗ[R] M') (l : M →₀ R)
Goal: f ((Finsupp.total M M R id) l) = (Finsupp.total M M' R ⇑f) l
-- Finsupp.total_unique
theorem total_unique [Unique α] (l : α →₀ R) (v) :
    Finsupp.total α M R v l = l default • v default 
Arguments: {α : Type u_1} {M : Type u_2} (R : Type u_5) [Semiring R] [AddCommMonoid M] [Module R M] [Unique α] (l : α →₀ R) (v : α → M)
Goal: (Finsupp.total α M R v) l = l default • v default
-- Finsupp.total_surjective
theorem total_surjective (h : Function.Surjective v) :
    Function.Surjective (Finsupp.total α M R v) 
Arguments: {α : Type u_1} {M : Type u_2} (R : Type u_5) [Semiring R] [AddCommMonoid M] [Module R M] {v : α → M} (h : Function.Surjective v)
Goal: Function.Surjective ⇑(Finsupp.total α M R v)