ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- isQuasiregular_iff_isUnit'
lemma isQuasiregular_iff_isUnit' (R : Type*) {A : Type*} [CommSemiring R] [NonUnitalSemiring A]
    [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] {x : A} :
    IsQuasiregular x ↔ IsUnit (1 + x : Unitization R A) 
Arguments: (R : Type u_1) {A : Type u_2} [CommSemiring R] [NonUnitalSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] {x : A}
Goal: IsQuasiregular x ↔ IsUnit (1 + ↑x)
-- quasispectrum.not_isUnit_mem
lemma quasispectrum.not_isUnit_mem (a : A) {r : R} (hr : ¬ IsUnit r) : r ∈ quasispectrum R a 
Arguments: {R : Type u_1} {A : Type u_2} [CommSemiring R] [NonUnitalRing A] [Module R A] (a : A) {r : R} (hr : ¬IsUnit r)
Goal: r ∈ quasispectrum R a
-- quasispectrum.zero_mem
@[simp]
lemma quasispectrum.zero_mem [Nontrivial R] (a : A) : 0 ∈ quasispectrum R a 
Arguments: (R : Type u_1) {A : Type u_2} [CommSemiring R] [NonUnitalRing A] [Module R A] [Nontrivial R] (a : A)
Goal: 0 ∈ quasispectrum R a
-- quasispectrum.coe_zero
@[simp]
lemma quasispectrum.coe_zero [Nontrivial R] (a : A) : (0 : quasispectrum R a) = (0 : R) 
Arguments: {R : Type u_1} {A : Type u_2} [CommSemiring R] [NonUnitalRing A] [Module R A] [Nontrivial R] (a : A)
Goal: ↑0 = 0
-- quasispectrum.mem_of_not_quasiregular
lemma quasispectrum.mem_of_not_quasiregular (a : A) {r : Rˣ}
    (hr : ¬ IsQuasiregular (-(r⁻¹ • a))) : (r : R) ∈ quasispectrum R a 
Arguments: {R : Type u_1} {A : Type u_2} [CommSemiring R] [NonUnitalRing A] [Module R A] (a : A) {r : Rˣ} (hr : ¬IsQuasiregular (-(r⁻¹ • a)))
Goal: ↑r ∈ quasispectrum R a
-- quasispectrum_eq_spectrum_union
lemma quasispectrum_eq_spectrum_union (R : Type*) {A : Type*} [CommSemiring R]
    [Ring A] [Algebra R A] (a : A) : quasispectrum R a = spectrum R a ∪ {r : R | ¬ IsUnit r} 
Arguments: (R : Type u_3) {A : Type u_4} [CommSemiring R] [Ring A] [Algebra R A] (a : A)
Goal: quasispectrum R a = spectrum R a ∪ {r | ¬IsUnit r}
-- spectrum_subset_quasispectrum
lemma spectrum_subset_quasispectrum (R : Type*) {A : Type*} [CommSemiring R] [Ring A] [Algebra R A]
    (a : A) : spectrum R a ⊆ quasispectrum R a 
Arguments: (R : Type u_3) {A : Type u_4} [CommSemiring R] [Ring A] [Algebra R A] (a : A)
Goal: spectrum R a ⊆ quasispectrum R a
-- quasispectrum_eq_spectrum_union_zero
lemma quasispectrum_eq_spectrum_union_zero (R : Type*) {A : Type*} [Semifield R] [Ring A]
    [Algebra R A] (a : A) : quasispectrum R a = spectrum R a ∪ {0} 
Arguments: (R : Type u_3) {A : Type u_4} [Semifield R] [Ring A] [Algebra R A] (a : A)
Goal: quasispectrum R a = spectrum R a ∪ {0}
-- mem_quasispectrum_iff
lemma mem_quasispectrum_iff {R A : Type*} [Semifield R] [Ring A]
    [Algebra R A] {a : A} {x : R} :
    x ∈ quasispectrum R a ↔ x = 0 ∨ x ∈ spectrum R a 
Arguments: {R : Type u_3} {A : Type u_4} [Semifield R] [Ring A] [Algebra R A] {a : A} {x : R}
Goal: x ∈ quasispectrum R a ↔ x = 0 ∨ x ∈ spectrum R a
-- Unitization.isQuasiregular_inr_iff
lemma isQuasiregular_inr_iff (a : A) :
    IsQuasiregular (a : Unitization R A) ↔ IsQuasiregular a 
Arguments: {R : Type u_1} {A : Type u_2} [CommSemiring R] [NonUnitalRing A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] (a : A)
Goal: IsQuasiregular ↑a ↔ IsQuasiregular a
-- Unitization.zero_mem_spectrum_inr
lemma zero_mem_spectrum_inr (R S : Type*) {A : Type*} [CommSemiring R]
    [CommRing S] [Nontrivial S] [NonUnitalRing A] [Algebra R S] [Module S A] [IsScalarTower S A A]
    [SMulCommClass S A A] [Module R A] [IsScalarTower R S A] (a : A) :
    0 ∈ spectrum R (a : Unitization S A) 
Arguments: (R : Type u_3) (S : Type u_4) {A : Type u_5} [CommSemiring R] [CommRing S] [Nontrivial S] [NonUnitalRing A] [Algebra R S] [Module S A] [IsScalarTower S A A] [SMulCommClass S A A] [Module R A] [IsScalarTower R S A] (a : A)
Goal: 0 ∈ spectrum R ↑a
-- Unitization.mem_spectrum_inr_of_not_isUnit
lemma mem_spectrum_inr_of_not_isUnit {R A : Type*} [CommRing R]
    [NonUnitalRing A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]
    (a : A) (r : R) (hr : ¬ IsUnit r) : r ∈ spectrum R (a : Unitization R A) 
Arguments: {R : Type u_3} {A : Type u_4} [CommRing R] [NonUnitalRing A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] (a : A) (r : R) (hr : ¬IsUnit r)
Goal: r ∈ spectrum R ↑a
-- Unitization.quasispectrum_eq_spectrum_inr
lemma quasispectrum_eq_spectrum_inr (R : Type*) {A : Type*} [CommRing R] [Ring A]
    [Algebra R A] (a : A) : quasispectrum R a = spectrum R (a : Unitization R A) 
Arguments: (R : Type u_3) {A : Type u_4} [CommRing R] [Ring A] [Algebra R A] (a : A)
Goal: quasispectrum R a = spectrum R ↑a
-- Unitization.quasispectrum_eq_spectrum_inr'
lemma quasispectrum_eq_spectrum_inr' (R S : Type*) {A : Type*} [Semifield R]
    [Field S] [NonUnitalRing A] [Algebra R S] [Module S A] [IsScalarTower S A A]
    [SMulCommClass S A A] [Module R A] [IsScalarTower R S A] (a : A) :
    quasispectrum R a = spectrum R (a : Unitization S A) 
Arguments: (R : Type u_3) (S : Type u_4) {A : Type u_5} [Semifield R] [Field S] [NonUnitalRing A] [Algebra R S] [Module S A] [IsScalarTower S A A] [SMulCommClass S A A] [Module R A] [IsScalarTower R S A] (a : A)
Goal: quasispectrum R a = spectrum R ↑a
-- NonnegSpectrumClass.quasispectrum_nonneg_of_nonneg
class NonnegSpectrumClass (𝕜 A : Type*) [OrderedCommSemiring 𝕜] [NonUnitalRing A] [PartialOrder A]
    [Module 𝕜 A] : Prop where
  quasispectrum_nonneg_of_nonneg : ∀ a : A, 0 ≤ a → ∀ x ∈ quasispectrum 𝕜 a, 0 ≤ x
Arguments: {𝕜 : Type u_3} {A : Type u_4} [OrderedCommSemiring 𝕜] [NonUnitalRing A] [PartialOrder A] [Module 𝕜 A] [self : NonnegSpectrumClass 𝕜 A] (a : A)
Goal: 0 ≤ a → ∀ x ∈ quasispectrum 𝕜 a, 0 ≤ x
-- NonnegSpectrumClass.iff_spectrum_nonneg
lemma iff_spectrum_nonneg {𝕜 A : Type*} [LinearOrderedSemifield 𝕜] [Ring A] [PartialOrder A]
    [Algebra 𝕜 A] : NonnegSpectrumClass 𝕜 A ↔ ∀ a : A, 0 ≤ a → ∀ x ∈ spectrum 𝕜 a, 0 ≤ x 
Arguments: {𝕜 : Type u_3} {A : Type u_4} [LinearOrderedSemifield 𝕜] [Ring A] [PartialOrder A] [Algebra 𝕜 A]
Goal: NonnegSpectrumClass 𝕜 A ↔ ∀ (a : A), 0 ≤ a → ∀ x ∈ spectrum 𝕜 a, 0 ≤ x
-- spectrum_nonneg_of_nonneg
lemma spectrum_nonneg_of_nonneg {𝕜 A : Type*} [OrderedCommSemiring 𝕜] [Ring A] [PartialOrder A]
    [Algebra 𝕜 A] [NonnegSpectrumClass 𝕜 A] ⦃a : A⦄ (ha : 0 ≤ a) ⦃x : 𝕜⦄ (hx : x ∈ spectrum 𝕜 a) :
    0 ≤ x 
Arguments: {𝕜 : Type u_3} {A : Type u_4} [OrderedCommSemiring 𝕜] [Ring A] [PartialOrder A] [Algebra 𝕜 A] [NonnegSpectrumClass 𝕜 A] ⦃a : A⦄ (ha : 0 ≤ a) ⦃x : 𝕜⦄ (hx : x ∈ spectrum 𝕜 a)
Goal: 0 ≤ x
-- QuasispectrumRestricts.rightInvOn
structure QuasispectrumRestricts
    {R S A : Type*} [CommSemiring R] [CommSemiring S] [NonUnitalRing A]
    [Module R A] [Module S A] [Algebra R S] (a : A) (f : S → R) : Prop where
    rightInvOn : (quasispectrum S a).RightInvOn f (algebraMap R S)
    left_inv : Function.LeftInverse f (algebraMap R S)
Arguments: {R : Type u_3} {S : Type u_4} {A : Type u_5} [CommSemiring R] [CommSemiring S] [NonUnitalRing A] [Module R A] [Module S A] [Algebra R S] {a : A} {f : S → R} (self : QuasispectrumRestricts a f)
Goal: Set.RightInvOn f (⇑(algebraMap R S)) (quasispectrum S a)
-- QuasispectrumRestricts.left_inv
structure QuasispectrumRestricts
    {R S A : Type*} [CommSemiring R] [CommSemiring S] [NonUnitalRing A]
    [Module R A] [Module S A] [Algebra R S] (a : A) (f : S → R) : Prop where
    rightInvOn : (quasispectrum S a).RightInvOn f (algebraMap R S)
    left_inv : Function.LeftInverse f (algebraMap R S)
Arguments: {R : Type u_3} {S : Type u_4} {A : Type u_5} [CommSemiring R] [CommSemiring S] [NonUnitalRing A] [Module R A] [Module S A] [Algebra R S] {a : A} {f : S → R} (self : QuasispectrumRestricts a f)
Goal: Function.LeftInverse f ⇑(algebraMap R S)
-- quasispectrumRestricts_iff
lemma quasispectrumRestricts_iff
    {R S A : Type*} [CommSemiring R] [CommSemiring S] [NonUnitalRing A]
    [Module R A] [Module S A] [Algebra R S] (a : A) (f : S → R) :
    QuasispectrumRestricts a f ↔ (quasispectrum S a).RightInvOn f (algebraMap R S) ∧
      Function.LeftInverse f (algebraMap R S) 
Arguments: {R : Type u_3} {S : Type u_4} {A : Type u_5} [CommSemiring R] [CommSemiring S] [NonUnitalRing A] [Module R A] [Module S A] [Algebra R S] (a : A) (f : S → R)
Goal: QuasispectrumRestricts a f ↔
  Set.RightInvOn f (⇑(algebraMap R S)) (quasispectrum S a) ∧ Function.LeftInverse f ⇑(algebraMap R S)
-- quasispectrum.algebraMap_mem_iff
@[simp]
theorem quasispectrum.algebraMap_mem_iff (S : Type*) {R A : Type*} [Semifield R] [Field S]
    [NonUnitalRing A] [Algebra R S] [Module S A] [IsScalarTower S A A]
    [SMulCommClass S A A] [Module R A] [IsScalarTower R S A] {a : A} {r : R} :
    algebraMap R S r ∈ quasispectrum S a ↔ r ∈ quasispectrum R a 
Arguments: (S : Type u_3) {R : Type u_4} {A : Type u_5} [Semifield R] [Field S] [NonUnitalRing A] [Algebra R S] [Module S A] [IsScalarTower S A A] [SMulCommClass S A A] [Module R A] [IsScalarTower R S A] {a : A} {r : R}
Goal: (algebraMap R S) r ∈ quasispectrum S a ↔ r ∈ quasispectrum R a
-- quasispectrum.preimage_algebraMap
@[simp]
theorem quasispectrum.preimage_algebraMap (S : Type*) {R A : Type*} [Semifield R] [Field S]
    [NonUnitalRing A] [Algebra R S] [Module S A] [IsScalarTower S A A]
    [SMulCommClass S A A] [Module R A] [IsScalarTower R S A] {a : A} :
    algebraMap R S ⁻¹' quasispectrum S a = quasispectrum R a 
Arguments: (S : Type u_3) {R : Type u_4} {A : Type u_5} [Semifield R] [Field S] [NonUnitalRing A] [Algebra R S] [Module S A] [IsScalarTower S A A] [SMulCommClass S A A] [Module R A] [IsScalarTower R S A] {a : A}
Goal: ⇑(algebraMap R S) ⁻¹' quasispectrum S a = quasispectrum R a
-- QuasispectrumRestricts.map_zero
protected theorem map_zero (h : QuasispectrumRestricts a f) : f 0 = 0 
Arguments: {R : Type u_3} {S : Type u_4} {A : Type u_5} [Semifield R] [Field S] [NonUnitalRing A] [Module R A] [Module S A] [Algebra R S] {a : A} {f : S → R} (h : QuasispectrumRestricts a f)
Goal: f 0 = 0
-- QuasispectrumRestricts.of_subset_range_algebraMap
theorem of_subset_range_algebraMap (hf : f.LeftInverse (algebraMap R S))
    (h : quasispectrum S a ⊆ Set.range (algebraMap R S)) : QuasispectrumRestricts a f 
Arguments: {R : Type u_3} {S : Type u_4} {A : Type u_5} [Semifield R] [Field S] [NonUnitalRing A] [Module R A] [Module S A] [Algebra R S] {a : A} {f : S → R} (hf : Function.LeftInverse f ⇑(algebraMap R S)) (h : quasispectrum S a ⊆ Set.range ⇑(algebraMap R S))
Goal: QuasispectrumRestricts a f
-- QuasispectrumRestricts.algebraMap_image
theorem algebraMap_image : algebraMap R S '' quasispectrum R a = quasispectrum S a 
Arguments: {R : Type u_3} {S : Type u_4} {A : Type u_5} [Semifield R] [Field S] [NonUnitalRing A] [Module R A] [Module S A] [IsScalarTower S A A] [SMulCommClass S A A] [Algebra R S] {a : A} {f : S → R} [IsScalarTower R S A] (h : QuasispectrumRestricts a f)
Goal: ⇑(algebraMap R S) '' quasispectrum R a = quasispectrum S a
-- QuasispectrumRestricts.image
theorem image : f '' quasispectrum S a = quasispectrum R a 
Arguments: {R : Type u_3} {S : Type u_4} {A : Type u_5} [Semifield R] [Field S] [NonUnitalRing A] [Module R A] [Module S A] [IsScalarTower S A A] [SMulCommClass S A A] [Algebra R S] {a : A} {f : S → R} [IsScalarTower R S A] (h : QuasispectrumRestricts a f)
Goal: f '' quasispectrum S a = quasispectrum R a
-- QuasispectrumRestricts.apply_mem
theorem apply_mem {s : S} (hs : s ∈ quasispectrum S a) : f s ∈ quasispectrum R a 
Arguments: {R : Type u_3} {S : Type u_4} {A : Type u_5} [Semifield R] [Field S] [NonUnitalRing A] [Module R A] [Module S A] [IsScalarTower S A A] [SMulCommClass S A A] [Algebra R S] {a : A} {f : S → R} [IsScalarTower R S A] (h : QuasispectrumRestricts a f) {s : S} (hs : s ∈ quasispectrum S a)
Goal: f s ∈ quasispectrum R a
-- QuasispectrumRestricts.subset_preimage
theorem subset_preimage : quasispectrum S a ⊆ f ⁻¹' quasispectrum R a 
Arguments: {R : Type u_3} {S : Type u_4} {A : Type u_5} [Semifield R] [Field S] [NonUnitalRing A] [Module R A] [Module S A] [IsScalarTower S A A] [SMulCommClass S A A] [Algebra R S] {a : A} {f : S → R} [IsScalarTower R S A] (h : QuasispectrumRestricts a f)
Goal: quasispectrum S a ⊆ f ⁻¹' quasispectrum R a
-- QuasispectrumRestricts.of_quasispectrum_eq
lemma of_quasispectrum_eq {a b : A} {f : S → R} (ha : QuasispectrumRestricts a f)
    (h : quasispectrum S a = quasispectrum S b) : QuasispectrumRestricts b f 
Arguments: {R : Type u_3} {S : Type u_4} {A : Type u_5} [Semifield R] [Field S] [NonUnitalRing A] [Module R A] [Module S A] [Algebra R S] {a b : A} {f : S → R} (ha : QuasispectrumRestricts a f) (h : quasispectrum S a = quasispectrum S b)
Goal: QuasispectrumRestricts b f
-- QuasispectrumRestricts.comp
protected lemma comp {R₁ R₂ R₃ A : Type*} [Semifield R₁] [Field R₂] [Field R₃]
    [NonUnitalRing A] [Module R₁ A] [Module R₂ A] [Module R₃ A] [Algebra R₁ R₂] [Algebra R₂ R₃]
    [Algebra R₁ R₃] [IsScalarTower R₁ R₂ R₃] [IsScalarTower R₂ R₃ A] [IsScalarTower R₃ A A]
    [SMulCommClass R₃ A A] {a : A} {f : R₃ → R₂} {g : R₂ → R₁} {e : R₃ → R₁} (hfge : g ∘ f = e)
    (hf : QuasispectrumRestricts a f) (hg : QuasispectrumRestricts a g) :
    QuasispectrumRestricts a e 
Arguments: {R₁ : Type u_6} {R₂ : Type u_7} {R₃ : Type u_8} {A : Type u_9} [Semifield R₁] [Field R₂] [Field R₃] [NonUnitalRing A] [Module R₁ A] [Module R₂ A] [Module R₃ A] [Algebra R₁ R₂] [Algebra R₂ R₃] [Algebra R₁ R₃] [IsScalarTower R₁ R₂ R₃] [IsScalarTower R₂ R₃ A] [IsScalarTower R₃ A A] [SMulCommClass R₃ A A] {a : A} {f : R₃ → R₂} {g : R₂ → R₁} {e : R₃ → R₁} (hfge : g ∘ f = e) (hf : QuasispectrumRestricts a f) (hg : QuasispectrumRestricts a g)
Goal: QuasispectrumRestricts a e
-- SpectrumRestricts.rightInvOn
theorem rightInvOn (h : SpectrumRestricts a f) : (spectrum S a).RightInvOn f (algebraMap R S) 
Arguments: {R : Type u_3} {S : Type u_4} {A : Type u_5} [Semifield R] [Semifield S] [Ring A] [Algebra R S] [Algebra R A] [Algebra S A] {a : A} {f : S → R} (h : SpectrumRestricts a f)
Goal: Set.RightInvOn f (⇑(algebraMap R S)) (spectrum S a)
-- SpectrumRestricts.of_rightInvOn
theorem of_rightInvOn (h₁ : Function.LeftInverse f (algebraMap R S))
    (h₂ : (spectrum S a).RightInvOn f (algebraMap R S)) : SpectrumRestricts a f 
Arguments: {R : Type u_3} {S : Type u_4} {A : Type u_5} [Semifield R] [Semifield S] [Ring A] [Algebra R S] [Algebra R A] [Algebra S A] {a : A} {f : S → R} (h₁ : Function.LeftInverse f ⇑(algebraMap R S)) (h₂ : Set.RightInvOn f (⇑(algebraMap R S)) (spectrum S a))
Goal: SpectrumRestricts a f
-- spectrumRestricts_iff
lemma _root_.spectrumRestricts_iff :
    SpectrumRestricts a f ↔ (spectrum S a).RightInvOn f (algebraMap R S) ∧
      Function.LeftInverse f (algebraMap R S) 
Arguments: {R : Type u_3} {S : Type u_4} {A : Type u_5} [Semifield R] [Semifield S] [Ring A] [Algebra R S] [Algebra R A] [Algebra S A] {a : A} {f : S → R}
Goal: SpectrumRestricts a f ↔ Set.RightInvOn f (⇑(algebraMap R S)) (spectrum S a) ∧ Function.LeftInverse f ⇑(algebraMap R S)
-- SpectrumRestricts.of_subset_range_algebraMap
theorem of_subset_range_algebraMap (hf : f.LeftInverse (algebraMap R S))
    (h : spectrum S a ⊆ Set.range (algebraMap R S)) : SpectrumRestricts a f 
Arguments: {R : Type u_3} {S : Type u_4} {A : Type u_5} [Semifield R] [Semifield S] [Ring A] [Algebra R S] [Algebra R A] [Algebra S A] {a : A} {f : S → R} (hf : Function.LeftInverse f ⇑(algebraMap R S)) (h : spectrum S a ⊆ Set.range ⇑(algebraMap R S))
Goal: SpectrumRestricts a f
-- SpectrumRestricts.algebraMap_image
theorem algebraMap_image : algebraMap R S '' spectrum R a = spectrum S a 
Arguments: {R : Type u_3} {S : Type u_4} {A : Type u_5} [Semifield R] [Semifield S] [Ring A] [Algebra R S] [Algebra R A] [Algebra S A] {a : A} {f : S → R} [IsScalarTower R S A] (h : SpectrumRestricts a f)
Goal: ⇑(algebraMap R S) '' spectrum R a = spectrum S a
-- SpectrumRestricts.image
theorem image : f '' spectrum S a = spectrum R a 
Arguments: {R : Type u_3} {S : Type u_4} {A : Type u_5} [Semifield R] [Semifield S] [Ring A] [Algebra R S] [Algebra R A] [Algebra S A] {a : A} {f : S → R} [IsScalarTower R S A] (h : SpectrumRestricts a f)
Goal: f '' spectrum S a = spectrum R a
-- SpectrumRestricts.apply_mem
theorem apply_mem {s : S} (hs : s ∈ spectrum S a) : f s ∈ spectrum R a 
Arguments: {R : Type u_3} {S : Type u_4} {A : Type u_5} [Semifield R] [Semifield S] [Ring A] [Algebra R S] [Algebra R A] [Algebra S A] {a : A} {f : S → R} [IsScalarTower R S A] (h : SpectrumRestricts a f) {s : S} (hs : s ∈ spectrum S a)
Goal: f s ∈ spectrum R a
-- SpectrumRestricts.subset_preimage
theorem subset_preimage : spectrum S a ⊆ f ⁻¹' spectrum R a 
Arguments: {R : Type u_3} {S : Type u_4} {A : Type u_5} [Semifield R] [Semifield S] [Ring A] [Algebra R S] [Algebra R A] [Algebra S A] {a : A} {f : S → R} [IsScalarTower R S A] (h : SpectrumRestricts a f)
Goal: spectrum S a ⊆ f ⁻¹' spectrum R a
-- SpectrumRestricts.of_spectrum_eq
lemma of_spectrum_eq {a b : A} {f : S → R} (ha : SpectrumRestricts a f)
    (h : spectrum S a = spectrum S b) : SpectrumRestricts b f 
Arguments: {R : Type u_3} {S : Type u_4} {A : Type u_5} [Semifield R] [Semifield S] [Ring A] [Algebra R S] [Algebra R A] [Algebra S A] {a b : A} {f : S → R} (ha : SpectrumRestricts a f) (h : spectrum S a = spectrum S b)
Goal: SpectrumRestricts b f
-- quasispectrumRestricts_iff_spectrumRestricts_inr
theorem quasispectrumRestricts_iff_spectrumRestricts_inr (S : Type*) {R A : Type*} [Semifield R]
    [Field S] [NonUnitalRing A] [Algebra R S] [Module R A] [Module S A] [IsScalarTower S A A]
    [SMulCommClass S A A] [IsScalarTower R S A] {a : A} {f : S → R} :
    QuasispectrumRestricts a f ↔ SpectrumRestricts (a : Unitization S A) f 
Arguments: (S : Type u_3) {R : Type u_4} {A : Type u_5} [Semifield R] [Field S] [NonUnitalRing A] [Algebra R S] [Module R A] [Module S A] [IsScalarTower S A A] [SMulCommClass S A A] [IsScalarTower R S A] {a : A} {f : S → R}
Goal: QuasispectrumRestricts a f ↔ SpectrumRestricts (↑a) f
-- quasispectrumRestricts_iff_spectrumRestricts
theorem quasispectrumRestricts_iff_spectrumRestricts {R S A : Type*} [Semifield R] [Semifield S]
    [Ring A] [Algebra R S] [Algebra R A] [Algebra S A] {a : A} {f : S → R} :
    QuasispectrumRestricts a f ↔ SpectrumRestricts a f 
Arguments: {R : Type u_3} {S : Type u_4} {A : Type u_5} [Semifield R] [Semifield S] [Ring A] [Algebra R S] [Algebra R A] [Algebra S A] {a : A} {f : S → R}
Goal: QuasispectrumRestricts a f ↔ SpectrumRestricts a f
-- MonoidAlgebra.mul_apply_mul_eq_mul_of_uniqueMul
theorem mul_apply_mul_eq_mul_of_uniqueMul [Mul A] {f g : MonoidAlgebra R A} {a0 b0 : A}
    (h : UniqueMul f.support g.support a0 b0) :
    (f * g) (a0 * b0) = f a0 * g b0 
Arguments: {R : Type u_1} {A : Type u_2} [Semiring R] [Mul A] {f g : MonoidAlgebra R A} {a0 b0 : A} (h : UniqueMul f.support g.support a0 b0)
Goal: (f * g) (a0 * b0) = f a0 * g b0
-- AddMonoidAlgebra.mul_apply_add_eq_mul_of_uniqueAdd
theorem mul_apply_add_eq_mul_of_uniqueAdd [Add A] {f g : R[A]} {a0 b0 : A}
    (h : UniqueAdd f.support g.support a0 b0) :
    (f * g) (a0 + b0) = f a0 * g b0 
Arguments: {R : Type u_1} {A : Type u_2} [Semiring R] [Add A] {f g : AddMonoidAlgebra R A} {a0 b0 : A} (h : UniqueAdd f.support g.support a0 b0)
Goal: (f * g) (a0 + b0) = f a0 * g b0
-- MonoidAlgebra.support_mul
theorem support_mul [Mul G] [DecidableEq G] (a b : MonoidAlgebra k G) :
    (a * b).support ⊆ a.support * b.support 
Arguments: {k : Type u₁} {G : Type u₂} [Semiring k] [Mul G] [DecidableEq G] (a b : MonoidAlgebra k G)
Goal: (a * b).support ⊆ a.support * b.support
-- MonoidAlgebra.support_single_mul_subset
theorem support_single_mul_subset [DecidableEq G] [Mul G] (f : MonoidAlgebra k G) (r : k) (a : G) :
    (single a r * f : MonoidAlgebra k G).support ⊆ Finset.image (a * ·) f.support 
Arguments: {k : Type u₁} {G : Type u₂} [Semiring k] [DecidableEq G] [Mul G] (f : MonoidAlgebra k G) (r : k) (a : G)
Goal: (MonoidAlgebra.single a r * f).support ⊆ Finset.image (fun x => a * x) f.support
-- MonoidAlgebra.support_mul_single_subset
theorem support_mul_single_subset [DecidableEq G] [Mul G] (f : MonoidAlgebra k G) (r : k) (a : G) :
    (f * single a r).support ⊆ Finset.image (· * a) f.support 
Arguments: {k : Type u₁} {G : Type u₂} [Semiring k] [DecidableEq G] [Mul G] (f : MonoidAlgebra k G) (r : k) (a : G)
Goal: (f * MonoidAlgebra.single a r).support ⊆ Finset.image (fun x => x * a) f.support
-- MonoidAlgebra.support_single_mul_eq_image
theorem support_single_mul_eq_image [DecidableEq G] [Mul G] (f : MonoidAlgebra k G) {r : k}
    (hr : ∀ y, r * y = 0 ↔ y = 0) {x : G} (lx : IsLeftRegular x) :
    (single x r * f : MonoidAlgebra k G).support = Finset.image (x * ·) f.support 
Arguments: {k : Type u₁} {G : Type u₂} [Semiring k] [DecidableEq G] [Mul G] (f : MonoidAlgebra k G) {r : k} (hr : ∀ (y : k), r * y = 0 ↔ y = 0) {x : G} (lx : IsLeftRegular x)
Goal: (MonoidAlgebra.single x r * f).support = Finset.image (fun x_1 => x * x_1) f.support
-- MonoidAlgebra.support_mul_single_eq_image
theorem support_mul_single_eq_image [DecidableEq G] [Mul G] (f : MonoidAlgebra k G) {r : k}
    (hr : ∀ y, y * r = 0 ↔ y = 0) {x : G} (rx : IsRightRegular x) :
    (f * single x r).support = Finset.image (· * x) f.support 
Arguments: {k : Type u₁} {G : Type u₂} [Semiring k] [DecidableEq G] [Mul G] (f : MonoidAlgebra k G) {r : k} (hr : ∀ (y : k), y * r = 0 ↔ y = 0) {x : G} (rx : IsRightRegular x)
Goal: (f * MonoidAlgebra.single x r).support = Finset.image (fun x_1 => x_1 * x) f.support
-- MonoidAlgebra.support_mul_single
theorem support_mul_single [Mul G] [IsRightCancelMul G] (f : MonoidAlgebra k G) (r : k)
    (hr : ∀ y, y * r = 0 ↔ y = 0) (x : G) :
    (f * single x r).support = f.support.map (mulRightEmbedding x) 
Arguments: {k : Type u₁} {G : Type u₂} [Semiring k] [Mul G] [IsRightCancelMul G] (f : MonoidAlgebra k G) (r : k) (hr : ∀ (y : k), y * r = 0 ↔ y = 0) (x : G)
Goal: (f * MonoidAlgebra.single x r).support = Finset.map (mulRightEmbedding x) f.support
-- MonoidAlgebra.support_single_mul
theorem support_single_mul [Mul G] [IsLeftCancelMul G] (f : MonoidAlgebra k G) (r : k)
    (hr : ∀ y, r * y = 0 ↔ y = 0) (x : G) :
    (single x r * f : MonoidAlgebra k G).support = f.support.map (mulLeftEmbedding x) 
Arguments: {k : Type u₁} {G : Type u₂} [Semiring k] [Mul G] [IsLeftCancelMul G] (f : MonoidAlgebra k G) (r : k) (hr : ∀ (y : k), r * y = 0 ↔ y = 0) (x : G)
Goal: (MonoidAlgebra.single x r * f).support = Finset.map (mulLeftEmbedding x) f.support
-- MonoidAlgebra.support_one_subset
lemma support_one_subset [One G] : (1 : MonoidAlgebra k G).support ⊆ 1 
Arguments: {k : Type u₁} {G : Type u₂} [Semiring k] [One G]
Goal: Finsupp.support 1 ⊆ 1
-- MonoidAlgebra.support_one
@[simp]
lemma support_one [One G] [NeZero (1 : k)] : (1 : MonoidAlgebra k G).support = 1 
Arguments: {k : Type u₁} {G : Type u₂} [Semiring k] [One G] [NeZero 1]
Goal: Finsupp.support 1 = 1
-- MonoidAlgebra.mem_span_support
theorem mem_span_support (f : MonoidAlgebra k G) :
    f ∈ Submodule.span k (of k G '' (f.support : Set G)) 
Arguments: {k : Type u₁} {G : Type u₂} [Semiring k] [MulOneClass G] (f : MonoidAlgebra k G)
Goal: f ∈ Submodule.span k (⇑(MonoidAlgebra.of k G) '' ↑f.support)
-- AddMonoidAlgebra.support_mul
theorem support_mul [DecidableEq G] [Add G] (a b : k[G]) :
    (a * b).support ⊆ a.support + b.support 
Arguments: {k : Type u₁} {G : Type u₂} [Semiring k] [DecidableEq G] [Add G] (a b : AddMonoidAlgebra k G)
Goal: (a * b).support ⊆ a.support + b.support
-- AddMonoidAlgebra.support_mul_single
theorem support_mul_single [Add G] [IsRightCancelAdd G] (f : k[G]) (r : k)
    (hr : ∀ y, y * r = 0 ↔ y = 0) (x : G) :
    (f * single x r : k[G]).support = f.support.map (addRightEmbedding x) 
Arguments: {k : Type u₁} {G : Type u₂} [Semiring k] [Add G] [IsRightCancelAdd G] (f : AddMonoidAlgebra k G) (r : k) (hr : ∀ (y : k), y * r = 0 ↔ y = 0) (x : G)
Goal: (f * AddMonoidAlgebra.single x r).support = Finset.map (addRightEmbedding x) f.support
-- AddMonoidAlgebra.support_single_mul
theorem support_single_mul [Add G] [IsLeftCancelAdd G] (f : k[G]) (r : k)
    (hr : ∀ y, r * y = 0 ↔ y = 0) (x : G) :
    (single x r * f : k[G]).support = f.support.map (addLeftEmbedding x) 
Arguments: {k : Type u₁} {G : Type u₂} [Semiring k] [Add G] [IsLeftCancelAdd G] (f : AddMonoidAlgebra k G) (r : k) (hr : ∀ (y : k), r * y = 0 ↔ y = 0) (x : G)
Goal: (AddMonoidAlgebra.single x r * f).support = Finset.map (addLeftEmbedding x) f.support