mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.CategoryTheory.CofilteredSystem

Goal state before:
pp: J : Type u
inst✝¹ : Category.{v_1, u} J
F : J ⥤ Type v
inst✝ : IsCofilteredOrEmpty J
Fsur : ∀ ⦃i j : J⦄ (f : i ⟶ j), Function.Surjective ⇑(ConcreteCategory.hom (F.map f))
i j : J
f g : i ⟶ j
k : J
φ : k ⟶ i
hφ : φ ≫ f = φ ≫ g
⊢ ConcreteCategory.hom (F.map f) = ConcreteCategory.hom (F.map g)

Tactic:
have := congrArg F.map hφ

Goal state after:
J : Type u
inst✝¹ : Category.{v_1, u} J
F : J ⥤ Type v
inst✝ : IsCofilteredOrEmpty J
Fsur : ∀ ⦃i j : J⦄ (f : i ⟶ j), Function.Surjective ⇑(ConcreteCategory.hom (F.map f))
i j : J
f g : i ⟶ j
k : J
φ : k ⟶ i
hφ : φ ≫ f = φ ≫ g
this : F.map (φ ≫ f) = F.map (φ ≫ g)
⊢ ConcreteCategory.hom (F.map f) = ConcreteCategory.hom (F.map g)
Module: Mathlib.CategoryTheory.CofilteredSystem

Goal state before:
pp: J : Type u
inst✝¹ : Category.{v_1, u} J
F : J ⥤ Type v
inst✝ : IsCofilteredOrEmpty J
j : J
Finj : ∀ (i : J) (f : i ⟶ j), Function.Injective ⇑(ConcreteCategory.hom (F.map f))
i : J
f : i ⟶ j
s₀ s₁ : ↑F.sections
k m : J
mi : m ⟶ i
h : (ConcreteCategory.hom (F.map (mi ≫ f))) (↑s₀ m) = (ConcreteCategory.hom (F.map (mi ≫ f))) (↑s₁ m)
mk : m ⟶ k
h✝ : True
⊢ (ConcreteCategory.hom (F.map mk)) (↑s₀ m) = ↑s₁ k

Tactic:
← s₁.prop mk

Goal state after:
J : Type u
inst✝¹ : Category.{v_1, u} J
F : J ⥤ Type v
inst✝ : IsCofilteredOrEmpty J
j : J
Finj : ∀ (i : J) (f : i ⟶ j), Function.Injective ⇑(ConcreteCategory.hom (F.map f))
i : J
f : i ⟶ j
s₀ s₁ : ↑F.sections
k m : J
mi : m ⟶ i
h : (ConcreteCategory.hom (F.map (mi ≫ f))) (↑s₀ m) = (ConcreteCategory.hom (F.map (mi ≫ f))) (↑s₁ m)
mk : m ⟶ k
h✝ : True
⊢ (ConcreteCategory.hom (F.map mk)) (↑s₀ m) = (ConcreteCategory.hom (F.map mk)) (↑s₁ m)
Module: Mathlib.Data.Matrix.PEquiv

Goal state before:
pp: m : Type u_3
n : Type u_4
α : Type u_5
inst✝³ : DecidableEq m
inst✝² : DecidableEq n
inst✝¹ : Zero α
inst✝ : One α
f : m ≃. n
i✝ : n
j✝ : m
⊢ f.symm.toMatrix i✝ j✝ = f.toMatrixᵀ i✝ j✝

Tactic:
simp only [transpose, mem_iff_mem f, toMatrix_apply]

Goal state after:
m : Type u_3
n : Type u_4
α : Type u_5
inst✝³ : DecidableEq m
inst✝² : DecidableEq n
inst✝¹ : Zero α
inst✝ : One α
f : m ≃. n
i✝ : n
j✝ : m
⊢ (if i✝ ∈ f j✝ then 1 else 0) = of (fun x y ↦ if x ∈ f y then 1 else 0) i✝ j✝
Module: Mathlib.Analysis.Convex.Between

Goal state before:
pp: case e'_9.convert_6
R : Type u_1
V : Type u_2
P : Type u_4
inst✝⁷ : Ring R
inst✝⁶ : PartialOrder R
inst✝⁵ : AddCommGroup V
inst✝⁴ : Module R V
inst✝³ : AffineSpace V P
inst✝² : IsOrderedRing R
inst✝¹ : IsDomain R
inst✝ : IsTorsionFree R V
n : ℕ
s : Simplex R P n
p : P
i j : Fin (n + 1)
h : i ≠ j
h' : Sbtw R (s.points i) p (s.points j) ↔ Sbtw R (s.points (min i j)) p (s.points (max i j))
⊢ 0 < 1 + 1

Tactic:
· lia

Goal state after:
Module: Mathlib.Analysis.Convex.Between

Goal state before:
pp: case e'_11.convert_6
R : Type u_1
V : Type u_2
P : Type u_4
inst✝⁷ : Ring R
inst✝⁶ : PartialOrder R
inst✝⁵ : AddCommGroup V
inst✝⁴ : Module R V
inst✝³ : AffineSpace V P
inst✝² : IsOrderedRing R
inst✝¹ : IsDomain R
inst✝ : IsTorsionFree R V
n : ℕ
s : Simplex R P n
p : P
i j : Fin (n + 1)
h : i ≠ j
h' : Sbtw R (s.points i) p (s.points j) ↔ Sbtw R (s.points (min i j)) p (s.points (max i j))
⊢ 0 < 1 + 1

Tactic:
· lia

Goal state after:
Module: Mathlib.Analysis.Convex.DoublyStochasticMatrix

Goal state before:
pp: R : Type u_1
n : Type u_2
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : Semiring R
inst✝¹ : PartialOrder R
inst✝ : IsOrderedRing R
M : Matrix n n R
x : n → R
hA : M ∈ doublyStochastic R n
⊢ ∑ i, (M *ᵥ x) i = ∑ i, x i

Tactic:
apply sum_mulVec_of_mem_colStochastic
  grind

Goal state after:
Module: Mathlib.Analysis.Convex.GaugeRescale

Goal state before:
pp: E : Type u_1
inst✝¹ : AddCommGroup E
inst✝ : Module ℝ E
s t : Set E
x : E
⊢ gauge t (gaugeRescale s t x) ≤ gauge s x

Tactic:
by_cases hx : gauge t x = 0

Goal state after:
case pos
E : Type u_1
inst✝¹ : AddCommGroup E
inst✝ : Module ℝ E
s t : Set E
x : E
hx : gauge t x = 0
⊢ gauge t (gaugeRescale s t x) ≤ gauge s x

case neg
E : Type u_1
inst✝¹ : AddCommGroup E
inst✝ : Module ℝ E
s t : Set E
x : E
hx : ¬gauge t x = 0
⊢ gauge t (gaugeRescale s t x) ≤ gauge s x
Module: Mathlib.Analysis.Normed.Affine.Convex

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
s t : Set E
hs₁ : Convex ℝ s
hs₂ : IsCompact s
ht : t ∈ 𝓝ˢ s
U : Set E
hU₁ : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] U
hU₂ : s ⊆ U
hU₃ : U ⊆ t
V : Set E
hV₁ : V ∈ 𝓝 0
hV₂ : V + s ⊆ U
⊢ ∃ u, s ⊆ interior ((convexHull ℝ) ↑u) ∧ (convexHull ℝ) ↑u ⊆ t

Tactic:
rcases exists_mem_interior_convexHull_affineBasis hV₁ with ⟨b, hb₁, hb₂⟩

Goal state after:
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
s t : Set E
hs₁ : Convex ℝ s
hs₂ : IsCompact s
ht : t ∈ 𝓝ˢ s
U : Set E
hU₁ : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] U
hU₂ : s ⊆ U
hU₃ : U ⊆ t
V : Set E
hV₁ : V ∈ 𝓝 0
hV₂ : V + s ⊆ U
b : AffineBasis (Fin (finrank ℝ E + 1)) ℝ E
hb₁ : 0 ∈ interior ((convexHull ℝ) (range ⇑b))
hb₂ : (convexHull ℝ) (range ⇑b) ⊆ V
⊢ ∃ u, s ⊆ interior ((convexHull ℝ) ↑u) ∧ (convexHull ℝ) ↑u ⊆ t
Module: Mathlib.Analysis.Convex.GaugeRescale

Goal state before:
pp: case refine_2
E : Type u_1
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module ℝ E
inst✝³ : TopologicalSpace E
inst✝² : IsTopologicalAddGroup E
inst✝¹ : ContinuousSMul ℝ E
inst✝ : T1Space E
s t : Set E
hsc : Convex ℝ s
hsb : IsVonNBounded ℝ s
hst : Convex ℝ t
htb : IsVonNBounded ℝ t
x : E
hx : x ∈ interior s
y : E
hy : y ∈ interior t
h : E ≃ₜ E := gaugeRescaleHomeomorph (-x +ᵥ s) (-y +ᵥ t) ⋯ ⋯ ⋯ ⋯ ⋯ ⋯
⊢ ⇑((Homeomorph.addLeft (-x)).trans (h.trans (Homeomorph.addLeft y))) '' closure[inst✝³] s = closure[inst✝³] t

Tactic:
calc
      (fun a ↦ y + h (-x + a)) '' closure s = y +ᵥ h '' closure (-x +ᵥ s) := by
        simp_rw [closure_vadd, ← image_vadd, image_image, vadd_eq_add]
      _ = _ := by rw [image_gaugeRescaleHomeomorph_closure, closure_vadd, vadd_neg_vadd]

Goal state after:
Module: Mathlib.Analysis.Convex.Continuous

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
C : Set E
f : E → ℝ
hf : ConvexOn ℝ C f
x₀ : E
hC : C ∈ 𝓝 x₀
r : ℝ
hr : ∀ᶠ (x : ℝ) in Filter.map f (𝓝 x₀), (fun x1 x2 ↦ x1 ≤ x2) x r
⊢ ∀ᶠ (x : ℝ) in Filter.map |f| (𝓝 x₀), (fun x1 x2 ↦ x1 ≤ x2) x (|r| + 2 * |f x₀|)

Tactic:
have : (𝓝 x₀).Tendsto (fun y => 2 • x₀ - y) (𝓝 x₀) :=
    tendsto_nhds_nhds.2 (⟨·, ·, by simp [two_nsmul, dist_comm]⟩)

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
C : Set E
f : E → ℝ
hf : ConvexOn ℝ C f
x₀ : E
hC : C ∈ 𝓝 x₀
r : ℝ
hr : ∀ᶠ (x : ℝ) in Filter.map f (𝓝 x₀), (fun x1 x2 ↦ x1 ≤ x2) x r
this : Filter.Tendsto (fun y ↦ 2 • x₀ - y) (𝓝 x₀) (𝓝 x₀)
⊢ ∀ᶠ (x : ℝ) in Filter.map |f| (𝓝 x₀), (fun x1 x2 ↦ x1 ≤ x2) x (|r| + 2 * |f x₀|)
Module: Mathlib.Analysis.Convex.Integral

Goal state before:
pp: α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
μ : Measure α
s : Set E
f : α → E
inst✝ : IsProbabilityMeasure μ
hs : Convex ℝ s
hsc : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
hf : ∀ᵐ (x : α) ∂μ, f x ∈ s
this✝¹ : MeasurableSpace E := borel E
this✝ : BorelSpace E
g : α → E
hfi : Integrable g μ
hgm : StronglyMeasurable g
hfg : f =ᵐ[μ] g
this : SeparableSpace ↑(range g ∩ s)
y₀ : E
h₀ : y₀ ∈ range g ∩ s
x : α
hx : g x ∈ s
⊢ g x ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (range g ∩ s)

Tactic:
apply subset_closure

Goal state after:
α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
μ : Measure α
s : Set E
f : α → E
inst✝ : IsProbabilityMeasure μ
hs : Convex ℝ s
hsc : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
hf : ∀ᵐ (x : α) ∂μ, f x ∈ s
this✝¹ : MeasurableSpace E := ⋯
this✝ : BorelSpace E
g : α → E
hfi : Integrable g μ
hgm : StronglyMeasurable g
hfg : f =ᵐ[μ] g
this : SeparableSpace ↑(range g ∩ s)
y₀ : E
h₀ : y₀ ∈ range g ∩ s
x : α
hx : g x ∈ s
⊢ g x ∈ range g ∩ s
Module: Mathlib.Analysis.Convex.Continuous

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
C : Set E
f : E → ℝ
hC : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] C
hC' : C.Nonempty
hf : ConvexOn ℝ C f
tfae_1_to_2 : LocallyLipschitzOn C f → ContinuousOn f C
tfae_2_to_3 : ContinuousOn f C → ∃ x₀ ∈ C, ContinuousAt f x₀
tfae_3_to_4 : (∃ x₀ ∈ C, ContinuousAt f x₀) → ∃ x₀ ∈ C, Filter.IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) (𝓝 x₀) f
x₀ : E
hx₀ : x₀ ∈ C
r : ℝ
hr : ∀ᶠ (x : ℝ) in Filter.map f (𝓝 x₀), (fun x1 x2 ↦ x1 ≤ x2) x r
x : E
hx : x ∈ C
⊢ Filter.IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) (𝓝 x) f

Tactic:
have : ∀ᶠ δ in 𝓝 (0 : ℝ), (1 - δ)⁻¹ • x - (δ / (1 - δ)) • x₀ ∈ C := by
      have h : ContinuousAt (fun δ : ℝ ↦ (1 - δ)⁻¹ • x - (δ / (1 - δ)) • x₀) 0 := by
        fun_prop (disch := norm_num)
      exact h (by simpa using! hC.mem_nhds hx)
    obtain ⟨δ, hδ₀, hy, hδ₁⟩ := (this.and <| eventually_lt_nhds zero_lt_one).exists_gt
    set y := (1 - δ)⁻¹ • x - (δ / (1 - δ)) • x₀
    refine ⟨max r (f y), ?_⟩
    simp only [Filter.eventually_map] at hr ⊢
    obtain ⟨ε, hε, hr⟩ := Metric.eventually_nhds_iff.1 <| hr.and (hC.eventually_mem hx₀)
    refine Metric.eventually_nhds_iff.2 ⟨ε * δ, by positivity, fun z hz ↦ ?_⟩
    have hx₀' : δ⁻¹ • (x - y) + y = x₀ := MulAction.injective₀ (sub_ne_zero.2 hδ₁.ne') <| by
      simp [y, smul_sub, smul_smul, hδ₀.ne', div_eq_mul_inv, sub_ne_zero.2 hδ₁.ne', mul_left_comm,
        sub_mul, sub_smul]
    let w := δ⁻¹ • (z - y) + y
    have hwyz : δ • w + (1 - δ) • y = z := by simp [w, hδ₀.ne', sub_smul]
    have hw : dist w x₀ < ε := by
      simpa [w, ← hx₀', dist_smul₀, abs_of_nonneg, hδ₀.le, inv_mul_lt_iff₀', hδ₀]
    calc
      f z ≤ max (f w) (f y) :=
        hf.le_max_of_mem_segment (hr hw).2 hy ⟨_, _, hδ₀.le, sub_nonneg.2 hδ₁.le, by simp, hwyz⟩
      _ ≤ max r (f y) := by gcongr; exact (hr hw).1

Goal state after:
Module: Mathlib.Analysis.Convex.KreinMilman

Goal state before:
pp: case inl
E : Type u_1
inst✝⁶ : AddCommGroup E
inst✝⁵ : Module ℝ E
inst✝⁴ : TopologicalSpace E
inst✝³ : T2Space E
inst✝² : IsTopologicalAddGroup E
inst✝¹ : ContinuousSMul ℝ E
inst✝ : LocallyConvexSpace ℝ E
s : Set E
hscomp : IsCompact s
hsnemp : s.Nonempty
S : Set (Set E) := {t | t.Nonempty ∧ IsClosed t ∧ IsExtreme ℝ s t}
hFS : ∅ ⊆ S
hF : IsChain (fun x1 x2 ↦ x1 ⊆ x2) ∅
⊢ ∃ lb ∈ S, ∀ s ∈ ∅, lb ⊆ s

pp: case inr
E : Type u_1
inst✝⁶ : AddCommGroup E
inst✝⁵ : Module ℝ E
inst✝⁴ : TopologicalSpace E
inst✝³ : T2Space E
inst✝² : IsTopologicalAddGroup E
inst✝¹ : ContinuousSMul ℝ E
inst✝ : LocallyConvexSpace ℝ E
s : Set E
hscomp : IsCompact s
hsnemp : s.Nonempty
S : Set (Set E) := {t | t.Nonempty ∧ IsClosed t ∧ IsExtreme ℝ s t}
F : Set (Set E)
hFS : F ⊆ S
hF : IsChain (fun x1 x2 ↦ x1 ⊆ x2) F
hFnemp : F.Nonempty
⊢ ∃ lb ∈ S, ∀ s ∈ F, lb ⊆ s

Tactic:
· exact ⟨s, ⟨hsnemp, hscomp.isClosed, IsExtreme.rfl⟩, fun _ => False.elim⟩

Goal state after:
case inr
E : Type u_1
inst✝⁶ : AddCommGroup E
inst✝⁵ : Module ℝ E
inst✝⁴ : TopologicalSpace E
inst✝³ : T2Space E
inst✝² : IsTopologicalAddGroup E
inst✝¹ : ContinuousSMul ℝ E
inst✝ : LocallyConvexSpace ℝ E
s : Set E
hscomp : IsCompact s
hsnemp : s.Nonempty
S : Set (Set E) := {t | t.Nonempty ∧ IsClosed t ∧ IsExtreme ℝ s t}
F : Set (Set E)
hFS : F ⊆ S
hF : IsChain (fun x1 x2 ↦ x1 ⊆ x2) F
hFnemp : F.Nonempty
⊢ ∃ lb ∈ S, ∀ s ∈ F, lb ⊆ s
Module: Mathlib.Analysis.Convex.KreinMilman

Goal state before:
pp: E : Type u_1
inst✝⁶ : AddCommGroup E
inst✝⁵ : Module ℝ E
inst✝⁴ : TopologicalSpace E
inst✝³ : T2Space E
inst✝² : IsTopologicalAddGroup E
inst✝¹ : ContinuousSMul ℝ E
inst✝ : LocallyConvexSpace ℝ E
s : Set E
hscomp : IsCompact s
hAconv : Convex ℝ s
hs : ¬s ⊆ closure ((convexHull ℝ) (extremePoints ℝ s))
x : E
hxA : x ∈ s
hxt : x ∉ closure ((convexHull ℝ) (extremePoints ℝ s))
l : StrongDual ℝ E
r : ℝ
hlr : ∀ a ∈ closure ((convexHull ℝ) (extremePoints ℝ s)), l a < r
hrx : r < l x
⊢ False

Tactic:
have h : IsExposed ℝ s ({ y ∈ s | ∀ z ∈ s, l z ≤ l y }) := fun _ => ⟨l, rfl⟩

Goal state after:
E : Type u_1
inst✝⁶ : AddCommGroup E
inst✝⁵ : Module ℝ E
inst✝⁴ : TopologicalSpace E
inst✝³ : T2Space E
inst✝² : IsTopologicalAddGroup E
inst✝¹ : ContinuousSMul ℝ E
inst✝ : LocallyConvexSpace ℝ E
s : Set E
hscomp : IsCompact s
hAconv : Convex ℝ s
hs : ¬s ⊆ closure ((convexHull ℝ) (extremePoints ℝ s))
x : E
hxA : x ∈ s
hxt : x ∉ closure ((convexHull ℝ) (extremePoints ℝ s))
l : StrongDual ℝ E
r : ℝ
hlr : ∀ a ∈ closure ((convexHull ℝ) (extremePoints ℝ s)), l a < r
hrx : r < l x
h : IsExposed ℝ s {y | y ∈ s ∧ ∀ z ∈ s, l z ≤ l y}
⊢ False
Module: Mathlib.Geometry.Convex.ConvexSpace.Defs

Goal state before:
pp: X : Type u_2
K : Type u_8
inst✝³ : Semifield K
inst✝² : LinearOrder K
inst✝¹ : IsStrictOrderedRing K
w : StdSimplex K X
p : X → Prop
inst✝ : DecidablePred p
⊢ ∀ i ∈ (filter p w.weights).support, 0 ≤ (filter p w.weights) i

Tactic:
by simp [filter_apply, apply_ite]

Goal state after:
Module: Mathlib.Geometry.Convex.ConvexSpace.Defs

Goal state before:
pp: R : Type u_1
X : Type u_2
M : Type u_3
N : Type u_4
P : Type u_5
I : Type u_6
J : Type u_7
K : Type u_8
inst✝² : Semifield K
inst✝¹ : LinearOrder K
inst✝ : IsStrictOrderedRing K
w : StdSimplex K X
s : Set X
hs : ∃ x ∈ s, w.weights x ≠ 0
x✝ : X
⊢ 0 x✝ ≤ (filter (fun x ↦ x ∈ s) w.weights) x✝

Tactic:
by simp [filter_apply, apply_ite]

Goal state after:
Module: Mathlib.Geometry.Convex.ConvexSpace.AffineSpace

Goal state before:
pp: case s
R : Type u_1
V : Type u_2
P : Type u_3
inst✝⁵ : Ring R
inst✝⁴ : PartialOrder R
inst✝³ : IsStrictOrderedRing R
inst✝² : AddCommGroup V
inst✝¹ : Module R V
inst✝ : AffineSpace V P
f : StdSimplex R (StdSimplex R P)
b : P
hL :
  (Finset.affineCombination R (StdSimplex.map convexCombination f).weights.support id)
      ⇑(StdSimplex.map convexCombination f).weights =
    ((StdSimplex.map convexCombination f).weights.support.weightedVSubOfPoint id b)
        ⇑(StdSimplex.map convexCombination f).weights +ᵥ
      b
hR :
  (Finset.affineCombination R f.join.weights.support id) ⇑f.join.weights =
    (f.join.weights.support.weightedVSubOfPoint id b) ⇑f.join.weights +ᵥ b
| f.weights.support

pp: case f
R : Type u_1
V : Type u_2
P : Type u_3
inst✝⁵ : Ring R
inst✝⁴ : PartialOrder R
inst✝³ : IsStrictOrderedRing R
inst✝² : AddCommGroup V
inst✝¹ : Module R V
inst✝ : AffineSpace V P
f : StdSimplex R (StdSimplex R P)
b : P
hL :
  (Finset.affineCombination R (StdSimplex.map convexCombination f).weights.support id)
      ⇑(StdSimplex.map convexCombination f).weights =
    ((StdSimplex.map convexCombination f).weights.support.weightedVSubOfPoint id b)
        ⇑(StdSimplex.map convexCombination f).weights +ᵥ
      b
hR :
  (Finset.affineCombination R f.join.weights.support id) ⇑f.join.weights =
    (f.join.weights.support.weightedVSubOfPoint id b) ⇑f.join.weights +ᵥ b
| fun a ↦ f.weights a • ((Finset.affineCombination R a.weights.support id) ⇑a.weights -ᵥ b)

Tactic:
· skip

Goal state after:
case f
R : Type u_1
V : Type u_2
P : Type u_3
inst✝⁵ : Ring R
inst✝⁴ : PartialOrder R
inst✝³ : IsStrictOrderedRing R
inst✝² : AddCommGroup V
inst✝¹ : Module R V
inst✝ : AffineSpace V P
f : StdSimplex R (StdSimplex R P)
b : P
hL :
  (Finset.affineCombination R (StdSimplex.map convexCombination f).weights.support id)
      ⇑(StdSimplex.map convexCombination f).weights =
    ((StdSimplex.map convexCombination f).weights.support.weightedVSubOfPoint id b)
        ⇑(StdSimplex.map convexCombination f).weights +ᵥ
      b
hR :
  (Finset.affineCombination R f.join.weights.support id) ⇑f.join.weights =
    (f.join.weights.support.weightedVSubOfPoint id b) ⇑f.join.weights +ᵥ b
| fun a ↦ f.weights a • ((Finset.affineCombination R a.weights.support id) ⇑a.weights -ᵥ b)
Module: Mathlib.Geometry.Convex.Set

Goal state before:
pp: R : Type u_3
X : Type u_5
inst✝⁴ : Semiring R
inst✝³ : PartialOrder R
inst✝² : IsStrictOrderedRing R
inst✝¹ : ConvexSpace R X
s : Set X
Y : Type u_7
inst✝ : ConvexSpace R Y
t : Set Y
hs : IsConvexSet R s
ht : IsConvexSet R t
⊢ IsConvexSet R (s ×ˢ t)

Tactic:
classical
  rintro w hw
  refine ⟨hs ?_, ht ?_⟩
  · grw [StdSimplex.weights_map, mapDomain_support, Finset.coe_image, hw, fst_image_prod_subset]
  · grw [StdSimplex.weights_map, mapDomain_support, Finset.coe_image, hw, snd_image_prod_subset]

Goal state after:
Module: Mathlib.Analysis.Convex.MetricSpace

Goal state before:
pp: X : Type u_2
inst✝³ : ConvexSpace ℝ X
inst✝² : MetricSpace X
inst✝¹ : IsConvexDist X
T : Type u_3
inst✝ : TopologicalSpace T
f : T → ℝ
hf : Continuous f
hf0 : ∀ (t : T), 0 ≤ f t
hf1 : ∀ (t : T), f t ≤ 1
x y : T → X
hx : ContinuousOn x (f ⁻¹' {0}ᶜ)
hy : ContinuousOn y (f ⁻¹' {1}ᶜ)
hx' : Bornology.IsBounded (Set.range x)
hy' : Bornology.IsBounded (Set.range y)
D : ℝ
hD' : 0 < D
hD : ∀ (t₁ t₂ : T), dist (x t₁) (y t₂) ≤ D
t : T
ht : f t ∉ Set.Ioo 0 1
⊢ f t = 0 ∨ f t = 1

Tactic:
by
    simpa [le_antisymm_iff, hf0, hf1, -not_and, not_and_or] using ht

Goal state after:
Module: Mathlib.Combinatorics.SimpleGraph.Basic

Goal state before:
pp: V : Type u
G : SimpleGraph V
⊢ G = ⊤ ↔ ∀ (a b : V), a ≠ b → G.Adj a b

Tactic:
by
  simp [← top_le_iff, le_iff_adj]

Goal state after:
Module: Mathlib.Combinatorics.SimpleGraph.Basic

Goal state before:
pp: V : Type u
G : SimpleGraph V
v : V
⊢ (G.neighborSet v).Nonempty ↔ ¬G.IsIsolated v

Tactic:
simp [Set.nonempty_iff_ne_empty]

Goal state after:
Module: Mathlib.Analysis.Convex.StrictConvexBetween

Goal state before:
pp: V : Type u_1
P : Type u_2
inst✝¹¹ : NormedAddCommGroup V
inst✝¹⁰ : NormedSpace ℝ V
inst✝⁹ : StrictConvexSpace ℝ V
E : Type u_3
F : Type u_4
PE : Type u_5
PF : Type u_6
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : NormedSpace ℝ F
inst✝⁴ : StrictConvexSpace ℝ E
inst✝³ : MetricSpace PE
inst✝² : MetricSpace PF
inst✝¹ : NormedAddTorsor E PE
inst✝ : NormedAddTorsor F PF
r : ℝ
f : PF → PE
x✝ y✝ z : PE
hi : Isometry f
x y : PF
⊢ f (midpoint ℝ x y) = midpoint ℝ (f x) (f y)

Tactic:
apply eq_midpoint_of_dist_eq_half

Goal state after:
case hx
V : Type u_1
P : Type u_2
inst✝¹¹ : NormedAddCommGroup V
inst✝¹⁰ : NormedSpace ℝ V
inst✝⁹ : StrictConvexSpace ℝ V
E : Type u_3
F : Type u_4
PE : Type u_5
PF : Type u_6
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : NormedSpace ℝ F
inst✝⁴ : StrictConvexSpace ℝ E
inst✝³ : MetricSpace PE
inst✝² : MetricSpace PF
inst✝¹ : NormedAddTorsor E PE
inst✝ : NormedAddTorsor F PF
r : ℝ
f : PF → PE
x✝ y✝ z : PE
hi : Isometry f
x y : PF
⊢ dist (f x) (f (midpoint ℝ x y)) = dist (f x) (f y) / 2

case hy
V : Type u_1
P : Type u_2
inst✝¹¹ : NormedAddCommGroup V
inst✝¹⁰ : NormedSpace ℝ V
inst✝⁹ : StrictConvexSpace ℝ V
E : Type u_3
F : Type u_4
PE : Type u_5
PF : Type u_6
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : NormedSpace ℝ F
inst✝⁴ : StrictConvexSpace ℝ E
inst✝³ : MetricSpace PE
inst✝² : MetricSpace PF
inst✝¹ : NormedAddTorsor E PE
inst✝ : NormedAddTorsor F PF
r : ℝ
f : PF → PE
x✝ y✝ z : PE
hi : Isometry f
x y : PF
⊢ dist (f (midpoint ℝ x y)) (f y) = dist (f x) (f y) / 2
Module: Mathlib.Analysis.Convex.StrictConvexBetween

Goal state before:
pp: case hx
V : Type u_1
P : Type u_2
inst✝¹¹ : NormedAddCommGroup V
inst✝¹⁰ : NormedSpace ℝ V
inst✝⁹ : StrictConvexSpace ℝ V
E : Type u_3
F : Type u_4
PE : Type u_5
PF : Type u_6
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : NormedSpace ℝ F
inst✝⁴ : StrictConvexSpace ℝ E
inst✝³ : MetricSpace PE
inst✝² : MetricSpace PF
inst✝¹ : NormedAddTorsor E PE
inst✝ : NormedAddTorsor F PF
r : ℝ
f : PF → PE
x✝ y✝ z : PE
hi : Isometry f
x y : PF
⊢ dist (f x) (f (midpoint ℝ x y)) = dist (f x) (f y) / 2

Tactic:
rw [hi.dist_eq, hi.dist_eq]

Goal state after:
case hx
V : Type u_1
P : Type u_2
inst✝¹¹ : NormedAddCommGroup V
inst✝¹⁰ : NormedSpace ℝ V
inst✝⁹ : StrictConvexSpace ℝ V
E : Type u_3
F : Type u_4
PE : Type u_5
PF : Type u_6
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : NormedSpace ℝ F
inst✝⁴ : StrictConvexSpace ℝ E
inst✝³ : MetricSpace PE
inst✝² : MetricSpace PF
inst✝¹ : NormedAddTorsor E PE
inst✝ : NormedAddTorsor F PF
r : ℝ
f : PF → PE
x✝ y✝ z : PE
hi : Isometry f
x y : PF
⊢ dist x (midpoint ℝ x y) = dist x y / 2
Module: Mathlib.Analysis.Convex.StrictConvexBetween

Goal state before:
pp: case hy
V : Type u_1
P : Type u_2
inst✝¹¹ : NormedAddCommGroup V
inst✝¹⁰ : NormedSpace ℝ V
inst✝⁹ : StrictConvexSpace ℝ V
E : Type u_3
F : Type u_4
PE : Type u_5
PF : Type u_6
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : NormedSpace ℝ F
inst✝⁴ : StrictConvexSpace ℝ E
inst✝³ : MetricSpace PE
inst✝² : MetricSpace PF
inst✝¹ : NormedAddTorsor E PE
inst✝ : NormedAddTorsor F PF
r : ℝ
f : PF → PE
x✝ y✝ z : PE
hi : Isometry f
x y : PF
⊢ dist (f (midpoint ℝ x y)) (f y) = dist (f x) (f y) / 2

Tactic:
rw [hi.dist_eq, hi.dist_eq]

Goal state after:
case hy
V : Type u_1
P : Type u_2
inst✝¹¹ : NormedAddCommGroup V
inst✝¹⁰ : NormedSpace ℝ V
inst✝⁹ : StrictConvexSpace ℝ V
E : Type u_3
F : Type u_4
PE : Type u_5
PF : Type u_6
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : NormedSpace ℝ F
inst✝⁴ : StrictConvexSpace ℝ E
inst✝³ : MetricSpace PE
inst✝² : MetricSpace PF
inst✝¹ : NormedAddTorsor E PE
inst✝ : NormedAddTorsor F PF
r : ℝ
f : PF → PE
x✝ y✝ z : PE
hi : Isometry f
x y : PF
⊢ dist (midpoint ℝ x y) y = dist x y / 2
Module: Mathlib.Analysis.Convex.Visible

Goal state before:
pp: 𝕜 : Type u_1
V : Type u_2
P : Type u_3
inst✝⁵ : Field 𝕜
inst✝⁴ : LinearOrder 𝕜
inst✝³ : IsOrderedRing 𝕜
inst✝² : AddCommGroup V
inst✝¹ : Module 𝕜 V
inst✝ : AddTorsor V P
s : Set P
x y : P
⊢ IsVisible 𝕜 s x y ↔ IsVisible 𝕜 s y x

Tactic:
by
  simp [IsVisible, sbtw_comm]

Goal state after:
Module: Mathlib.Analysis.Convex.Side

Goal state before:
pp: case neg
V : Type u_2
P : Type u_4
inst✝³ : SeminormedAddCommGroup V
inst✝² : NormedSpace ℝ V
inst✝¹ : PseudoMetricSpace P
inst✝ : NormedAddTorsor V P
s : AffineSubspace ℝ P
x : P
h : (↑s).Nonempty
hx : x ∉ s
⊢ IsPreconnected {y | s.SSameSide x y}

Tactic:
exact (isConnected_setOfPred_sSameSide hx h).isPreconnected

Goal state after:
Module: Mathlib.Analysis.Convex.Visible

Goal state before:
pp: 𝕜 : Type u_1
V : Type u_2
inst✝⁹ : Field 𝕜
inst✝⁸ : LinearOrder 𝕜
inst✝⁷ : IsStrictOrderedRing 𝕜
inst✝⁶ : AddCommGroup V
inst✝⁵ : Module 𝕜 V
s : Set V
x y : V
inst✝⁴ : TopologicalSpace 𝕜
inst✝³ : OrderTopology 𝕜
inst✝² : TopologicalSpace V
inst✝¹ : IsTopologicalAddGroup V
inst✝ : ContinuousSMul 𝕜 V
hsxy : IsVisible 𝕜 s x y
hy : y ∈ interior s
hxy : x ≠ y
hmem✝ : ∀ᶠ (δ : 𝕜) in 𝓝[>] 0, (lineMap y x) δ ∈ s
δ : 𝕜
hmem : (lineMap y x) δ ∈ s
⊢ δ ∈ Set.Ioo 0 1 → False

Tactic:
hsbt

Goal state after:
𝕜 : Type u_1
V : Type u_2
inst✝⁹ : Field 𝕜
inst✝⁸ : LinearOrder 𝕜
inst✝⁷ : IsStrictOrderedRing 𝕜
inst✝⁶ : AddCommGroup V
inst✝⁵ : Module 𝕜 V
s : Set V
x y : V
inst✝⁴ : TopologicalSpace 𝕜
inst✝³ : OrderTopology 𝕜
inst✝² : TopologicalSpace V
inst✝¹ : IsTopologicalAddGroup V
inst✝ : ContinuousSMul 𝕜 V
hsxy : IsVisible 𝕜 s x y
hy : y ∈ interior s
hxy : x ≠ y
hmem✝ : ∀ᶠ (δ : 𝕜) in 𝓝[>] 0, (lineMap y x) δ ∈ s
δ : 𝕜
hmem : (lineMap y x) δ ∈ s
hsbt : δ ∈ Set.Ioo 0 1
⊢ False
Module: Mathlib.Analysis.Convex.Side

Goal state before:
pp: case refine_2
R : Type u_1
V : Type u_2
P : Type u_4
inst✝⁶ : Field R
inst✝⁵ : LinearOrder R
inst✝⁴ : IsStrictOrderedRing R
inst✝³ : AddCommGroup V
inst✝² : Module R V
inst✝¹ : AffineSpace V P
n : ℕ
inst✝ : NeZero n
s : Simplex R P n
w₁ w₂ : Fin (n + 1) → R
hw₁ : ∑ j, w₁ j = 1
hw₂ : ∑ j, w₂ j = 1
i : Fin (n + 1)
h : SignType.sign (w₁ i) = SignType.sign (w₂ i) ∨ w₁ i = 0 ∨ w₂ i = 0
⊢ (affineSpan R (Set.range (s.faceOpposite i).points)).WSameSide ((Finset.affineCombination R Finset.univ s.points) w₁)
    ((Finset.affineCombination R Finset.univ s.points) w₂)

Tactic:
by_cases h0 : w₁ i = 0
    · exact wSameSide_of_left_mem _ ((s.affineCombination_mem_affineSpan_faceOpposite_iff hw₁).2 h0)
    · by_cases h0' : w₂ i = 0
      · exact wSameSide_of_right_mem _
          ((s.affineCombination_mem_affineSpan_faceOpposite_iff hw₂).2 h0')
      simp only [h0, h0', or_self, or_false] at h
      exact (s.sSameSide_affineSpan_faceOpposite_of_sign_eq hw₁ hw₂ h h0).wSameSide

Goal state after: