mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Topology.Sion

Goal state before:
pp: case a.inl
E : Type u_1
F : Type u_2
β : Type u_3
inst✝¹¹ : LinearOrder β
X : Set E
f : E → F → β
inst✝¹⁰ : TopologicalSpace E
inst✝⁹ : AddCommGroup E
inst✝⁸ : Module ℝ E
inst✝⁷ : IsTopologicalAddGroup E
inst✝⁶ : ContinuousSMul ℝ E
ne_X : X.Nonempty
kX : IsCompact X
inst✝⁵ : TopologicalSpace F
inst✝⁴ : AddCommGroup F
inst✝³ : Module ℝ F
inst✝² : DenselyOrdered β
inst✝¹ : IsTopologicalAddGroup F
inst✝ : ContinuousSMul ℝ F
cX : Convex ℝ X
sup_y : E → β
inf_sup : β
hinf_sup : IsGLB {x | ∃ x_1 ∈ X, sup_y x_1 = x} inf_sup
inf_x : F → β
sup_inf : β
hfy : ∀ y ∈ ∅, LowerSemicontinuousOn (fun x ↦ f x y) X
hfy' : ∀ y ∈ ∅, QuasiconvexOn ℝ X fun x ↦ f x y
cY : Convex ℝ ∅
hfx : ∀ x ∈ X, UpperSemicontinuousOn (fun y ↦ f x y) ∅
hfx' : ∀ x ∈ X, QuasiconcaveOn ℝ ∅ fun y ↦ f x y
hsup_y : ∀ x ∈ X, IsLUB {x_1 | ∃ y ∈ ∅, f x y = x_1} (sup_y x)
hinf_x : ∀ y ∈ ∅, IsGLB {x | ∃ x_1 ∈ X, f x_1 y = x} (inf_x y)
hsup_inf : IsLUB {x | ∃ y ∈ ∅, inf_x y = x} sup_inf
⊢ inf_sup ≤ sup_inf

pp: case a.inr
E : Type u_1
F : Type u_2
β : Type u_3
inst✝¹¹ : LinearOrder β
X : Set E
Y : Set F
f : E → F → β
inst✝¹⁰ : TopologicalSpace E
inst✝⁹ : AddCommGroup E
inst✝⁸ : Module ℝ E
inst✝⁷ : IsTopologicalAddGroup E
inst✝⁶ : ContinuousSMul ℝ E
ne_X : X.Nonempty
kX : IsCompact X
hfy : ∀ y ∈ Y, LowerSemicontinuousOn (fun x ↦ f x y) X
hfy' : ∀ y ∈ Y, QuasiconvexOn ℝ X fun x ↦ f x y
inst✝⁵ : TopologicalSpace F
inst✝⁴ : AddCommGroup F
inst✝³ : Module ℝ F
cY : Convex ℝ Y
hfx : ∀ x ∈ X, UpperSemicontinuousOn (fun y ↦ f x y) Y
hfx' : ∀ x ∈ X, QuasiconcaveOn ℝ Y fun y ↦ f x y
inst✝² : DenselyOrdered β
inst✝¹ : IsTopologicalAddGroup F
inst✝ : ContinuousSMul ℝ F
cX : Convex ℝ X
sup_y : E → β
hsup_y : ∀ x ∈ X, IsLUB {x_1 | ∃ y ∈ Y, f x y = x_1} (sup_y x)
inf_sup : β
hinf_sup : IsGLB {x | ∃ x_1 ∈ X, sup_y x_1 = x} inf_sup
inf_x : F → β
hinf_x : ∀ y ∈ Y, IsGLB {x | ∃ x_1 ∈ X, f x_1 y = x} (inf_x y)
sup_inf : β
hsup_inf : IsLUB {x | ∃ y ∈ Y, inf_x y = x} sup_inf
ne_Y : Y.Nonempty
⊢ inf_sup ≤ sup_inf

Tactic:
·

Goal state after:
case a.inl
E : Type u_1
F : Type u_2
β : Type u_3
inst✝¹¹ : LinearOrder β
X : Set E
f : E → F → β
inst✝¹⁰ : TopologicalSpace E
inst✝⁹ : AddCommGroup E
inst✝⁸ : Module ℝ E
inst✝⁷ : IsTopologicalAddGroup E
inst✝⁶ : ContinuousSMul ℝ E
ne_X : X.Nonempty
kX : IsCompact X
inst✝⁵ : TopologicalSpace F
inst✝⁴ : AddCommGroup F
inst✝³ : Module ℝ F
inst✝² : DenselyOrdered β
inst✝¹ : IsTopologicalAddGroup F
inst✝ : ContinuousSMul ℝ F
cX : Convex ℝ X
sup_y : E → β
inf_sup : β
hinf_sup : IsGLB {x | ∃ x_1 ∈ X, sup_y x_1 = x} inf_sup
inf_x : F → β
sup_inf : β
hfy : ∀ y ∈ ∅, LowerSemicontinuousOn (fun x ↦ f x y) X
hfy' : ∀ y ∈ ∅, QuasiconvexOn ℝ X fun x ↦ f x y
cY : Convex ℝ ∅
hfx : ∀ x ∈ X, UpperSemicontinuousOn (fun y ↦ f x y) ∅
hfx' : ∀ x ∈ X, QuasiconcaveOn ℝ ∅ fun y ↦ f x y
hsup_y : ∀ x ∈ X, IsLUB {x_1 | ∃ y ∈ ∅, f x y = x_1} (sup_y x)
hinf_x : ∀ y ∈ ∅, IsGLB {x | ∃ x_1 ∈ X, f x_1 y = x} (inf_x y)
hsup_inf : IsLUB {x | ∃ y ∈ ∅, inf_x y = x} sup_inf
⊢ inf_sup ≤ sup_inf
Module: Mathlib.Topology.Sion

Goal state before:
pp: case a.inl.mp
E : Type u_1
F : Type u_2
β : Type u_3
inst✝¹¹ : LinearOrder β
X : Set E
f : E → F → β
inst✝¹⁰ : TopologicalSpace E
inst✝⁹ : AddCommGroup E
inst✝⁸ : Module ℝ E
inst✝⁷ : IsTopologicalAddGroup E
inst✝⁶ : ContinuousSMul ℝ E
ne_X : X.Nonempty
kX : IsCompact X
inst✝⁵ : TopologicalSpace F
inst✝⁴ : AddCommGroup F
inst✝³ : Module ℝ F
inst✝² : DenselyOrdered β
inst✝¹ : IsTopologicalAddGroup F
inst✝ : ContinuousSMul ℝ F
cX : Convex ℝ X
sup_y : E → β
inf_sup : β
hinf_sup : IsGLB {x | ∃ x_1 ∈ X, sup_y x_1 = x} inf_sup
inf_x : F → β
sup_inf : β
cY : Convex ℝ ∅
hfx : ∀ x ∈ X, UpperSemicontinuousOn (fun y ↦ f x y) ∅
hfx' : ∀ x ∈ X, QuasiconcaveOn ℝ ∅ fun y ↦ f x y
hfy hfy' hinf_x : True
hsup_inf : IsBot sup_inf
hsup_y : ∀ x ∈ X, sup_y x = sup_inf
t : β
⊢ (∃ x ∈ X, sup_y x = t) → t = sup_inf

pp: case a.inl.mpr
E : Type u_1
F : Type u_2
β : Type u_3
inst✝¹¹ : LinearOrder β
X : Set E
f : E → F → β
inst✝¹⁰ : TopologicalSpace E
inst✝⁹ : AddCommGroup E
inst✝⁸ : Module ℝ E
inst✝⁷ : IsTopologicalAddGroup E
inst✝⁶ : ContinuousSMul ℝ E
ne_X : X.Nonempty
kX : IsCompact X
inst✝⁵ : TopologicalSpace F
inst✝⁴ : AddCommGroup F
inst✝³ : Module ℝ F
inst✝² : DenselyOrdered β
inst✝¹ : IsTopologicalAddGroup F
inst✝ : ContinuousSMul ℝ F
cX : Convex ℝ X
sup_y : E → β
inf_sup : β
hinf_sup : IsGLB {x | ∃ x_1 ∈ X, sup_y x_1 = x} inf_sup
inf_x : F → β
sup_inf : β
cY : Convex ℝ ∅
hfx : ∀ x ∈ X, UpperSemicontinuousOn (fun y ↦ f x y) ∅
hfx' : ∀ x ∈ X, QuasiconcaveOn ℝ ∅ fun y ↦ f x y
hfy hfy' hinf_x : True
hsup_inf : IsBot sup_inf
hsup_y : ∀ x ∈ X, sup_y x = sup_inf
t : β
⊢ t = sup_inf → ∃ x ∈ X, sup_y x = t

Tactic:
·

Goal state after:
case a.inl.mp
E : Type u_1
F : Type u_2
β : Type u_3
inst✝¹¹ : LinearOrder β
X : Set E
f : E → F → β
inst✝¹⁰ : TopologicalSpace E
inst✝⁹ : AddCommGroup E
inst✝⁸ : Module ℝ E
inst✝⁷ : IsTopologicalAddGroup E
inst✝⁶ : ContinuousSMul ℝ E
ne_X : X.Nonempty
kX : IsCompact X
inst✝⁵ : TopologicalSpace F
inst✝⁴ : AddCommGroup F
inst✝³ : Module ℝ F
inst✝² : DenselyOrdered β
inst✝¹ : IsTopologicalAddGroup F
inst✝ : ContinuousSMul ℝ F
cX : Convex ℝ X
sup_y : E → β
inf_sup : β
hinf_sup : IsGLB {x | ∃ x_1 ∈ X, sup_y x_1 = x} inf_sup
inf_x : F → β
sup_inf : β
cY : Convex ℝ ∅
hfx : ∀ x ∈ X, UpperSemicontinuousOn (fun y ↦ f x y) ∅
hfx' : ∀ x ∈ X, QuasiconcaveOn ℝ ∅ fun y ↦ f x y
hfy hfy' hinf_x : True
hsup_inf : IsBot sup_inf
hsup_y : ∀ x ∈ X, sup_y x = sup_inf
t : β
⊢ (∃ x ∈ X, sup_y x = t) → t = sup_inf
Module: Mathlib.Topology.Sion

Goal state before:
pp: case a.inl.mpr
E : Type u_1
F : Type u_2
β : Type u_3
inst✝¹¹ : LinearOrder β
X : Set E
f : E → F → β
inst✝¹⁰ : TopologicalSpace E
inst✝⁹ : AddCommGroup E
inst✝⁸ : Module ℝ E
inst✝⁷ : IsTopologicalAddGroup E
inst✝⁶ : ContinuousSMul ℝ E
ne_X : X.Nonempty
kX : IsCompact X
inst✝⁵ : TopologicalSpace F
inst✝⁴ : AddCommGroup F
inst✝³ : Module ℝ F
inst✝² : DenselyOrdered β
inst✝¹ : IsTopologicalAddGroup F
inst✝ : ContinuousSMul ℝ F
cX : Convex ℝ X
sup_y : E → β
inf_sup : β
hinf_sup : IsGLB {x | ∃ x_1 ∈ X, sup_y x_1 = x} inf_sup
inf_x : F → β
sup_inf : β
cY : Convex ℝ ∅
hfx : ∀ x ∈ X, UpperSemicontinuousOn (fun y ↦ f x y) ∅
hfx' : ∀ x ∈ X, QuasiconcaveOn ℝ ∅ fun y ↦ f x y
hfy hfy' hinf_x : True
hsup_inf : IsBot sup_inf
hsup_y : ∀ x ∈ X, sup_y x = sup_inf
t : β
⊢ t = sup_inf → ∃ x ∈ X, sup_y x = t

Tactic:
·

Goal state after:
case a.inl.mpr
E : Type u_1
F : Type u_2
β : Type u_3
inst✝¹¹ : LinearOrder β
X : Set E
f : E → F → β
inst✝¹⁰ : TopologicalSpace E
inst✝⁹ : AddCommGroup E
inst✝⁸ : Module ℝ E
inst✝⁷ : IsTopologicalAddGroup E
inst✝⁶ : ContinuousSMul ℝ E
ne_X : X.Nonempty
kX : IsCompact X
inst✝⁵ : TopologicalSpace F
inst✝⁴ : AddCommGroup F
inst✝³ : Module ℝ F
inst✝² : DenselyOrdered β
inst✝¹ : IsTopologicalAddGroup F
inst✝ : ContinuousSMul ℝ F
cX : Convex ℝ X
sup_y : E → β
inf_sup : β
hinf_sup : IsGLB {x | ∃ x_1 ∈ X, sup_y x_1 = x} inf_sup
inf_x : F → β
sup_inf : β
cY : Convex ℝ ∅
hfx : ∀ x ∈ X, UpperSemicontinuousOn (fun y ↦ f x y) ∅
hfx' : ∀ x ∈ X, QuasiconcaveOn ℝ ∅ fun y ↦ f x y
hfy hfy' hinf_x : True
hsup_inf : IsBot sup_inf
hsup_y : ∀ x ∈ X, sup_y x = sup_inf
t : β
⊢ t = sup_inf → ∃ x ∈ X, sup_y x = t
Module: Mathlib.Topology.VectorBundle.ContinuousAlternatingMap

Goal state before:
pp: case intro.refine_2
𝕜 : Type u_1
ι : Type u_2
inst✝²² : NontriviallyNormedField 𝕜
B : Type u_3
inst✝²¹ : TopologicalSpace B
F₁ : Type u_4
inst✝²⁰ : NormedAddCommGroup F₁
inst✝¹⁹ : NormedSpace 𝕜 F₁
E₁ : B → Type u_5
inst✝¹⁸ : (x : B) → AddCommGroup (E₁ x)
inst✝¹⁷ : (x : B) → Module 𝕜 (E₁ x)
inst✝¹⁶ : TopologicalSpace (TotalSpace F₁ E₁)
F₂ : Type u_6
inst✝¹⁵ : NormedAddCommGroup F₂
inst✝¹⁴ : NormedSpace 𝕜 F₂
E₂ : B → Type u_7
inst✝¹³ : (x : B) → AddCommGroup (E₂ x)
inst✝¹² : (x : B) → Module 𝕜 (E₂ x)
inst✝¹¹ : TopologicalSpace (TotalSpace F₂ E₂)
inst✝¹⁰ : (x : B) → TopologicalSpace (E₁ x)
inst✝⁹ : FiberBundle F₁ E₁
inst✝⁸ : (x : B) → TopologicalSpace (E₂ x)
inst✝⁷ : FiberBundle F₂ E₂
e₁ e₁' : Trivialization F₁ TotalSpace.proj
e₂ e₂' : Trivialization F₂ TotalSpace.proj
inst✝⁶ : Finite ι
inst✝⁵ : VectorBundle 𝕜 F₁ E₁
inst✝⁴ : VectorBundle 𝕜 F₂ E₂
inst✝³ : MemTrivializationAtlas e₁
inst✝² : MemTrivializationAtlas e₁'
inst✝¹ : MemTrivializationAtlas e₂
inst✝ : MemTrivializationAtlas e₂'
val✝ : Fintype ι
⊢ ContinuousOn (fun b ↦ ContinuousAlternatingMap.compContinuousLinearMapCLM ↑(Trivialization.coordChangeL 𝕜 e₁' e₁ b))
    (e₁.baseSet ∩ e₂.baseSet ∩ (e₁'.baseSet ∩ e₂'.baseSet))

Tactic:
·

Goal state after:
case intro.refine_2
𝕜 : Type u_1
ι : Type u_2
inst✝²² : NontriviallyNormedField 𝕜
B : Type u_3
inst✝²¹ : TopologicalSpace B
F₁ : Type u_4
inst✝²⁰ : NormedAddCommGroup F₁
inst✝¹⁹ : NormedSpace 𝕜 F₁
E₁ : B → Type u_5
inst✝¹⁸ : (x : B) → AddCommGroup (E₁ x)
inst✝¹⁷ : (x : B) → Module 𝕜 (E₁ x)
inst✝¹⁶ : TopologicalSpace (TotalSpace F₁ E₁)
F₂ : Type u_6
inst✝¹⁵ : NormedAddCommGroup F₂
inst✝¹⁴ : NormedSpace 𝕜 F₂
E₂ : B → Type u_7
inst✝¹³ : (x : B) → AddCommGroup (E₂ x)
inst✝¹² : (x : B) → Module 𝕜 (E₂ x)
inst✝¹¹ : TopologicalSpace (TotalSpace F₂ E₂)
inst✝¹⁰ : (x : B) → TopologicalSpace (E₁ x)
inst✝⁹ : FiberBundle F₁ E₁
inst✝⁸ : (x : B) → TopologicalSpace (E₂ x)
inst✝⁷ : FiberBundle F₂ E₂
e₁ e₁' : Trivialization F₁ TotalSpace.proj
e₂ e₂' : Trivialization F₂ TotalSpace.proj
inst✝⁶ : Finite ι
inst✝⁵ : VectorBundle 𝕜 F₁ E₁
inst✝⁴ : VectorBundle 𝕜 F₂ E₂
inst✝³ : MemTrivializationAtlas e₁
inst✝² : MemTrivializationAtlas e₁'
inst✝¹ : MemTrivializationAtlas e₂
inst✝ : MemTrivializationAtlas e₂'
val✝ : Fintype ι
⊢ ContinuousOn (fun b ↦ ContinuousAlternatingMap.compContinuousLinearMapCLM ↑(Trivialization.coordChangeL 𝕜 e₁' e₁ b))
    (e₁.baseSet ∩ e₂.baseSet ∩ (e₁'.baseSet ∩ e₂'.baseSet))
Module: Mathlib.Topology.VectorBundle.ContinuousAlternatingMap

Goal state before:
pp: 𝕜 : Type u_1
ι : Type u_2
inst✝¹⁹ : NontriviallyNormedField 𝕜
B : Type u_3
inst✝¹⁸ : TopologicalSpace B
F₁ : Type u_4
inst✝¹⁷ : NormedAddCommGroup F₁
inst✝¹⁶ : NormedSpace 𝕜 F₁
E₁ : B → Type u_5
inst✝¹⁵ : (x : B) → AddCommGroup (E₁ x)
inst✝¹⁴ : (x : B) → Module 𝕜 (E₁ x)
inst✝¹³ : TopologicalSpace (TotalSpace F₁ E₁)
F₂ : Type u_6
inst✝¹² : NormedAddCommGroup F₂
inst✝¹¹ : NormedSpace 𝕜 F₂
E₂ : B → Type u_7
inst✝¹⁰ : (x : B) → AddCommGroup (E₂ x)
inst✝⁹ : (x : B) → Module 𝕜 (E₂ x)
inst✝⁸ : TopologicalSpace (TotalSpace F₂ E₂)
inst✝⁷ : (x : B) → TopologicalSpace (E₁ x)
inst✝⁶ : FiberBundle F₁ E₁
inst✝⁵ : (x : B) → TopologicalSpace (E₂ x)
inst✝⁴ : FiberBundle F₂ E₂
e₁ e₁' : Trivialization F₁ TotalSpace.proj
e₂ e₂' : Trivialization F₂ TotalSpace.proj
inst✝³ : Trivialization.IsLinear 𝕜 e₁
inst✝² : Trivialization.IsLinear 𝕜 e₁'
inst✝¹ : Trivialization.IsLinear 𝕜 e₂
inst✝ : Trivialization.IsLinear 𝕜 e₂'
⊢ ∀ ⦃x : B × F₁ [⋀^ι]→L[𝕜] F₂⦄,
    x ∈ (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ →
      ({ proj := x.1,
              snd :=
                (Trivialization.symmL 𝕜 e₂ x.1).compContinuousAlternatingMap
                  (x.2.compContinuousLinearMap (Trivialization.continuousLinearMapAt 𝕜 e₁ x.1)) }.proj,
          (Trivialization.continuousLinearMapAt 𝕜 e₂
                { proj := x.1,
                    snd :=
                      (Trivialization.symmL 𝕜 e₂ x.1).compContinuousAlternatingMap
                        (x.2.compContinuousLinearMap
                          (Trivialization.continuousLinearMapAt 𝕜 e₁ x.1)) }.proj).compContinuousAlternatingMap
            ({ proj := x.1,
                    snd :=
                      (Trivialization.symmL 𝕜 e₂ x.1).compContinuousAlternatingMap
                        (x.2.compContinuousLinearMap
                          (Trivialization.continuousLinearMapAt 𝕜 e₁ x.1)) }.snd.compContinuousLinearMap
              (Trivialization.symmL 𝕜 e₁
                { proj := x.1,
                    snd :=
                      (Trivialization.symmL 𝕜 e₂ x.1).compContinuousAlternatingMap
                        (x.2.compContinuousLinearMap (Trivialization.continuousLinearMapAt 𝕜 e₁ x.1)) }.proj))) =
        x

Tactic:
by
    rintro ⟨x, f⟩ ⟨⟨h₁, h₂⟩, -⟩
    simp only [Prod.mk_right_inj]
    ext v
    simp [Function.comp_def, h₁, h₂]

Goal state after:
Module: Mathlib.Topology.VectorBundle.ContinuousAlternatingMap

Goal state before:
pp: case hb
𝕜 : Type u_1
ι : Type u_2
inst✝¹⁷ : NontriviallyNormedField 𝕜
B : Type u_3
inst✝¹⁶ : TopologicalSpace B
F₁ : Type u_4
inst✝¹⁵ : NormedAddCommGroup F₁
inst✝¹⁴ : NormedSpace 𝕜 F₁
E₁ : B → Type u_5
inst✝¹³ : (x : B) → AddCommGroup (E₁ x)
inst✝¹² : (x : B) → Module 𝕜 (E₁ x)
inst✝¹¹ : TopologicalSpace (TotalSpace F₁ E₁)
F₂ : Type u_6
inst✝¹⁰ : NormedAddCommGroup F₂
inst✝⁹ : NormedSpace 𝕜 F₂
E₂ : B → Type u_7
inst✝⁸ : (x : B) → AddCommGroup (E₂ x)
inst✝⁷ : (x : B) → Module 𝕜 (E₂ x)
inst✝⁶ : TopologicalSpace (TotalSpace F₂ E₂)
inst✝⁵ : (x : B) → TopologicalSpace (E₁ x)
inst✝⁴ : FiberBundle F₁ E₁
inst✝³ : (x : B) → TopologicalSpace (E₂ x)
inst✝² : FiberBundle F₂ E₂
e₁ : Trivialization F₁ TotalSpace.proj
e₂ : Trivialization F₂ TotalSpace.proj
inst✝¹ : Trivialization.IsLinear 𝕜 e₁
inst✝ : Trivialization.IsLinear 𝕜 e₂
b : B
hb : b ∈ e₁.baseSet ∩ e₂.baseSet
L : F₁ [⋀^ι]→L[𝕜] F₂
⊢ b ∈ (continuousAlternatingMap 𝕜 ι e₁ e₂).baseSet

Tactic:
·

Goal state after:
case hb
𝕜 : Type u_1
ι : Type u_2
inst✝¹⁷ : NontriviallyNormedField 𝕜
B : Type u_3
inst✝¹⁶ : TopologicalSpace B
F₁ : Type u_4
inst✝¹⁵ : NormedAddCommGroup F₁
inst✝¹⁴ : NormedSpace 𝕜 F₁
E₁ : B → Type u_5
inst✝¹³ : (x : B) → AddCommGroup (E₁ x)
inst✝¹² : (x : B) → Module 𝕜 (E₁ x)
inst✝¹¹ : TopologicalSpace (TotalSpace F₁ E₁)
F₂ : Type u_6
inst✝¹⁰ : NormedAddCommGroup F₂
inst✝⁹ : NormedSpace 𝕜 F₂
E₂ : B → Type u_7
inst✝⁸ : (x : B) → AddCommGroup (E₂ x)
inst✝⁷ : (x : B) → Module 𝕜 (E₂ x)
inst✝⁶ : TopologicalSpace (TotalSpace F₂ E₂)
inst✝⁵ : (x : B) → TopologicalSpace (E₁ x)
inst✝⁴ : FiberBundle F₁ E₁
inst✝³ : (x : B) → TopologicalSpace (E₂ x)
inst✝² : FiberBundle F₂ E₂
e₁ : Trivialization F₁ TotalSpace.proj
e₂ : Trivialization F₂ TotalSpace.proj
inst✝¹ : Trivialization.IsLinear 𝕜 e₁
inst✝ : Trivialization.IsLinear 𝕜 e₂
b : B
hb : b ∈ e₁.baseSet ∩ e₂.baseSet
L : F₁ [⋀^ι]→L[𝕜] F₂
⊢ b ∈ (continuousAlternatingMap 𝕜 ι e₁ e₂).baseSet