ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Convex.contractibleSpace
protected theorem Convex.contractibleSpace (hs : Convex ℝ s) (hne : s.Nonempty) :
    ContractibleSpace s 
Arguments: {E : Type u_1} [AddCommGroup E] [Module ℝ E] [TopologicalSpace E] [ContinuousAdd E] [ContinuousSMul ℝ E] {s : Set E} (hs : Convex ℝ s) (hne : s.Nonempty)
Goal: ContractibleSpace ↑s
-- SpectrumRestricts.compactSpace
lemma compactSpace {R S A : Type*} [Semifield R] [Semifield S] [Ring A]
    [Algebra R S] [Algebra R A] [Algebra S A] [IsScalarTower R S A] [TopologicalSpace R]
    [TopologicalSpace S] {a : A} (f : C(S, R)) (h : SpectrumRestricts a f)
    [h_cpct : CompactSpace (spectrum S a)] : CompactSpace (spectrum R a) 
Arguments: {R : Type u_1} {S : Type u_2} {A : Type u_3} [Semifield R] [Semifield S] [Ring A] [Algebra R S] [Algebra R A] [Algebra S A] [IsScalarTower R S A] [TopologicalSpace R] [TopologicalSpace S] {a : A} (f : C(S, R)) (h : SpectrumRestricts a ⇑f) [h_cpct : CompactSpace ↑(spectrum S a)]
Goal: CompactSpace ↑(spectrum R a)
-- SpectrumRestricts.starAlgHom_apply
@[simps!]
def starAlgHom {R : Type u} {S : Type v} {A : Type w} [Semifield R]
    [StarRing R] [TopologicalSpace R] [TopologicalSemiring R] [ContinuousStar R] [Semifield S]
    [StarRing S] [TopologicalSpace S] [TopologicalSemiring S] [ContinuousStar S] [Ring A]
    [StarRing A] [Algebra R S] [Algebra R A] [Algebra S A]
    [IsScalarTower R S A] [StarModule R S] [ContinuousSMul R S] {a : A}
    (φ : C(spectrum S a, S) →⋆ₐ[S] A) {f : C(S, R)} (h : SpectrumRestricts a f) :
    C(spectrum R a, R) →⋆ₐ[R] A 
Arguments: {R : Type u} {S : Type v} {A : Type w} [Semifield R] [StarRing R] [TopologicalSpace R] [TopologicalSemiring R] [ContinuousStar R] [Semifield S] [StarRing S] [TopologicalSpace S] [TopologicalSemiring S] [ContinuousStar S] [Ring A] [StarRing A] [Algebra R S] [Algebra R A] [Algebra S A] [IsScalarTower R S A] [StarModule R S] [ContinuousSMul R S] {a : A} (φ : C(↑(spectrum S a), S) →⋆ₐ[S] A) {f : C(S, R)} (h : SpectrumRestricts a ⇑f)
Goal: ∀ (a_1 : C(↑(spectrum R a), R)),
  (SpectrumRestricts.starAlgHom φ h) a_1 =
    φ
      ({ toFun := ⇑(StarAlgHom.ofId R S), continuous_toFun := ⋯ }.comp
        (a_1.comp { toFun := Subtype.map ⇑f ⋯, continuous_toFun := ⋯ }))
-- SpectrumRestricts.closedEmbedding_starAlgHom
lemma closedEmbedding_starAlgHom {a : A} {φ : C(spectrum S a, S) →⋆ₐ[S] A}
    (hφ : ClosedEmbedding φ) {f : C(S, R)} (h : SpectrumRestricts a f)
    (halg : UniformEmbedding (algebraMap R S)) [CompactSpace (spectrum S a)] :
    ClosedEmbedding (h.starAlgHom φ) 
Arguments: {R : Type u_1} {S : Type u_2} {A : Type u_3} [Semifield R] [StarRing R] [MetricSpace R] [TopologicalSemiring R] [ContinuousStar R] [Semifield S] [StarRing S] [MetricSpace S] [TopologicalSemiring S] [ContinuousStar S] [TopologicalSpace A] [Ring A] [StarRing A] [Algebra S A] [Algebra R S] [Algebra R A] [IsScalarTower R S A] [StarModule R S] [ContinuousSMul R S] [CompleteSpace R] {a : A} {φ : C(↑(spectrum S a), S) →⋆ₐ[S] A} (hφ : ClosedEmbedding ⇑φ) {f : C(S, R)} (h : SpectrumRestricts a ⇑f) (halg : UniformEmbedding ⇑(algebraMap R S)) [CompactSpace ↑(spectrum S a)]
Goal: ClosedEmbedding ⇑(SpectrumRestricts.starAlgHom φ h)
-- SpectrumRestricts.starAlgHom_id
lemma starAlgHom_id {a : A} {φ : C(spectrum S a, S) →⋆ₐ[S] A} {f : C(S, R)}
    (h : SpectrumRestricts a f) (h_id : φ (.restrict (spectrum S a) <| .id S) = a) :
    h.starAlgHom φ (.restrict (spectrum R a) <| .id R) = a 
Arguments: {R : Type u_1} {S : Type u_2} {A : Type u_3} [Semifield R] [StarRing R] [MetricSpace R] [TopologicalSemiring R] [ContinuousStar R] [Semifield S] [StarRing S] [MetricSpace S] [TopologicalSemiring S] [ContinuousStar S] [Ring A] [StarRing A] [Algebra S A] [Algebra R S] [Algebra R A] [IsScalarTower R S A] [StarModule R S] [ContinuousSMul R S] {a : A} {φ : C(↑(spectrum S a), S) →⋆ₐ[S] A} {f : C(S, R)} (h : SpectrumRestricts a ⇑f) (h_id : φ (ContinuousMap.restrict (spectrum S a) (ContinuousMap.id S)) = a)
Goal: (SpectrumRestricts.starAlgHom φ h) (ContinuousMap.restrict (spectrum R a) (ContinuousMap.id R)) = a
-- SpectrumRestricts.cfc
protected theorem cfc (f : C(S, R)) (halg : UniformEmbedding (algebraMap R S)) (h0 : p 0)
    (h : ∀ a, p a ↔ q a ∧ SpectrumRestricts a f) (h_cpct : ∀ a, q a → CompactSpace (spectrum S a)) :
    ContinuousFunctionalCalculus R p 
Arguments: {R : Type u_1} {S : Type u_2} {A : Type u_3} {p q : A → Prop} [Semifield R] [StarRing R] [MetricSpace R] [TopologicalSemiring R] [ContinuousStar R] [Semifield S] [StarRing S] [MetricSpace S] [TopologicalSemiring S] [ContinuousStar S] [TopologicalSpace A] [Ring A] [StarRing A] [Algebra S A] [ContinuousFunctionalCalculus S q] [Algebra R S] [Algebra R A] [IsScalarTower R S A] [StarModule R S] [ContinuousSMul R S] [CompleteSpace R] (f : C(S, R)) (halg : UniformEmbedding ⇑(algebraMap R S)) (h0 : p 0) (h : ∀ (a : A), p a ↔ q a ∧ SpectrumRestricts a ⇑f) (h_cpct : ∀ (a : A), q a → CompactSpace ↑(spectrum S a))
Goal: ContinuousFunctionalCalculus R p
-- SpectrumRestricts.cfcHom_eq_restrict
lemma cfcHom_eq_restrict (f : C(S, R)) (halg : UniformEmbedding (algebraMap R S))
    {a : A} (hpa : p a) (hqa : q a) (h : SpectrumRestricts a f) [CompactSpace (spectrum S a)] :
    cfcHom hpa = h.starAlgHom (cfcHom hqa) 
Arguments: {R : Type u_1} {S : Type u_2} {A : Type u_3} {p q : A → Prop} [Semifield R] [StarRing R] [MetricSpace R] [TopologicalSemiring R] [ContinuousStar R] [Semifield S] [StarRing S] [MetricSpace S] [TopologicalSemiring S] [ContinuousStar S] [TopologicalSpace A] [Ring A] [StarRing A] [Algebra S A] [ContinuousFunctionalCalculus S q] [Algebra R S] [Algebra R A] [IsScalarTower R S A] [StarModule R S] [ContinuousSMul R S] [CompleteSpace R] [ContinuousFunctionalCalculus R p] [UniqueContinuousFunctionalCalculus R A] (f : C(S, R)) (halg : UniformEmbedding ⇑(algebraMap R S)) {a : A} (hpa : p a) (hqa : q a) (h : SpectrumRestricts a ⇑f) [CompactSpace ↑(spectrum S a)]
Goal: cfcHom hpa = SpectrumRestricts.starAlgHom (cfcHom hqa) h
-- SpectrumRestricts.cfc_eq_restrict
lemma cfc_eq_restrict (f : C(S, R)) (halg : UniformEmbedding (algebraMap R S)) {a : A} (hpa : p a)
    (hqa : q a) (h : SpectrumRestricts a f) [CompactSpace (spectrum S a)] (g : R → R) :
    cfc g a = cfc (fun x ↦ algebraMap R S (g (f x))) a 
Arguments: {R : Type u_1} {S : Type u_2} {A : Type u_3} {p q : A → Prop} [Semifield R] [StarRing R] [MetricSpace R] [TopologicalSemiring R] [ContinuousStar R] [Semifield S] [StarRing S] [MetricSpace S] [TopologicalSemiring S] [ContinuousStar S] [TopologicalSpace A] [Ring A] [StarRing A] [Algebra S A] [ContinuousFunctionalCalculus S q] [Algebra R S] [Algebra R A] [IsScalarTower R S A] [StarModule R S] [ContinuousSMul R S] [CompleteSpace R] [ContinuousFunctionalCalculus R p] [UniqueContinuousFunctionalCalculus R A] (f : C(S, R)) (halg : UniformEmbedding ⇑(algebraMap R S)) {a : A} (hpa : p a) (hqa : q a) (h : SpectrumRestricts a ⇑f) [CompactSpace ↑(spectrum S a)] (g : R → R)
Goal: cfc g a = cfc (fun x => (algebraMap R S) (g (f x))) a
-- QuasispectrumRestricts.nonUnitalStarAlgHom_apply
@[simps!]
def nonUnitalStarAlgHom {R : Type u} {S : Type v} {A : Type w} [Semifield R]
    [StarRing R] [TopologicalSpace R] [TopologicalSemiring R] [ContinuousStar R] [Field S]
    [StarRing S] [TopologicalSpace S] [TopologicalRing S] [ContinuousStar S] [NonUnitalRing A]
    [StarRing A] [Algebra R S] [Module R A] [Module S A] [IsScalarTower S A A] [SMulCommClass S A A]
    [IsScalarTower R S A] [StarModule R S] [ContinuousSMul R S] {a : A}
    (φ : C(σₙ S a, S)₀ →⋆ₙₐ[S] A) {f : C(S, R)} (h : QuasispectrumRestricts a f) :
    C(σₙ R a, R)₀ →⋆ₙₐ[R] A 
Arguments: {R : Type u} {S : Type v} {A : Type w} [Semifield R] [StarRing R] [TopologicalSpace R] [TopologicalSemiring R] [ContinuousStar R] [Field S] [StarRing S] [TopologicalSpace S] [TopologicalRing S] [ContinuousStar S] [NonUnitalRing A] [StarRing A] [Algebra R S] [Module R A] [Module S A] [IsScalarTower S A A] [SMulCommClass S A A] [IsScalarTower R S A] [StarModule R S] [ContinuousSMul R S] {a : A} (φ : ContinuousMapZero (↑(quasispectrum S a)) S →⋆ₙₐ[S] A) {f : C(S, R)} (h : QuasispectrumRestricts a ⇑f)
Goal: ∀ (a_1 : ContinuousMapZero (↑(quasispectrum R a)) R),
  (QuasispectrumRestricts.nonUnitalStarAlgHom φ h) a_1 =
    φ
      ({ toFun := ⇑(StarAlgHom.ofId R S), continuous_toFun := ⋯, map_zero' := ⋯ }.comp
        (a_1.comp { toFun := Subtype.map ⇑f ⋯, continuous_toFun := ⋯, map_zero' := ⋯ }))
-- QuasispectrumRestricts.closedEmbedding_nonUnitalStarAlgHom
lemma closedEmbedding_nonUnitalStarAlgHom {a : A} {φ : C(σₙ S a, S)₀ →⋆ₙₐ[S] A}
    (hφ : ClosedEmbedding φ) {f : C(S, R)} (h : QuasispectrumRestricts a f)
    (halg : UniformEmbedding (algebraMap R S)) [h_cpct : CompactSpace (σₙ S a)] :
    ClosedEmbedding (h.nonUnitalStarAlgHom φ) 
Arguments: {R : Type u_1} {S : Type u_2} {A : Type u_3} [Semifield R] [StarRing R] [MetricSpace R] [TopologicalSemiring R] [ContinuousStar R] [Field S] [StarRing S] [MetricSpace S] [TopologicalRing S] [ContinuousStar S] [TopologicalSpace A] [NonUnitalRing A] [StarRing A] [Module S A] [IsScalarTower S A A] [SMulCommClass S A A] [Algebra R S] [Module R A] [IsScalarTower R S A] [StarModule R S] [ContinuousSMul R S] [CompleteSpace R] {a : A} {φ : ContinuousMapZero (↑(quasispectrum S a)) S →⋆ₙₐ[S] A} (hφ : ClosedEmbedding ⇑φ) {f : C(S, R)} (h : QuasispectrumRestricts a ⇑f) (halg : UniformEmbedding ⇑(algebraMap R S)) [h_cpct : CompactSpace ↑(quasispectrum S a)]
Goal: ClosedEmbedding ⇑(QuasispectrumRestricts.nonUnitalStarAlgHom φ h)
-- QuasispectrumRestricts.nonUnitalStarAlgHom_id
lemma nonUnitalStarAlgHom_id {a : A} {φ : C(σₙ S a, S)₀ →⋆ₙₐ[S] A} {f : C(S, R)}
    (h : QuasispectrumRestricts a f) (h_id : φ (.id rfl) = a) :
    h.nonUnitalStarAlgHom φ (.id rfl) = a 
Arguments: {R : Type u_1} {S : Type u_2} {A : Type u_3} [Semifield R] [StarRing R] [MetricSpace R] [TopologicalSemiring R] [ContinuousStar R] [Field S] [StarRing S] [MetricSpace S] [TopologicalRing S] [ContinuousStar S] [NonUnitalRing A] [StarRing A] [Module S A] [IsScalarTower S A A] [SMulCommClass S A A] [Algebra R S] [Module R A] [IsScalarTower R S A] [StarModule R S] [ContinuousSMul R S] {a : A} {φ : ContinuousMapZero (↑(quasispectrum S a)) S →⋆ₙₐ[S] A} {f : C(S, R)} (h : QuasispectrumRestricts a ⇑f) (h_id : φ (ContinuousMapZero.id ⋯) = a)
Goal: (QuasispectrumRestricts.nonUnitalStarAlgHom φ h) (ContinuousMapZero.id ⋯) = a
-- QuasispectrumRestricts.cfc
protected theorem cfc (f : C(S, R)) (halg : UniformEmbedding (algebraMap R S)) (h0 : p 0)
    (h : ∀ a, p a ↔ q a ∧ QuasispectrumRestricts a f) (h_cpct : ∀ a, q a → CompactSpace (σₙ S a)) :
    NonUnitalContinuousFunctionalCalculus R p 
Arguments: {R : Type u_1} {S : Type u_2} {A : Type u_3} {p q : A → Prop} [Semifield R] [StarRing R] [MetricSpace R] [TopologicalSemiring R] [ContinuousStar R] [Field S] [StarRing S] [MetricSpace S] [TopologicalRing S] [ContinuousStar S] [TopologicalSpace A] [NonUnitalRing A] [StarRing A] [Module S A] [IsScalarTower S A A] [SMulCommClass S A A] [NonUnitalContinuousFunctionalCalculus S q] [Algebra R S] [Module R A] [IsScalarTower R S A] [StarModule R S] [ContinuousSMul R S] [CompleteSpace R] [IsScalarTower R A A] [SMulCommClass R A A] (f : C(S, R)) (halg : UniformEmbedding ⇑(algebraMap R S)) (h0 : p 0) (h : ∀ (a : A), p a ↔ q a ∧ QuasispectrumRestricts a ⇑f) (h_cpct : ∀ (a : A), q a → CompactSpace ↑(quasispectrum S a))
Goal: NonUnitalContinuousFunctionalCalculus R p
-- QuasispectrumRestricts.cfcₙHom_eq_restrict
lemma cfcₙHom_eq_restrict (f : C(S, R)) (halg : UniformEmbedding (algebraMap R S)) {a : A}
    (hpa : p a) (hqa : q a) (h : QuasispectrumRestricts a f) [CompactSpace (σₙ S a)] :
    cfcₙHom hpa = h.nonUnitalStarAlgHom (cfcₙHom hqa) 
Arguments: {R : Type u_1} {S : Type u_2} {A : Type u_3} {p q : A → Prop} [Semifield R] [StarRing R] [MetricSpace R] [TopologicalSemiring R] [ContinuousStar R] [Field S] [StarRing S] [MetricSpace S] [TopologicalRing S] [ContinuousStar S] [TopologicalSpace A] [NonUnitalRing A] [StarRing A] [Module S A] [IsScalarTower S A A] [SMulCommClass S A A] [NonUnitalContinuousFunctionalCalculus S q] [Algebra R S] [Module R A] [IsScalarTower R S A] [StarModule R S] [ContinuousSMul R S] [CompleteSpace R] [IsScalarTower R A A] [SMulCommClass R A A] [NonUnitalContinuousFunctionalCalculus R p] [UniqueNonUnitalContinuousFunctionalCalculus R A] (f : C(S, R)) (halg : UniformEmbedding ⇑(algebraMap R S)) {a : A} (hpa : p a) (hqa : q a) (h : QuasispectrumRestricts a ⇑f) [CompactSpace ↑(quasispectrum S a)]
Goal: cfcₙHom hpa = QuasispectrumRestricts.nonUnitalStarAlgHom (cfcₙHom hqa) h
-- QuasispectrumRestricts.cfcₙ_eq_restrict
lemma cfcₙ_eq_restrict (f : C(S, R)) (halg : UniformEmbedding (algebraMap R S)) {a : A} (hpa : p a)
    (hqa : q a) (h : QuasispectrumRestricts a f) [CompactSpace (σₙ S a)] (g : R → R) :
    cfcₙ g a = cfcₙ (fun x ↦ algebraMap R S (g (f x))) a 
Arguments: {R : Type u_1} {S : Type u_2} {A : Type u_3} {p q : A → Prop} [Semifield R] [StarRing R] [MetricSpace R] [TopologicalSemiring R] [ContinuousStar R] [Field S] [StarRing S] [MetricSpace S] [TopologicalRing S] [ContinuousStar S] [TopologicalSpace A] [NonUnitalRing A] [StarRing A] [Module S A] [IsScalarTower S A A] [SMulCommClass S A A] [NonUnitalContinuousFunctionalCalculus S q] [Algebra R S] [Module R A] [IsScalarTower R S A] [StarModule R S] [ContinuousSMul R S] [CompleteSpace R] [IsScalarTower R A A] [SMulCommClass R A A] [NonUnitalContinuousFunctionalCalculus R p] [UniqueNonUnitalContinuousFunctionalCalculus R A] (f : C(S, R)) (halg : UniformEmbedding ⇑(algebraMap R S)) {a : A} (hpa : p a) (hqa : q a) (h : QuasispectrumRestricts a ⇑f) [CompactSpace ↑(quasispectrum S a)] (g : R → R)
Goal: cfcₙ g a = cfcₙ (fun x => (algebraMap R S) (g (f x))) a
-- not_disjoint_segment_convexHull_triple
theorem not_disjoint_segment_convexHull_triple {p q u v x y z : E} (hz : z ∈ segment 𝕜 x y)
    (hu : u ∈ segment 𝕜 x p) (hv : v ∈ segment 𝕜 y q) :
    ¬Disjoint (segment 𝕜 u v) (convexHull 𝕜 {p, q, z}) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [LinearOrderedField 𝕜] [AddCommGroup E] [Module 𝕜 E] {p q u v x y z : E} (hz : z ∈ segment 𝕜 x y) (hu : u ∈ segment 𝕜 x p) (hv : v ∈ segment 𝕜 y q)
Goal: ¬Disjoint (segment 𝕜 u v) ((convexHull 𝕜) {p, q, z})
-- exists_convex_convex_compl_subset
theorem exists_convex_convex_compl_subset (hs : Convex 𝕜 s) (ht : Convex 𝕜 t) (hst : Disjoint s t) :
    ∃ C : Set E, Convex 𝕜 C ∧ Convex 𝕜 Cᶜ ∧ s ⊆ C ∧ t ⊆ Cᶜ 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [LinearOrderedField 𝕜] [AddCommGroup E] [Module 𝕜 E] {s t : Set E} (hs : Convex 𝕜 s) (ht : Convex 𝕜 t) (hst : Disjoint s t)
Goal: ∃ C, Convex 𝕜 C ∧ Convex 𝕜 Cᶜ ∧ s ⊆ C ∧ t ⊆ Cᶜ
-- IsCompact.extremePoints_nonempty
theorem IsCompact.extremePoints_nonempty (hscomp : IsCompact s) (hsnemp : s.Nonempty) :
    (s.extremePoints ℝ).Nonempty 
Arguments: {E : Type u_1} [AddCommGroup E] [Module ℝ E] [TopologicalSpace E] [T2Space E] [TopologicalAddGroup E] [ContinuousSMul ℝ E] [LocallyConvexSpace ℝ E] {s : Set E} (hscomp : IsCompact s) (hsnemp : s.Nonempty)
Goal: (Set.extremePoints ℝ s).Nonempty
-- closure_convexHull_extremePoints
theorem closure_convexHull_extremePoints (hscomp : IsCompact s) (hAconv : Convex ℝ s) :
    closure (convexHull ℝ <| s.extremePoints ℝ) = s 
Arguments: {E : Type u_1} [AddCommGroup E] [Module ℝ E] [TopologicalSpace E] [T2Space E] [TopologicalAddGroup E] [ContinuousSMul ℝ E] [LocallyConvexSpace ℝ E] {s : Set E} (hscomp : IsCompact s) (hAconv : Convex ℝ s)
Goal: closure ((convexHull ℝ) (Set.extremePoints ℝ s)) = s
-- surjOn_extremePoints_image
lemma surjOn_extremePoints_image (f : E →ᴬ[ℝ] F) (hs : IsCompact s) :
    SurjOn f (extremePoints ℝ s) (extremePoints ℝ (f '' s)) 
Arguments: {E : Type u_1} {F : Type u_2} [AddCommGroup E] [Module ℝ E] [TopologicalSpace E] [T2Space E] [TopologicalAddGroup E] [ContinuousSMul ℝ E] [LocallyConvexSpace ℝ E] {s : Set E} [AddCommGroup F] [Module ℝ F] [TopologicalSpace F] [T1Space F] (f : E →ᴬ[ℝ] F) (hs : IsCompact s)
Goal: Set.SurjOn (⇑f) (Set.extremePoints ℝ s) (Set.extremePoints ℝ (⇑f '' s))
-- segment_eq_image₂
theorem segment_eq_image₂ (x y : E) :
    [x -[𝕜] y] =
      (fun p : 𝕜 × 𝕜 => p.1 • x + p.2 • y) '' { p | 0 ≤ p.1 ∧ 0 ≤ p.2 ∧ p.1 + p.2 = 1 } 
Arguments: (𝕜 : Type u_1) {E : Type u_2} [OrderedSemiring 𝕜] [AddCommMonoid E] [SMul 𝕜 E] (x y : E)
Goal: [x-[𝕜]y] = (fun p => p.1 • x + p.2 • y) '' {p | 0 ≤ p.1 ∧ 0 ≤ p.2 ∧ p.1 + p.2 = 1}
-- openSegment_eq_image₂
theorem openSegment_eq_image₂ (x y : E) :
    openSegment 𝕜 x y =
      (fun p : 𝕜 × 𝕜 => p.1 • x + p.2 • y) '' { p | 0 < p.1 ∧ 0 < p.2 ∧ p.1 + p.2 = 1 } 
Arguments: (𝕜 : Type u_1) {E : Type u_2} [OrderedSemiring 𝕜] [AddCommMonoid E] [SMul 𝕜 E] (x y : E)
Goal: openSegment 𝕜 x y = (fun p => p.1 • x + p.2 • y) '' {p | 0 < p.1 ∧ 0 < p.2 ∧ p.1 + p.2 = 1}
-- segment_symm
theorem segment_symm (x y : E) : [x -[𝕜] y] = [y -[𝕜] x] 
Arguments: (𝕜 : Type u_1) {E : Type u_2} [OrderedSemiring 𝕜] [AddCommMonoid E] [SMul 𝕜 E] (x y : E)
Goal: [x-[𝕜]y] = [y-[𝕜]x]
-- openSegment_symm
theorem openSegment_symm (x y : E) : openSegment 𝕜 x y = openSegment 𝕜 y x 
Arguments: (𝕜 : Type u_1) {E : Type u_2} [OrderedSemiring 𝕜] [AddCommMonoid E] [SMul 𝕜 E] (x y : E)
Goal: openSegment 𝕜 x y = openSegment 𝕜 y x
-- openSegment_subset_segment
theorem openSegment_subset_segment (x y : E) : openSegment 𝕜 x y ⊆ [x -[𝕜] y] 
Arguments: (𝕜 : Type u_1) {E : Type u_2} [OrderedSemiring 𝕜] [AddCommMonoid E] [SMul 𝕜 E] (x y : E)
Goal: openSegment 𝕜 x y ⊆ [x-[𝕜]y]
-- segment_subset_iff
theorem segment_subset_iff :
    [x -[𝕜] y] ⊆ s ↔ ∀ a b : 𝕜, 0 ≤ a → 0 ≤ b → a + b = 1 → a • x + b • y ∈ s 
Arguments: (𝕜 : Type u_1) {E : Type u_2} [OrderedSemiring 𝕜] [AddCommMonoid E] [SMul 𝕜 E] {s : Set E} {x y : E}
Goal: [x-[𝕜]y] ⊆ s ↔ ∀ (a b : 𝕜), 0 ≤ a → 0 ≤ b → a + b = 1 → a • x + b • y ∈ s
-- openSegment_subset_iff
theorem openSegment_subset_iff :
    openSegment 𝕜 x y ⊆ s ↔ ∀ a b : 𝕜, 0 < a → 0 < b → a + b = 1 → a • x + b • y ∈ s 
Arguments: (𝕜 : Type u_1) {E : Type u_2} [OrderedSemiring 𝕜] [AddCommMonoid E] [SMul 𝕜 E] {s : Set E} {x y : E}
Goal: openSegment 𝕜 x y ⊆ s ↔ ∀ (a b : 𝕜), 0 < a → 0 < b → a + b = 1 → a • x + b • y ∈ s
-- left_mem_segment
theorem left_mem_segment (x y : E) : x ∈ [x -[𝕜] y] 
Arguments: (𝕜 : Type u_1) {E : Type u_2} [OrderedSemiring 𝕜] [AddCommMonoid E] [MulActionWithZero 𝕜 E] (x y : E)
Goal: x ∈ [x-[𝕜]y]
-- right_mem_segment
theorem right_mem_segment (x y : E) : y ∈ [x -[𝕜] y] 
Arguments: (𝕜 : Type u_1) {E : Type u_2} [OrderedSemiring 𝕜] [AddCommMonoid E] [MulActionWithZero 𝕜 E] (x y : E)
Goal: y ∈ [x-[𝕜]y]
-- segment_same
@[simp]
theorem segment_same (x : E) : [x -[𝕜] x] = {x} 
Arguments: (𝕜 : Type u_1) {E : Type u_2} [OrderedSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E] (x : E)
Goal: [x-[𝕜]x] = {x}
-- insert_endpoints_openSegment
theorem insert_endpoints_openSegment (x y : E) :
    insert x (insert y (openSegment 𝕜 x y)) = [x -[𝕜] y] 
Arguments: (𝕜 : Type u_1) {E : Type u_2} [OrderedSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E] (x y : E)
Goal: insert x (insert y (openSegment 𝕜 x y)) = [x-[𝕜]y]
-- mem_openSegment_of_ne_left_right
theorem mem_openSegment_of_ne_left_right (hx : x ≠ z) (hy : y ≠ z) (hz : z ∈ [x -[𝕜] y]) :
    z ∈ openSegment 𝕜 x y 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E] {x y z : E} (hx : x ≠ z) (hy : y ≠ z) (hz : z ∈ [x-[𝕜]y])
Goal: z ∈ openSegment 𝕜 x y
-- openSegment_subset_iff_segment_subset
theorem openSegment_subset_iff_segment_subset (hx : x ∈ s) (hy : y ∈ s) :
    openSegment 𝕜 x y ⊆ s ↔ [x -[𝕜] y] ⊆ s 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E] {s : Set E} {x y : E} (hx : x ∈ s) (hy : y ∈ s)
Goal: openSegment 𝕜 x y ⊆ s ↔ [x-[𝕜]y] ⊆ s
-- openSegment_same
@[simp]
theorem openSegment_same (x : E) : openSegment 𝕜 x x = {x} 
Arguments: (𝕜 : Type u_1) {E : Type u_2} [OrderedRing 𝕜] [AddCommGroup E] [Module 𝕜 E] [Nontrivial 𝕜] [DenselyOrdered 𝕜] (x : E)
Goal: openSegment 𝕜 x x = {x}
-- segment_eq_image
theorem segment_eq_image (x y : E) :
    [x -[𝕜] y] = (fun θ : 𝕜 => (1 - θ) • x + θ • y) '' Icc (0 : 𝕜) 1 
Arguments: (𝕜 : Type u_1) {E : Type u_2} [OrderedRing 𝕜] [AddCommGroup E] [Module 𝕜 E] (x y : E)
Goal: [x-[𝕜]y] = (fun θ => (1 - θ) • x + θ • y) '' Set.Icc 0 1
-- openSegment_eq_image
theorem openSegment_eq_image (x y : E) :
    openSegment 𝕜 x y = (fun θ : 𝕜 => (1 - θ) • x + θ • y) '' Ioo (0 : 𝕜) 1 
Arguments: (𝕜 : Type u_1) {E : Type u_2} [OrderedRing 𝕜] [AddCommGroup E] [Module 𝕜 E] (x y : E)
Goal: openSegment 𝕜 x y = (fun θ => (1 - θ) • x + θ • y) '' Set.Ioo 0 1
-- segment_eq_image'
theorem segment_eq_image' (x y : E) :
    [x -[𝕜] y] = (fun θ : 𝕜 => x + θ • (y - x)) '' Icc (0 : 𝕜) 1 
Arguments: (𝕜 : Type u_1) {E : Type u_2} [OrderedRing 𝕜] [AddCommGroup E] [Module 𝕜 E] (x y : E)
Goal: [x-[𝕜]y] = (fun θ => x + θ • (y - x)) '' Set.Icc 0 1
-- openSegment_eq_image'
theorem openSegment_eq_image' (x y : E) :
    openSegment 𝕜 x y = (fun θ : 𝕜 => x + θ • (y - x)) '' Ioo (0 : 𝕜) 1 
Arguments: (𝕜 : Type u_1) {E : Type u_2} [OrderedRing 𝕜] [AddCommGroup E] [Module 𝕜 E] (x y : E)
Goal: openSegment 𝕜 x y = (fun θ => x + θ • (y - x)) '' Set.Ioo 0 1
-- segment_eq_image_lineMap
theorem segment_eq_image_lineMap (x y : E) : [x -[𝕜] y] =
    AffineMap.lineMap x y '' Icc (0 : 𝕜) 1 
Arguments: (𝕜 : Type u_1) {E : Type u_2} [OrderedRing 𝕜] [AddCommGroup E] [Module 𝕜 E] (x y : E)
Goal: [x-[𝕜]y] = ⇑(AffineMap.lineMap x y) '' Set.Icc 0 1
-- openSegment_eq_image_lineMap
theorem openSegment_eq_image_lineMap (x y : E) :
    openSegment 𝕜 x y = AffineMap.lineMap x y '' Ioo (0 : 𝕜) 1 
Arguments: (𝕜 : Type u_1) {E : Type u_2} [OrderedRing 𝕜] [AddCommGroup E] [Module 𝕜 E] (x y : E)
Goal: openSegment 𝕜 x y = ⇑(AffineMap.lineMap x y) '' Set.Ioo 0 1
-- image_segment
@[simp]
theorem image_segment (f : E →ᵃ[𝕜] F) (a b : E) : f '' [a -[𝕜] b] = [f a -[𝕜] f b] 
Arguments: (𝕜 : Type u_1) {E : Type u_2} {F : Type u_3} [OrderedRing 𝕜] [AddCommGroup E] [AddCommGroup F] [Module 𝕜 E] [Module 𝕜 F] (f : E →ᵃ[𝕜] F) (a b : E)
Goal: ⇑f '' [a-[𝕜]b] = [f a-[𝕜]f b]
-- image_openSegment
@[simp]
theorem image_openSegment (f : E →ᵃ[𝕜] F) (a b : E) :
    f '' openSegment 𝕜 a b = openSegment 𝕜 (f a) (f b) 
Arguments: (𝕜 : Type u_1) {E : Type u_2} {F : Type u_3} [OrderedRing 𝕜] [AddCommGroup E] [AddCommGroup F] [Module 𝕜 E] [Module 𝕜 F] (f : E →ᵃ[𝕜] F) (a b : E)
Goal: ⇑f '' openSegment 𝕜 a b = openSegment 𝕜 (f a) (f b)
-- vadd_segment
@[simp]
theorem vadd_segment [AddTorsor G E] [VAddCommClass G E E] (a : G) (b c : E) :
    a +ᵥ [b -[𝕜] c] = [a +ᵥ b -[𝕜] a +ᵥ c] 
Arguments: (𝕜 : Type u_1) {E : Type u_2} {G : Type u_4} [OrderedRing 𝕜] [AddCommGroup E] [AddCommGroup G] [Module 𝕜 E] [AddTorsor G E] [VAddCommClass G E E] (a : G) (b c : E)
Goal: a +ᵥ [b-[𝕜]c] = [a +ᵥ b-[𝕜]a +ᵥ c]
-- vadd_openSegment
@[simp]
theorem vadd_openSegment [AddTorsor G E] [VAddCommClass G E E] (a : G) (b c : E) :
    a +ᵥ openSegment 𝕜 b c = openSegment 𝕜 (a +ᵥ b) (a +ᵥ c) 
Arguments: (𝕜 : Type u_1) {E : Type u_2} {G : Type u_4} [OrderedRing 𝕜] [AddCommGroup E] [AddCommGroup G] [Module 𝕜 E] [AddTorsor G E] [VAddCommClass G E E] (a : G) (b c : E)
Goal: a +ᵥ openSegment 𝕜 b c = openSegment 𝕜 (a +ᵥ b) (a +ᵥ c)
-- mem_segment_translate
@[simp]
theorem mem_segment_translate (a : E) {x b c} : a + x ∈ [a + b -[𝕜] a + c] ↔ x ∈ [b -[𝕜] c] 
Arguments: (𝕜 : Type u_1) {E : Type u_2} [OrderedRing 𝕜] [AddCommGroup E] [Module 𝕜 E] (a : E) {x b c : E}
Goal: a + x ∈ [a + b-[𝕜]a + c] ↔ x ∈ [b-[𝕜]c]
-- mem_openSegment_translate
@[simp]
theorem mem_openSegment_translate (a : E) {x b c : E} :
    a + x ∈ openSegment 𝕜 (a + b) (a + c) ↔ x ∈ openSegment 𝕜 b c 
Arguments: (𝕜 : Type u_1) {E : Type u_2} [OrderedRing 𝕜] [AddCommGroup E] [Module 𝕜 E] (a : E) {x b c : E}
Goal: a + x ∈ openSegment 𝕜 (a + b) (a + c) ↔ x ∈ openSegment 𝕜 b c
-- segment_translate_preimage
theorem segment_translate_preimage (a b c : E) :
    (fun x => a + x) ⁻¹' [a + b -[𝕜] a + c] = [b -[𝕜] c] 
Arguments: (𝕜 : Type u_1) {E : Type u_2} [OrderedRing 𝕜] [AddCommGroup E] [Module 𝕜 E] (a b c : E)
Goal: (fun x => a + x) ⁻¹' [a + b-[𝕜]a + c] = [b-[𝕜]c]
-- openSegment_translate_preimage
theorem openSegment_translate_preimage (a b c : E) :
    (fun x => a + x) ⁻¹' openSegment 𝕜 (a + b) (a + c) = openSegment 𝕜 b c 
Arguments: (𝕜 : Type u_1) {E : Type u_2} [OrderedRing 𝕜] [AddCommGroup E] [Module 𝕜 E] (a b c : E)
Goal: (fun x => a + x) ⁻¹' openSegment 𝕜 (a + b) (a + c) = openSegment 𝕜 b c