ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- convex_sum
theorem convex_sum {ι} {s : Finset ι} (t : ι → Set E) (h : ∀ i ∈ s, Convex 𝕜 (t i)) :
    Convex 𝕜 (∑ i ∈ s, t i) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E] {ι : Type u_5} {s : Finset ι} (t : ι → Set E) (h : ∀ i ∈ s, Convex 𝕜 (t i))
Goal: Convex 𝕜 (∑ i ∈ s, t i)
-- Convex.vadd
theorem Convex.vadd (hs : Convex 𝕜 s) (z : E) : Convex 𝕜 (z +ᵥ s) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E] {s : Set E} (hs : Convex 𝕜 s) (z : E)
Goal: Convex 𝕜 (z +ᵥ s)
-- Convex.translate
theorem Convex.translate (hs : Convex 𝕜 s) (z : E) : Convex 𝕜 ((fun x => z + x) '' s) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E] {s : Set E} (hs : Convex 𝕜 s) (z : E)
Goal: Convex 𝕜 ((fun x => z + x) '' s)
-- Convex.translate_preimage_right
theorem Convex.translate_preimage_right (hs : Convex 𝕜 s) (z : E) :
    Convex 𝕜 ((fun x => z + x) ⁻¹' s) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E] {s : Set E} (hs : Convex 𝕜 s) (z : E)
Goal: Convex 𝕜 ((fun x => z + x) ⁻¹' s)
-- Convex.translate_preimage_left
theorem Convex.translate_preimage_left (hs : Convex 𝕜 s) (z : E) :
    Convex 𝕜 ((fun x => x + z) ⁻¹' s) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E] {s : Set E} (hs : Convex 𝕜 s) (z : E)
Goal: Convex 𝕜 ((fun x => x + z) ⁻¹' s)
-- convex_Iic
theorem convex_Iic (r : β) : Convex 𝕜 (Iic r) 
Arguments: {𝕜 : Type u_1} {β : Type u_4} [OrderedSemiring 𝕜] [OrderedAddCommMonoid β] [Module 𝕜 β] [OrderedSMul 𝕜 β] (r : β)
Goal: Convex 𝕜 (Set.Iic r)
-- convex_Ici
theorem convex_Ici (r : β) : Convex 𝕜 (Ici r) 
Arguments: {𝕜 : Type u_1} {β : Type u_4} [OrderedSemiring 𝕜] [OrderedAddCommMonoid β] [Module 𝕜 β] [OrderedSMul 𝕜 β] (r : β)
Goal: Convex 𝕜 (Set.Ici r)
-- convex_Icc
theorem convex_Icc (r s : β) : Convex 𝕜 (Icc r s) 
Arguments: {𝕜 : Type u_1} {β : Type u_4} [OrderedSemiring 𝕜] [OrderedAddCommMonoid β] [Module 𝕜 β] [OrderedSMul 𝕜 β] (r s : β)
Goal: Convex 𝕜 (Set.Icc r s)
-- convex_halfspace_le
theorem convex_halfspace_le {f : E → β} (h : IsLinearMap 𝕜 f) (r : β) : Convex 𝕜 { w | f w ≤ r } 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {β : Type u_4} [OrderedSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E] [OrderedAddCommMonoid β] [Module 𝕜 β] [OrderedSMul 𝕜 β] {f : E → β} (h : IsLinearMap 𝕜 f) (r : β)
Goal: Convex 𝕜 {w | f w ≤ r}
-- convex_halfspace_ge
theorem convex_halfspace_ge {f : E → β} (h : IsLinearMap 𝕜 f) (r : β) : Convex 𝕜 { w | r ≤ f w } 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {β : Type u_4} [OrderedSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E] [OrderedAddCommMonoid β] [Module 𝕜 β] [OrderedSMul 𝕜 β] {f : E → β} (h : IsLinearMap 𝕜 f) (r : β)
Goal: Convex 𝕜 {w | r ≤ f w}
-- convex_hyperplane
theorem convex_hyperplane {f : E → β} (h : IsLinearMap 𝕜 f) (r : β) : Convex 𝕜 { w | f w = r } 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {β : Type u_4} [OrderedSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E] [OrderedAddCommMonoid β] [Module 𝕜 β] [OrderedSMul 𝕜 β] {f : E → β} (h : IsLinearMap 𝕜 f) (r : β)
Goal: Convex 𝕜 {w | f w = r}
-- convex_Iio
theorem convex_Iio (r : β) : Convex 𝕜 (Iio r) 
Arguments: {𝕜 : Type u_1} {β : Type u_4} [OrderedSemiring 𝕜] [OrderedCancelAddCommMonoid β] [Module 𝕜 β] [OrderedSMul 𝕜 β] (r : β)
Goal: Convex 𝕜 (Set.Iio r)
-- convex_Ioi
theorem convex_Ioi (r : β) : Convex 𝕜 (Ioi r) 
Arguments: {𝕜 : Type u_1} {β : Type u_4} [OrderedSemiring 𝕜] [OrderedCancelAddCommMonoid β] [Module 𝕜 β] [OrderedSMul 𝕜 β] (r : β)
Goal: Convex 𝕜 (Set.Ioi r)
-- convex_Ioo
theorem convex_Ioo (r s : β) : Convex 𝕜 (Ioo r s) 
Arguments: {𝕜 : Type u_1} {β : Type u_4} [OrderedSemiring 𝕜] [OrderedCancelAddCommMonoid β] [Module 𝕜 β] [OrderedSMul 𝕜 β] (r s : β)
Goal: Convex 𝕜 (Set.Ioo r s)
-- convex_Ico
theorem convex_Ico (r s : β) : Convex 𝕜 (Ico r s) 
Arguments: {𝕜 : Type u_1} {β : Type u_4} [OrderedSemiring 𝕜] [OrderedCancelAddCommMonoid β] [Module 𝕜 β] [OrderedSMul 𝕜 β] (r s : β)
Goal: Convex 𝕜 (Set.Ico r s)
-- convex_Ioc
theorem convex_Ioc (r s : β) : Convex 𝕜 (Ioc r s) 
Arguments: {𝕜 : Type u_1} {β : Type u_4} [OrderedSemiring 𝕜] [OrderedCancelAddCommMonoid β] [Module 𝕜 β] [OrderedSMul 𝕜 β] (r s : β)
Goal: Convex 𝕜 (Set.Ioc r s)
-- convex_halfspace_lt
theorem convex_halfspace_lt {f : E → β} (h : IsLinearMap 𝕜 f) (r : β) : Convex 𝕜 { w | f w < r } 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {β : Type u_4} [OrderedSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E] [OrderedCancelAddCommMonoid β] [Module 𝕜 β] [OrderedSMul 𝕜 β] {f : E → β} (h : IsLinearMap 𝕜 f) (r : β)
Goal: Convex 𝕜 {w | f w < r}
-- convex_halfspace_gt
theorem convex_halfspace_gt {f : E → β} (h : IsLinearMap 𝕜 f) (r : β) : Convex 𝕜 { w | r < f w } 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {β : Type u_4} [OrderedSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E] [OrderedCancelAddCommMonoid β] [Module 𝕜 β] [OrderedSMul 𝕜 β] {f : E → β} (h : IsLinearMap 𝕜 f) (r : β)
Goal: Convex 𝕜 {w | r < f w}
-- convex_uIcc
theorem convex_uIcc (r s : β) : Convex 𝕜 (uIcc r s) 
Arguments: {𝕜 : Type u_1} {β : Type u_4} [OrderedSemiring 𝕜] [LinearOrderedAddCommMonoid β] [Module 𝕜 β] [OrderedSMul 𝕜 β] (r s : β)
Goal: Convex 𝕜 (Set.uIcc r s)
-- MonotoneOn.convex_le
theorem MonotoneOn.convex_le (hf : MonotoneOn f s) (hs : Convex 𝕜 s) (r : β) :
    Convex 𝕜 ({ x ∈ s | f x ≤ r }) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {β : Type u_4} [OrderedSemiring 𝕜] [LinearOrderedAddCommMonoid E] [OrderedAddCommMonoid β] [Module 𝕜 E] [OrderedSMul 𝕜 E] {s : Set E} {f : E → β} (hf : MonotoneOn f s) (hs : Convex 𝕜 s) (r : β)
Goal: Convex 𝕜 {x | x ∈ s ∧ f x ≤ r}
-- MonotoneOn.convex_lt
theorem MonotoneOn.convex_lt (hf : MonotoneOn f s) (hs : Convex 𝕜 s) (r : β) :
    Convex 𝕜 ({ x ∈ s | f x < r }) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {β : Type u_4} [OrderedSemiring 𝕜] [LinearOrderedAddCommMonoid E] [OrderedAddCommMonoid β] [Module 𝕜 E] [OrderedSMul 𝕜 E] {s : Set E} {f : E → β} (hf : MonotoneOn f s) (hs : Convex 𝕜 s) (r : β)
Goal: Convex 𝕜 {x | x ∈ s ∧ f x < r}
-- MonotoneOn.convex_ge
theorem MonotoneOn.convex_ge (hf : MonotoneOn f s) (hs : Convex 𝕜 s) (r : β) :
    Convex 𝕜 ({ x ∈ s | r ≤ f x }) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {β : Type u_4} [OrderedSemiring 𝕜] [LinearOrderedAddCommMonoid E] [OrderedAddCommMonoid β] [Module 𝕜 E] [OrderedSMul 𝕜 E] {s : Set E} {f : E → β} (hf : MonotoneOn f s) (hs : Convex 𝕜 s) (r : β)
Goal: Convex 𝕜 {x | x ∈ s ∧ r ≤ f x}
-- MonotoneOn.convex_gt
theorem MonotoneOn.convex_gt (hf : MonotoneOn f s) (hs : Convex 𝕜 s) (r : β) :
    Convex 𝕜 ({ x ∈ s | r < f x }) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {β : Type u_4} [OrderedSemiring 𝕜] [LinearOrderedAddCommMonoid E] [OrderedAddCommMonoid β] [Module 𝕜 E] [OrderedSMul 𝕜 E] {s : Set E} {f : E → β} (hf : MonotoneOn f s) (hs : Convex 𝕜 s) (r : β)
Goal: Convex 𝕜 {x | x ∈ s ∧ r < f x}
-- AntitoneOn.convex_le
theorem AntitoneOn.convex_le (hf : AntitoneOn f s) (hs : Convex 𝕜 s) (r : β) :
    Convex 𝕜 ({ x ∈ s | f x ≤ r }) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {β : Type u_4} [OrderedSemiring 𝕜] [LinearOrderedAddCommMonoid E] [OrderedAddCommMonoid β] [Module 𝕜 E] [OrderedSMul 𝕜 E] {s : Set E} {f : E → β} (hf : AntitoneOn f s) (hs : Convex 𝕜 s) (r : β)
Goal: Convex 𝕜 {x | x ∈ s ∧ f x ≤ r}
-- AntitoneOn.convex_lt
theorem AntitoneOn.convex_lt (hf : AntitoneOn f s) (hs : Convex 𝕜 s) (r : β) :
    Convex 𝕜 ({ x ∈ s | f x < r }) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {β : Type u_4} [OrderedSemiring 𝕜] [LinearOrderedAddCommMonoid E] [OrderedAddCommMonoid β] [Module 𝕜 E] [OrderedSMul 𝕜 E] {s : Set E} {f : E → β} (hf : AntitoneOn f s) (hs : Convex 𝕜 s) (r : β)
Goal: Convex 𝕜 {x | x ∈ s ∧ f x < r}
-- AntitoneOn.convex_ge
theorem AntitoneOn.convex_ge (hf : AntitoneOn f s) (hs : Convex 𝕜 s) (r : β) :
    Convex 𝕜 ({ x ∈ s | r ≤ f x }) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {β : Type u_4} [OrderedSemiring 𝕜] [LinearOrderedAddCommMonoid E] [OrderedAddCommMonoid β] [Module 𝕜 E] [OrderedSMul 𝕜 E] {s : Set E} {f : E → β} (hf : AntitoneOn f s) (hs : Convex 𝕜 s) (r : β)
Goal: Convex 𝕜 {x | x ∈ s ∧ r ≤ f x}
-- AntitoneOn.convex_gt
theorem AntitoneOn.convex_gt (hf : AntitoneOn f s) (hs : Convex 𝕜 s) (r : β) :
    Convex 𝕜 ({ x ∈ s | r < f x }) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {β : Type u_4} [OrderedSemiring 𝕜] [LinearOrderedAddCommMonoid E] [OrderedAddCommMonoid β] [Module 𝕜 E] [OrderedSMul 𝕜 E] {s : Set E} {f : E → β} (hf : AntitoneOn f s) (hs : Convex 𝕜 s) (r : β)
Goal: Convex 𝕜 {x | x ∈ s ∧ r < f x}
-- Monotone.convex_le
theorem Monotone.convex_le (hf : Monotone f) (r : β) : Convex 𝕜 { x | f x ≤ r } 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {β : Type u_4} [OrderedSemiring 𝕜] [LinearOrderedAddCommMonoid E] [OrderedAddCommMonoid β] [Module 𝕜 E] [OrderedSMul 𝕜 E] {f : E → β} (hf : Monotone f) (r : β)
Goal: Convex 𝕜 {x | f x ≤ r}
-- Monotone.convex_lt
theorem Monotone.convex_lt (hf : Monotone f) (r : β) : Convex 𝕜 { x | f x ≤ r } 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {β : Type u_4} [OrderedSemiring 𝕜] [LinearOrderedAddCommMonoid E] [OrderedAddCommMonoid β] [Module 𝕜 E] [OrderedSMul 𝕜 E] {f : E → β} (hf : Monotone f) (r : β)
Goal: Convex 𝕜 {x | f x ≤ r}
-- Monotone.convex_ge
theorem Monotone.convex_ge (hf : Monotone f) (r : β) : Convex 𝕜 { x | r ≤ f x } 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {β : Type u_4} [OrderedSemiring 𝕜] [LinearOrderedAddCommMonoid E] [OrderedAddCommMonoid β] [Module 𝕜 E] [OrderedSMul 𝕜 E] {f : E → β} (hf : Monotone f) (r : β)
Goal: Convex 𝕜 {x | r ≤ f x}
-- Monotone.convex_gt
theorem Monotone.convex_gt (hf : Monotone f) (r : β) : Convex 𝕜 { x | f x ≤ r } 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {β : Type u_4} [OrderedSemiring 𝕜] [LinearOrderedAddCommMonoid E] [OrderedAddCommMonoid β] [Module 𝕜 E] [OrderedSMul 𝕜 E] {f : E → β} (hf : Monotone f) (r : β)
Goal: Convex 𝕜 {x | f x ≤ r}
-- Antitone.convex_le
theorem Antitone.convex_le (hf : Antitone f) (r : β) : Convex 𝕜 { x | f x ≤ r } 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {β : Type u_4} [OrderedSemiring 𝕜] [LinearOrderedAddCommMonoid E] [OrderedAddCommMonoid β] [Module 𝕜 E] [OrderedSMul 𝕜 E] {f : E → β} (hf : Antitone f) (r : β)
Goal: Convex 𝕜 {x | f x ≤ r}
-- Antitone.convex_lt
theorem Antitone.convex_lt (hf : Antitone f) (r : β) : Convex 𝕜 { x | f x < r } 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {β : Type u_4} [OrderedSemiring 𝕜] [LinearOrderedAddCommMonoid E] [OrderedAddCommMonoid β] [Module 𝕜 E] [OrderedSMul 𝕜 E] {f : E → β} (hf : Antitone f) (r : β)
Goal: Convex 𝕜 {x | f x < r}
-- Antitone.convex_ge
theorem Antitone.convex_ge (hf : Antitone f) (r : β) : Convex 𝕜 { x | r ≤ f x } 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {β : Type u_4} [OrderedSemiring 𝕜] [LinearOrderedAddCommMonoid E] [OrderedAddCommMonoid β] [Module 𝕜 E] [OrderedSMul 𝕜 E] {f : E → β} (hf : Antitone f) (r : β)
Goal: Convex 𝕜 {x | r ≤ f x}
-- Antitone.convex_gt
theorem Antitone.convex_gt (hf : Antitone f) (r : β) : Convex 𝕜 { x | r < f x } 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {β : Type u_4} [OrderedSemiring 𝕜] [LinearOrderedAddCommMonoid E] [OrderedAddCommMonoid β] [Module 𝕜 E] [OrderedSMul 𝕜 E] {f : E → β} (hf : Antitone f) (r : β)
Goal: Convex 𝕜 {x | r < f x}
-- Convex.smul
theorem Convex.smul (hs : Convex 𝕜 s) (c : 𝕜) : Convex 𝕜 (c • s) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedCommSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E] {s : Set E} (hs : Convex 𝕜 s) (c : 𝕜)
Goal: Convex 𝕜 (c • s)
-- Convex.smul_preimage
theorem Convex.smul_preimage (hs : Convex 𝕜 s) (c : 𝕜) : Convex 𝕜 ((fun z => c • z) ⁻¹' s) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedCommSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E] {s : Set E} (hs : Convex 𝕜 s) (c : 𝕜)
Goal: Convex 𝕜 ((fun z => c • z) ⁻¹' s)
-- Convex.affinity
theorem Convex.affinity (hs : Convex 𝕜 s) (z : E) (c : 𝕜) :
    Convex 𝕜 ((fun x => z + c • x) '' s) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedCommSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E] {s : Set E} (hs : Convex 𝕜 s) (z : E) (c : 𝕜)
Goal: Convex 𝕜 ((fun x => z + c • x) '' s)
-- convex_openSegment
theorem convex_openSegment (a b : E) : Convex 𝕜 (openSegment 𝕜 a b) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [StrictOrderedCommSemiring 𝕜] [AddCommGroup E] [Module 𝕜 E] (a b : E)
Goal: Convex 𝕜 (openSegment 𝕜 a b)
-- convex_vadd
@[simp]
theorem convex_vadd (a : E) : Convex 𝕜 (a +ᵥ s) ↔ Convex 𝕜 s 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedRing 𝕜] [AddCommGroup E] [Module 𝕜 E] {s : Set E} (a : E)
Goal: Convex 𝕜 (a +ᵥ s) ↔ Convex 𝕜 s
-- Convex.add_smul_mem
theorem Convex.add_smul_mem (hs : Convex 𝕜 s) {x y : E} (hx : x ∈ s) (hy : x + y ∈ s) {t : 𝕜}
    (ht : t ∈ Icc (0 : 𝕜) 1) : x + t • y ∈ s 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedRing 𝕜] [AddCommGroup E] [Module 𝕜 E] {s : Set E} (hs : Convex 𝕜 s) {x y : E} (hx : x ∈ s) (hy : x + y ∈ s) {t : 𝕜} (ht : t ∈ Set.Icc 0 1)
Goal: x + t • y ∈ s
-- Convex.smul_mem_of_zero_mem
theorem Convex.smul_mem_of_zero_mem (hs : Convex 𝕜 s) {x : E} (zero_mem : (0 : E) ∈ s) (hx : x ∈ s)
    {t : 𝕜} (ht : t ∈ Icc (0 : 𝕜) 1) : t • x ∈ s 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedRing 𝕜] [AddCommGroup E] [Module 𝕜 E] {s : Set E} (hs : Convex 𝕜 s) {x : E} (zero_mem : 0 ∈ s) (hx : x ∈ s) {t : 𝕜} (ht : t ∈ Set.Icc 0 1)
Goal: t • x ∈ s
-- Convex.mapsTo_lineMap
theorem Convex.mapsTo_lineMap (h : Convex 𝕜 s) {x y : E} (hx : x ∈ s) (hy : y ∈ s) :
    MapsTo (AffineMap.lineMap x y) (Icc (0 : 𝕜) 1) s 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedRing 𝕜] [AddCommGroup E] [Module 𝕜 E] {s : Set E} (h : Convex 𝕜 s) {x y : E} (hx : x ∈ s) (hy : y ∈ s)
Goal: Set.MapsTo (⇑(AffineMap.lineMap x y)) (Set.Icc 0 1) s
-- Convex.lineMap_mem
theorem Convex.lineMap_mem (h : Convex 𝕜 s) {x y : E} (hx : x ∈ s) (hy : y ∈ s) {t : 𝕜}
    (ht : t ∈ Icc 0 1) : AffineMap.lineMap x y t ∈ s 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedRing 𝕜] [AddCommGroup E] [Module 𝕜 E] {s : Set E} (h : Convex 𝕜 s) {x y : E} (hx : x ∈ s) (hy : y ∈ s) {t : 𝕜} (ht : t ∈ Set.Icc 0 1)
Goal: (AffineMap.lineMap x y) t ∈ s
-- Convex.add_smul_sub_mem
theorem Convex.add_smul_sub_mem (h : Convex 𝕜 s) {x y : E} (hx : x ∈ s) (hy : y ∈ s) {t : 𝕜}
    (ht : t ∈ Icc (0 : 𝕜) 1) : x + t • (y - x) ∈ s 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedRing 𝕜] [AddCommGroup E] [Module 𝕜 E] {s : Set E} (h : Convex 𝕜 s) {x y : E} (hx : x ∈ s) (hy : y ∈ s) {t : 𝕜} (ht : t ∈ Set.Icc 0 1)
Goal: x + t • (y - x) ∈ s
-- AffineSubspace.convex
theorem AffineSubspace.convex (Q : AffineSubspace 𝕜 E) : Convex 𝕜 (Q : Set E) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedRing 𝕜] [AddCommGroup E] [Module 𝕜 E] (Q : AffineSubspace 𝕜 E)
Goal: Convex 𝕜 ↑Q
-- Convex.affine_preimage
theorem Convex.affine_preimage (f : E →ᵃ[𝕜] F) {s : Set F} (hs : Convex 𝕜 s) : Convex 𝕜 (f ⁻¹' s) 
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) {s : Set F} (hs : Convex 𝕜 s)
Goal: Convex 𝕜 (⇑f ⁻¹' s)
-- Convex.affine_image
theorem Convex.affine_image (f : E →ᵃ[𝕜] F) (hs : Convex 𝕜 s) : Convex 𝕜 (f '' s) 
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] {s : Set E} (f : E →ᵃ[𝕜] F) (hs : Convex 𝕜 s)
Goal: Convex 𝕜 (⇑f '' s)
-- Convex.neg
theorem Convex.neg (hs : Convex 𝕜 s) : Convex 𝕜 (-s) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedRing 𝕜] [AddCommGroup E] [Module 𝕜 E] {s : Set E} (hs : Convex 𝕜 s)
Goal: Convex 𝕜 (-s)
-- Convex.sub
theorem Convex.sub (hs : Convex 𝕜 s) (ht : Convex 𝕜 t) : Convex 𝕜 (s - t) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedRing 𝕜] [AddCommGroup E] [Module 𝕜 E] {s t : Set E} (hs : Convex 𝕜 s) (ht : Convex 𝕜 t)
Goal: Convex 𝕜 (s - t)
-- Convex_subadditive_le
theorem Convex_subadditive_le [SMul 𝕜 E] {f : E → 𝕜} (hf1 : ∀ x y, f (x + y) ≤ (f x) + (f y))
    (hf2 : ∀ ⦃c⦄ x, 0 ≤ c → f (c • x) ≤ c * f x) (B : 𝕜) :
    Convex 𝕜 { x | f x ≤ B } 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [LinearOrderedRing 𝕜] [AddCommMonoid E] [SMul 𝕜 E] {f : E → 𝕜} (hf1 : ∀ (x y : E), f (x + y) ≤ f x + f y) (hf2 : ∀ ⦃c : 𝕜⦄ (x : E), 0 ≤ c → f (c • x) ≤ c * f x) (B : 𝕜)
Goal: Convex 𝕜 {x | f x ≤ B}
-- convex_iff_div
theorem convex_iff_div :
    Convex 𝕜 s ↔ ∀ ⦃x⦄, x ∈ s → ∀ ⦃y⦄, y ∈ s →
      ∀ ⦃a b : 𝕜⦄, 0 ≤ a → 0 ≤ b → 0 < a + b → (a / (a + b)) • x + (b / (a + b)) • y ∈ s 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [LinearOrderedField 𝕜] [AddCommGroup E] [Module 𝕜 E] {s : Set E}
Goal: Convex 𝕜 s ↔
  ∀ ⦃x : E⦄,
    x ∈ s → ∀ ⦃y : E⦄, y ∈ s → ∀ ⦃a b : 𝕜⦄, 0 ≤ a → 0 ≤ b → 0 < a + b → (a / (a + b)) • x + (b / (a + b)) • y ∈ s
-- Convex.mem_smul_of_zero_mem
theorem Convex.mem_smul_of_zero_mem (h : Convex 𝕜 s) {x : E} (zero_mem : (0 : E) ∈ s) (hx : x ∈ s)
    {t : 𝕜} (ht : 1 ≤ t) : x ∈ t • s 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [LinearOrderedField 𝕜] [AddCommGroup E] [Module 𝕜 E] {s : Set E} (h : Convex 𝕜 s) {x : E} (zero_mem : 0 ∈ s) (hx : x ∈ s) {t : 𝕜} (ht : 1 ≤ t)
Goal: x ∈ t • s
-- Convex.exists_mem_add_smul_eq
theorem Convex.exists_mem_add_smul_eq (h : Convex 𝕜 s) {x y : E} {p q : 𝕜} (hx : x ∈ s) (hy : y ∈ s)
    (hp : 0 ≤ p) (hq : 0 ≤ q) : ∃ z ∈ s, (p + q) • z = p • x + q • y 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [LinearOrderedField 𝕜] [AddCommGroup E] [Module 𝕜 E] {s : Set E} (h : Convex 𝕜 s) {x y : E} {p q : 𝕜} (hx : x ∈ s) (hy : y ∈ s) (hp : 0 ≤ p) (hq : 0 ≤ q)
Goal: ∃ z ∈ s, (p + q) • z = p • x + q • y
-- Convex.add_smul
theorem Convex.add_smul (h_conv : Convex 𝕜 s) {p q : 𝕜} (hp : 0 ≤ p) (hq : 0 ≤ q) :
    (p + q) • s = p • s + q • s 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [LinearOrderedField 𝕜] [AddCommGroup E] [Module 𝕜 E] {s : Set E} (h_conv : Convex 𝕜 s) {p q : 𝕜} (hp : 0 ≤ p) (hq : 0 ≤ q)
Goal: (p + q) • s = p • s + q • s
-- Set.OrdConnected.convex_of_chain
theorem Set.OrdConnected.convex_of_chain [OrderedSemiring 𝕜] [OrderedAddCommMonoid E] [Module 𝕜 E]
    [OrderedSMul 𝕜 E] {s : Set E} (hs : s.OrdConnected) (h : IsChain (· ≤ ·) s) : Convex 𝕜 s 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedSemiring 𝕜] [OrderedAddCommMonoid E] [Module 𝕜 E] [OrderedSMul 𝕜 E] {s : Set E} (hs : s.OrdConnected) (h : IsChain (fun x x_1 => x ≤ x_1) s)
Goal: Convex 𝕜 s
-- Set.OrdConnected.convex
theorem Set.OrdConnected.convex [OrderedSemiring 𝕜] [LinearOrderedAddCommMonoid E] [Module 𝕜 E]
    [OrderedSMul 𝕜 E] {s : Set E} (hs : s.OrdConnected) : Convex 𝕜 s 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedSemiring 𝕜] [LinearOrderedAddCommMonoid E] [Module 𝕜 E] [OrderedSMul 𝕜 E] {s : Set E} (hs : s.OrdConnected)
Goal: Convex 𝕜 s
-- convex_iff_ordConnected
theorem convex_iff_ordConnected [LinearOrderedField 𝕜] {s : Set 𝕜} :
    Convex 𝕜 s ↔ s.OrdConnected 
Arguments: {𝕜 : Type u_1} [LinearOrderedField 𝕜] {s : Set 𝕜}
Goal: Convex 𝕜 s ↔ s.OrdConnected
-- Submodule.convex
protected theorem convex (K : Submodule 𝕜 E) : Convex 𝕜 (↑K : Set E) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E] (K : Submodule 𝕜 E)
Goal: Convex 𝕜 ↑K
-- Submodule.starConvex
protected theorem starConvex (K : Submodule 𝕜 E) : StarConvex 𝕜 (0 : E) K 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E] (K : Submodule 𝕜 E)
Goal: StarConvex 𝕜 0 ↑K
-- stdSimplex_eq_inter
theorem stdSimplex_eq_inter : stdSimplex 𝕜 ι = (⋂ x, { f | 0 ≤ f x }) ∩ { f | ∑ x, f x = 1 } 
Arguments: (𝕜 : Type u_1) (ι : Type u_5) [OrderedSemiring 𝕜] [Fintype ι]
Goal: stdSimplex 𝕜 ι = (⋂ x, {f | 0 ≤ f x}) ∩ {f | ∑ x : ι, f x = 1}
-- convex_stdSimplex
theorem convex_stdSimplex : Convex 𝕜 (stdSimplex 𝕜 ι) 
Arguments: (𝕜 : Type u_1) (ι : Type u_5) [OrderedSemiring 𝕜] [Fintype ι]
Goal: Convex 𝕜 (stdSimplex 𝕜 ι)
-- stdSimplex_of_subsingleton
@[nontriviality] lemma stdSimplex_of_subsingleton [Subsingleton 𝕜] : stdSimplex 𝕜 ι = univ 
Arguments: (𝕜 : Type u_1) (ι : Type u_5) [OrderedSemiring 𝕜] [Fintype ι] [Subsingleton 𝕜]
Goal: stdSimplex 𝕜 ι = Set.univ
-- stdSimplex_of_isEmpty_index
lemma stdSimplex_of_isEmpty_index [IsEmpty ι] [Nontrivial 𝕜] : stdSimplex 𝕜 ι = ∅ 
Arguments: (𝕜 : Type u_1) (ι : Type u_5) [OrderedSemiring 𝕜] [Fintype ι] [IsEmpty ι] [Nontrivial 𝕜]
Goal: stdSimplex 𝕜 ι = ∅
-- stdSimplex_unique
lemma stdSimplex_unique [Unique ι] : stdSimplex 𝕜 ι = {fun _ ↦ 1} 
Arguments: (𝕜 : Type u_1) (ι : Type u_5) [OrderedSemiring 𝕜] [Fintype ι] [Unique ι]
Goal: stdSimplex 𝕜 ι = {fun x => 1}
-- single_mem_stdSimplex
theorem single_mem_stdSimplex (i : ι) : Pi.single i 1 ∈ stdSimplex 𝕜 ι 
Arguments: (𝕜 : Type u_1) {ι : Type u_5} [OrderedSemiring 𝕜] [Fintype ι] [DecidableEq ι] (i : ι)
Goal: Pi.single i 1 ∈ stdSimplex 𝕜 ι
-- ite_eq_mem_stdSimplex
theorem ite_eq_mem_stdSimplex (i : ι) : (if i = · then (1 : 𝕜) else 0) ∈ stdSimplex 𝕜 ι 
Arguments: (𝕜 : Type u_1) {ι : Type u_5} [OrderedSemiring 𝕜] [Fintype ι] [DecidableEq ι] (i : ι)
Goal: (fun x => if i = x then 1 else 0) ∈ stdSimplex 𝕜 ι
-- segment_single_subset_stdSimplex
lemma segment_single_subset_stdSimplex (i j : ι) :
    ([Pi.single i 1 -[𝕜] Pi.single j 1] : Set (ι → 𝕜)) ⊆ stdSimplex 𝕜 ι 
Arguments: (𝕜 : Type u_1) {ι : Type u_5} [OrderedSemiring 𝕜] [Fintype ι] [DecidableEq ι] (i j : ι)
Goal: [Pi.single i 1-[𝕜]Pi.single j 1] ⊆ stdSimplex 𝕜 ι
-- stdSimplex_fin_two
lemma stdSimplex_fin_two :
    stdSimplex 𝕜 (Fin 2) = ([Pi.single 0 1 -[𝕜] Pi.single 1 1] : Set (Fin 2 → 𝕜)) 
Arguments: (𝕜 : Type u_1) [OrderedSemiring 𝕜]
Goal: stdSimplex 𝕜 (Fin 2) = [Pi.single 0 1-[𝕜]Pi.single 1 1]
-- stdSimplexEquivIcc_apply_coe
@[simps (config := .asFn)]
def stdSimplexEquivIcc : stdSimplex 𝕜 (Fin 2) ≃ Icc (0 : 𝕜) 1 
Arguments: (𝕜 : Type u_1) [OrderedRing 𝕜] (f : ↑(stdSimplex 𝕜 (Fin 2)))
Goal: ↑((stdSimplexEquivIcc 𝕜) f) = ↑f 0
-- stdSimplexEquivIcc_symm_apply_coe
@[simps (config := .asFn)]
def stdSimplexEquivIcc : stdSimplex 𝕜 (Fin 2) ≃ Icc (0 : 𝕜) 1 
Arguments: (𝕜 : Type u_1) [OrderedRing 𝕜] (x : ↑(Set.Icc 0 1))
Goal: ↑((stdSimplexEquivIcc 𝕜).symm x) = ![↑x, 1 - ↑x]
-- entry_norm_bound_of_unitary
theorem entry_norm_bound_of_unitary {U : Matrix n n 𝕜} (hU : U ∈ Matrix.unitaryGroup n 𝕜)
    (i j : n) : ‖U i j‖ ≤ 1 
Arguments: {𝕜 : Type u_1} {n : Type u_3} [RCLike 𝕜] [Fintype n] [DecidableEq n] {U : Matrix n n 𝕜} (hU : U ∈ Matrix.unitaryGroup n 𝕜) (i j : n)
Goal: ‖U i j‖ ≤ 1
-- entrywise_sup_norm_bound_of_unitary
theorem entrywise_sup_norm_bound_of_unitary {U : Matrix n n 𝕜} (hU : U ∈ Matrix.unitaryGroup n 𝕜) :
    ‖U‖ ≤ 1 
Arguments: {𝕜 : Type u_1} {n : Type u_3} [RCLike 𝕜] [Fintype n] [DecidableEq n] {U : Matrix n n 𝕜} (hU : U ∈ Matrix.unitaryGroup n 𝕜)
Goal: ‖U‖ ≤ 1
-- Matrix.coe_toEuclideanCLM_eq_toEuclideanLin
lemma coe_toEuclideanCLM_eq_toEuclideanLin (A : Matrix n n 𝕜) :
    (toEuclideanCLM (n := n) (𝕜 := 𝕜) A : _ →ₗ[𝕜] _) = toEuclideanLin A 
Arguments: {𝕜 : Type u_1} {n : Type u_3} [RCLike 𝕜] [Fintype n] [DecidableEq n] (A : Matrix n n 𝕜)
Goal: ↑(Matrix.toEuclideanCLM A) = Matrix.toEuclideanLin A
-- Matrix.toEuclideanCLM_piLp_equiv_symm
@[simp]
lemma toEuclideanCLM_piLp_equiv_symm (A : Matrix n n 𝕜) (x : n → 𝕜) :
    toEuclideanCLM (n := n) (𝕜 := 𝕜) A ((WithLp.equiv _ _).symm x) =
      (WithLp.equiv _ _).symm (toLin' A x) 
Arguments: {𝕜 : Type u_1} {n : Type u_3} [RCLike 𝕜] [Fintype n] [DecidableEq n] (A : Matrix n n 𝕜) (x : n → 𝕜)
Goal: (Matrix.toEuclideanCLM A) ((WithLp.equiv 2 ((i : n) → (fun x => 𝕜) i)).symm x) =
  (WithLp.equiv 2 (n → 𝕜)).symm ((Matrix.toLin' A) x)
-- Matrix.piLp_equiv_toEuclideanCLM
@[simp]
lemma piLp_equiv_toEuclideanCLM (A : Matrix n n 𝕜) (x : EuclideanSpace 𝕜 n) :
    WithLp.equiv _ _ (toEuclideanCLM (n := n) (𝕜 := 𝕜) A x) =
      toLin' A (WithLp.equiv _ _ x) 
Arguments: {𝕜 : Type u_1} {n : Type u_3} [RCLike 𝕜] [Fintype n] [DecidableEq n] (A : Matrix n n 𝕜) (x : EuclideanSpace 𝕜 n)
Goal: (WithLp.equiv 2 ((i : n) → (fun x => 𝕜) i)) ((Matrix.toEuclideanCLM A) x) =
  (Matrix.toLin' A) ((WithLp.equiv 2 (n → 𝕜)) x)
-- Matrix.l2_opNorm_def
lemma l2_opNorm_def (A : Matrix m n 𝕜) :
    ‖A‖ = ‖(toEuclideanLin (𝕜 := 𝕜) (m := m) (n := n)).trans toContinuousLinearMap A‖ 
Arguments: {𝕜 : Type u_1} {m : Type u_2} {n : Type u_3} [RCLike 𝕜] [Fintype m] [Fintype n] [DecidableEq n] (A : Matrix m n 𝕜)
Goal: ‖A‖ = ‖(Matrix.toEuclideanLin ≪≫ₗ LinearMap.toContinuousLinearMap) A‖