mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Normed.Operator.Bilinear

Goal state before:
pp: 𝕜 : Type u_1
𝕜₂ : Type u_2
𝕜₃ : Type u_3
E : Type u_4
Eₗ : Type u_5
F : Type u_6
Fₗ : Type u_7
G : Type u_8
Gₗ : Type u_9
𝓕 : Type u_10
inst✝¹⁹ : SeminormedAddCommGroup E
inst✝¹⁸ : SeminormedAddCommGroup Eₗ
inst✝¹⁷ : SeminormedAddCommGroup F
inst✝¹⁶ : SeminormedAddCommGroup Fₗ
inst✝¹⁵ : SeminormedAddCommGroup G
inst✝¹⁴ : SeminormedAddCommGroup Gₗ
inst✝¹³ : NontriviallyNormedField 𝕜
inst✝¹² : NontriviallyNormedField 𝕜₂
inst✝¹¹ : NontriviallyNormedField 𝕜₃
inst✝¹⁰ : NormedSpace 𝕜 E
inst✝⁹ : NormedSpace 𝕜 Eₗ
inst✝⁸ : NormedSpace 𝕜₂ F
inst✝⁷ : NormedSpace 𝕜 Fₗ
inst✝⁶ : NormedSpace 𝕜₃ G
inst✝⁵ : NormedSpace 𝕜 Gₗ
σ₁₂ : 𝕜 →+* 𝕜₂
σ₂₃ : 𝕜₂ →+* 𝕜₃
σ₁₃ : 𝕜 →+* 𝕜₃
inst✝⁴ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃
inst✝³ : FunLike 𝓕 E F
inst✝² : RingHomIsometric σ₂₃
inst✝¹ : RingHomIsometric σ₁₃
inst✝ : RingHomIsometric σ₁₂
f : F →SL[σ₂₃] G
g : E →SL[σ₁₂] F
⊢ ‖((LinearMap.mk₂'ₛₗ (RingHom.id 𝕜₃) σ₂₃ comp ⋯ ⋯ ⋯ ⋯) f) g‖ ≤ 1 * ‖f‖ * ‖g‖

Tactic:
by

Goal state after:
Module: Mathlib.Topology.UniformSpace.CompactConvergence

Goal state before:
pp: α : Type u₁
β : Type u₂
inst✝² : TopologicalSpace α
inst✝¹ : UniformSpace β
s : Set α
inst✝ : CompactSpace ↑s
f : α → β
hf : ContinuousOn f s
ι : Type u_1
p : Filter ι
F : ι → α → β
hF : ∀ (i : ι), ContinuousOn (F i) s
⊢ Tendsto (fun i ↦ { toFun := s.domRestrict (F i), continuous_toFun := ⋯ }) p
      (𝓝 { toFun := s.domRestrict f, continuous_toFun := ⋯ }) ↔
    TendstoUniformlyOn F f p s

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Operator.Bilinear

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_4
Eₗ : Type u_5
Fₗ : Type u_7
Gₗ : Type u_9
inst✝⁸ : SeminormedAddCommGroup E
inst✝⁷ : SeminormedAddCommGroup Eₗ
inst✝⁶ : SeminormedAddCommGroup Fₗ
inst✝⁵ : SeminormedAddCommGroup Gₗ
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedSpace 𝕜 Eₗ
inst✝¹ : NormedSpace 𝕜 Fₗ
inst✝ : NormedSpace 𝕜 Gₗ
L : E →L[𝕜] Fₗ →L[𝕜] Gₗ
⊢ ‖compL 𝕜 Eₗ Fₗ Gₗ‖ * ‖L‖ ≤ 1 * ‖L‖

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Operator.Bilinear

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_4
Eₗ : Type u_5
Fₗ : Type u_7
Gₗ : Type u_9
inst✝⁸ : SeminormedAddCommGroup E
inst✝⁷ : SeminormedAddCommGroup Eₗ
inst✝⁶ : SeminormedAddCommGroup Fₗ
inst✝⁵ : SeminormedAddCommGroup Gₗ
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedSpace 𝕜 Eₗ
inst✝¹ : NormedSpace 𝕜 Fₗ
inst✝ : NormedSpace 𝕜 Gₗ
L : E →L[𝕜] Fₗ →L[𝕜] Gₗ
⊢ 1 * ‖L‖ = ‖L‖

Tactic:
by

Goal state after:
Module: Mathlib.Topology.UniformSpace.CompactConvergence

Goal state before:
pp: α : Type u₁
β : Type u₂
inst✝³ : TopologicalSpace α
inst✝² : UniformSpace β
X : Type u_1
inst✝¹ : TopologicalSpace X
f : X → α → β
s : Set α
hf : ∀ (x : X), ContinuousOn (f x) s
inst✝ : CompactSpace ↑s
⊢ (Continuous[inst✝¹, _] fun x ↦ { toFun := s.domRestrict (f x), continuous_toFun := ⋯ }) ↔
    Continuous[inst✝¹, topologicalSpace α β {s}] fun x ↦ (ofFun {s}) (f x)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Operator.Bilinear

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_4
Eₗ : Type u_5
Fₗ : Type u_7
Gₗ : Type u_9
inst✝⁸ : SeminormedAddCommGroup E
inst✝⁷ : SeminormedAddCommGroup Eₗ
inst✝⁶ : SeminormedAddCommGroup Fₗ
inst✝⁵ : SeminormedAddCommGroup Gₗ
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedSpace 𝕜 Eₗ
inst✝¹ : NormedSpace 𝕜 Fₗ
inst✝ : NormedSpace 𝕜 Gₗ
L : E →L[𝕜] Fₗ →L[𝕜] Gₗ
⊢ ‖precompL Eₗ L‖ ≤ ‖L‖

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Operator.Bilinear

Goal state before:
pp: 𝕜 : Type u_1
𝕜₂ : Type u_2
𝕜₃ : Type u_3
E : Type u_4
F : Type u_6
G : Type u_8
inst✝¹⁹ : SeminormedAddCommGroup E
inst✝¹⁸ : SeminormedAddCommGroup F
inst✝¹⁷ : SeminormedAddCommGroup G
inst✝¹⁶ : NontriviallyNormedField 𝕜
inst✝¹⁵ : NontriviallyNormedField 𝕜₂
inst✝¹⁴ : NontriviallyNormedField 𝕜₃
inst✝¹³ : NormedSpace 𝕜 E
inst✝¹² : NormedSpace 𝕜₂ F
inst✝¹¹ : NormedSpace 𝕜₃ G
σ₂₃ : 𝕜₂ →+* 𝕜₃
σ₁₃ : 𝕜 →+* 𝕜₃
E' : Type u_11
F' : Type u_12
inst✝¹⁰ : SeminormedAddCommGroup E'
inst✝⁹ : SeminormedAddCommGroup F'
𝕜₁' : Type u_13
𝕜₂' : Type u_14
inst✝⁸ : NontriviallyNormedField 𝕜₁'
inst✝⁷ : NontriviallyNormedField 𝕜₂'
inst✝⁶ : NormedSpace 𝕜₁' E'
inst✝⁵ : NormedSpace 𝕜₂' F'
σ₁' : 𝕜₁' →+* 𝕜
σ₁₃' : 𝕜₁' →+* 𝕜₃
σ₂' : 𝕜₂' →+* 𝕜₂
σ₂₃' : 𝕜₂' →+* 𝕜₃
inst✝⁴ : RingHomCompTriple σ₁' σ₁₃ σ₁₃'
inst✝³ : RingHomCompTriple σ₂' σ₂₃ σ₂₃'
inst✝² : RingHomIsometric σ₂₃
inst✝¹ : RingHomIsometric σ₁₃'
inst✝ : RingHomIsometric σ₂₃'
f : E →SL[σ₁₃] F →SL[σ₂₃] G
gF : F' →SL[σ₂'] F
⊢ f.bilinearComp 0 gF = 0

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Operator.Bilinear

Goal state before:
pp: 𝕜 : Type u_1
𝕜₂ : Type u_2
𝕜₃ : Type u_3
E : Type u_4
F : Type u_6
G : Type u_8
inst✝¹⁹ : SeminormedAddCommGroup E
inst✝¹⁸ : SeminormedAddCommGroup F
inst✝¹⁷ : SeminormedAddCommGroup G
inst✝¹⁶ : NontriviallyNormedField 𝕜
inst✝¹⁵ : NontriviallyNormedField 𝕜₂
inst✝¹⁴ : NontriviallyNormedField 𝕜₃
inst✝¹³ : NormedSpace 𝕜 E
inst✝¹² : NormedSpace 𝕜₂ F
inst✝¹¹ : NormedSpace 𝕜₃ G
σ₂₃ : 𝕜₂ →+* 𝕜₃
σ₁₃ : 𝕜 →+* 𝕜₃
E' : Type u_11
F' : Type u_12
inst✝¹⁰ : SeminormedAddCommGroup E'
inst✝⁹ : SeminormedAddCommGroup F'
𝕜₁' : Type u_13
𝕜₂' : Type u_14
inst✝⁸ : NontriviallyNormedField 𝕜₁'
inst✝⁷ : NontriviallyNormedField 𝕜₂'
inst✝⁶ : NormedSpace 𝕜₁' E'
inst✝⁵ : NormedSpace 𝕜₂' F'
σ₁' : 𝕜₁' →+* 𝕜
σ₁₃' : 𝕜₁' →+* 𝕜₃
σ₂' : 𝕜₂' →+* 𝕜₂
σ₂₃' : 𝕜₂' →+* 𝕜₃
inst✝⁴ : RingHomCompTriple σ₁' σ₁₃ σ₁₃'
inst✝³ : RingHomCompTriple σ₂' σ₂₃ σ₂₃'
inst✝² : RingHomIsometric σ₂₃
inst✝¹ : RingHomIsometric σ₁₃'
inst✝ : RingHomIsometric σ₂₃'
f : E →SL[σ₁₃] F →SL[σ₂₃] G
gE : E' →SL[σ₁'] E
⊢ f.bilinearComp gE 0 = 0

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Operator.Bilinear

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_4
Fₗ : Type u_7
Gₗ : Type u_9
inst✝⁶ : SeminormedAddCommGroup E
inst✝⁵ : SeminormedAddCommGroup Fₗ
inst✝⁴ : SeminormedAddCommGroup Gₗ
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedSpace 𝕜 Fₗ
inst✝ : NormedSpace 𝕜 Gₗ
f : E →L[𝕜] Fₗ →L[𝕜] Gₗ
x x' : E
y y' : Fₗ
⊢ (f (x + x')) (y + y') = (f x) y + (f.deriv₂ (x, y)) (x', y') + (f x') y'

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Operator.Bilinear

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_4
Fₗ : Type u_7
inst✝⁴ : SeminormedAddCommGroup E
inst✝³ : SeminormedAddCommGroup Fₗ
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedSpace 𝕜 E
inst✝ : NormedSpace 𝕜 Fₗ
c : StrongDual 𝕜 E
f : Fₗ
⊢ 0 ≤ ‖c‖ * ‖f‖

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Operator.Bilinear

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_4
Fₗ : Type u_7
inst✝⁴ : SeminormedAddCommGroup E
inst✝³ : SeminormedAddCommGroup Fₗ
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedSpace 𝕜 E
inst✝ : NormedSpace 𝕜 Fₗ
c : StrongDual 𝕜 E
f : Fₗ
x : E
⊢ ‖c x‖ * ‖f‖ ≤ ‖c‖ * ‖x‖ * ‖f‖

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Operator.Bilinear

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_4
Fₗ : Type u_7
inst✝⁴ : SeminormedAddCommGroup E
inst✝³ : SeminormedAddCommGroup Fₗ
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedSpace 𝕜 E
inst✝ : NormedSpace 𝕜 Fₗ
c : StrongDual 𝕜 E
f : Fₗ
x : E
⊢ ‖c‖ * ‖x‖ * ‖f‖ = ‖c‖ * ‖f‖ * ‖x‖

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Operator.Bilinear

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_4
Fₗ : Type u_7
inst✝⁴ : SeminormedAddCommGroup E
inst✝³ : SeminormedAddCommGroup Fₗ
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedSpace 𝕜 E
inst✝ : NormedSpace 𝕜 Fₗ
c : StrongDual 𝕜 E
f : Fₗ
hf : 0 < ‖f‖
⊢ 0 ≤ ‖smulRight c f‖ / ‖f‖

Tactic:
by

Goal state after:
Module: Mathlib.Topology.UniformSpace.CompactConvergence

Goal state before:
pp: α : Type u₁
β : Type u₂
inst✝³ : TopologicalSpace α
inst✝² : UniformSpace β
δ₁ : Type u_1
δ₂ : Type u_2
inst✝¹ : TopologicalSpace δ₁
inst✝ : TopologicalSpace δ₂
φ₁ : C(δ₁, α)
φ₂ : C(δ₂, α)
h_proper₁ : IsProperMap ⇑φ₁
h_proper₂ : IsProperMap ⇑φ₂
h_cover : range ⇑φ₁ ∪ range ⇑φ₂ = univ
⊢ inferInstance =
    UniformSpace.comap (fun x ↦ x.comp φ₁) inferInstance ⊓ UniformSpace.comap (fun x ↦ x.comp φ₂) inferInstance

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Operator.Bilinear

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_4
Fₗ : Type u_7
inst✝⁴ : SeminormedAddCommGroup E
inst✝³ : SeminormedAddCommGroup Fₗ
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedSpace 𝕜 E
inst✝ : NormedSpace 𝕜 Fₗ
c : StrongDual 𝕜 E
f : Fₗ
⊢ ‖smulRight c f‖ = ‖c‖ * ‖f‖

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Operator.Bilinear

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_4
inst✝² : SeminormedAddCommGroup E
inst✝¹ : NontriviallyNormedField 𝕜
inst✝ : NormedSpace 𝕜 E
x : E
⊢ ‖toSpanSingleton 𝕜 x‖ = ‖x‖

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Operator.Bilinear

Goal state before:
pp: 𝕜 : Type u_1
𝕜₂ : Type u_2
𝕜₃ : Type u_3
E : Type u_4
Eₗ : Type u_5
F : Type u_6
Fₗ : Type u_7
G : Type u_8
Gₗ : Type u_9
𝓕 : Type u_10
inst✝³² : SeminormedAddCommGroup E
inst✝³¹ : SeminormedAddCommGroup Eₗ
inst✝³⁰ : SeminormedAddCommGroup F
inst✝²⁹ : SeminormedAddCommGroup Fₗ
inst✝²⁸ : SeminormedAddCommGroup G
inst✝²⁷ : SeminormedAddCommGroup Gₗ
inst✝²⁶ : NontriviallyNormedField 𝕜
inst✝²⁵ : NontriviallyNormedField 𝕜₂
inst✝²⁴ : NontriviallyNormedField 𝕜₃
inst✝²³ : NormedSpace 𝕜 E
inst✝²² : NormedSpace 𝕜 Eₗ
inst✝²¹ : NormedSpace 𝕜₂ F
inst✝²⁰ : NormedSpace 𝕜 Fₗ
inst✝¹⁹ : NormedSpace 𝕜₃ G
inst✝¹⁸ : NormedSpace 𝕜 Gₗ
σ₁₂ : 𝕜 →+* 𝕜₂
σ₂₃ : 𝕜₂ →+* 𝕜₃
σ₁₃ : 𝕜 →+* 𝕜₃
inst✝¹⁷ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃
inst✝¹⁶ : FunLike 𝓕 E F
σ₂₁ : 𝕜₂ →+* 𝕜
inst✝¹⁵ : RingHomInvPair σ₁₂ σ₂₁
inst✝¹⁴ : RingHomInvPair σ₂₁ σ₁₂
E' : Type u_11
F' : Type u_12
inst✝¹³ : SeminormedAddCommGroup E'
inst✝¹² : SeminormedAddCommGroup F'
𝕜₁' : Type u_13
𝕜₂' : Type u_14
inst✝¹¹ : NontriviallyNormedField 𝕜₁'
inst✝¹⁰ : NontriviallyNormedField 𝕜₂'
inst✝⁹ : NormedSpace 𝕜₁' E'
inst✝⁸ : NormedSpace 𝕜₂' F'
σ₁' : 𝕜₁' →+* 𝕜
σ₁₃' : 𝕜₁' →+* 𝕜₃
σ₂' : 𝕜₂' →+* 𝕜₂
σ₂₃' : 𝕜₂' →+* 𝕜₃
inst✝⁷ : RingHomCompTriple σ₁' σ₁₃ σ₁₃'
inst✝⁶ : RingHomCompTriple σ₂' σ₂₃ σ₂₃'
inst✝⁵ : RingHomIsometric σ₂₃
inst✝⁴ : RingHomIsometric σ₁₃'
inst✝³ : RingHomIsometric σ₂₃'
inst✝² : RingHomIsometric σ₁₃
inst✝¹ : RingHomIsometric σ₁'
inst✝ : RingHomIsometric σ₂'
c₁ c₂ : StrongDual 𝕜 E
⊢ smulRightₗ (c₁ + c₂) = smulRightₗ c₁ + smulRightₗ c₂

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Operator.Bilinear

Goal state before:
pp: 𝕜 : Type u_1
𝕜₂ : Type u_2
𝕜₃ : Type u_3
E : Type u_4
Eₗ : Type u_5
F : Type u_6
Fₗ : Type u_7
G : Type u_8
Gₗ : Type u_9
𝓕 : Type u_10
inst✝³² : SeminormedAddCommGroup E
inst✝³¹ : SeminormedAddCommGroup Eₗ
inst✝³⁰ : SeminormedAddCommGroup F
inst✝²⁹ : SeminormedAddCommGroup Fₗ
inst✝²⁸ : SeminormedAddCommGroup G
inst✝²⁷ : SeminormedAddCommGroup Gₗ
inst✝²⁶ : NontriviallyNormedField 𝕜
inst✝²⁵ : NontriviallyNormedField 𝕜₂
inst✝²⁴ : NontriviallyNormedField 𝕜₃
inst✝²³ : NormedSpace 𝕜 E
inst✝²² : NormedSpace 𝕜 Eₗ
inst✝²¹ : NormedSpace 𝕜₂ F
inst✝²⁰ : NormedSpace 𝕜 Fₗ
inst✝¹⁹ : NormedSpace 𝕜₃ G
inst✝¹⁸ : NormedSpace 𝕜 Gₗ
σ₁₂ : 𝕜 →+* 𝕜₂
σ₂₃ : 𝕜₂ →+* 𝕜₃
σ₁₃ : 𝕜 →+* 𝕜₃
inst✝¹⁷ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃
inst✝¹⁶ : FunLike 𝓕 E F
σ₂₁ : 𝕜₂ →+* 𝕜
inst✝¹⁵ : RingHomInvPair σ₁₂ σ₂₁
inst✝¹⁴ : RingHomInvPair σ₂₁ σ₁₂
E' : Type u_11
F' : Type u_12
inst✝¹³ : SeminormedAddCommGroup E'
inst✝¹² : SeminormedAddCommGroup F'
𝕜₁' : Type u_13
𝕜₂' : Type u_14
inst✝¹¹ : NontriviallyNormedField 𝕜₁'
inst✝¹⁰ : NontriviallyNormedField 𝕜₂'
inst✝⁹ : NormedSpace 𝕜₁' E'
inst✝⁸ : NormedSpace 𝕜₂' F'
σ₁' : 𝕜₁' →+* 𝕜
σ₁₃' : 𝕜₁' →+* 𝕜₃
σ₂' : 𝕜₂' →+* 𝕜₂
σ₂₃' : 𝕜₂' →+* 𝕜₃
inst✝⁷ : RingHomCompTriple σ₁' σ₁₃ σ₁₃'
inst✝⁶ : RingHomCompTriple σ₂' σ₂₃ σ₂₃'
inst✝⁵ : RingHomIsometric σ₂₃
inst✝⁴ : RingHomIsometric σ₁₃'
inst✝³ : RingHomIsometric σ₂₃'
inst✝² : RingHomIsometric σ₁₃
inst✝¹ : RingHomIsometric σ₁'
inst✝ : RingHomIsometric σ₂'
m : 𝕜
c : StrongDual 𝕜 E
⊢ smulRightₗ (m • c) = (RingHom.id 𝕜) m • smulRightₗ c

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Operator.Bilinear

Goal state before:
pp: 𝕜 : Type u_1
𝕜₂ : Type u_2
𝕜₃ : Type u_3
E : Type u_4
Eₗ : Type u_5
F : Type u_6
Fₗ : Type u_7
G : Type u_8
Gₗ : Type u_9
𝓕 : Type u_10
inst✝³² : SeminormedAddCommGroup E
inst✝³¹ : SeminormedAddCommGroup Eₗ
inst✝³⁰ : SeminormedAddCommGroup F
inst✝²⁹ : SeminormedAddCommGroup Fₗ
inst✝²⁸ : SeminormedAddCommGroup G
inst✝²⁷ : SeminormedAddCommGroup Gₗ
inst✝²⁶ : NontriviallyNormedField 𝕜
inst✝²⁵ : NontriviallyNormedField 𝕜₂
inst✝²⁴ : NontriviallyNormedField 𝕜₃
inst✝²³ : NormedSpace 𝕜 E
inst✝²² : NormedSpace 𝕜 Eₗ
inst✝²¹ : NormedSpace 𝕜₂ F
inst✝²⁰ : NormedSpace 𝕜 Fₗ
inst✝¹⁹ : NormedSpace 𝕜₃ G
inst✝¹⁸ : NormedSpace 𝕜 Gₗ
σ₁₂ : 𝕜 →+* 𝕜₂
σ₂₃ : 𝕜₂ →+* 𝕜₃
σ₁₃ : 𝕜 →+* 𝕜₃
inst✝¹⁷ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃
inst✝¹⁶ : FunLike 𝓕 E F
σ₂₁ : 𝕜₂ →+* 𝕜
inst✝¹⁵ : RingHomInvPair σ₁₂ σ₂₁
inst✝¹⁴ : RingHomInvPair σ₂₁ σ₁₂
E' : Type u_11
F' : Type u_12
inst✝¹³ : SeminormedAddCommGroup E'
inst✝¹² : SeminormedAddCommGroup F'
𝕜₁' : Type u_13
𝕜₂' : Type u_14
inst✝¹¹ : NontriviallyNormedField 𝕜₁'
inst✝¹⁰ : NontriviallyNormedField 𝕜₂'
inst✝⁹ : NormedSpace 𝕜₁' E'
inst✝⁸ : NormedSpace 𝕜₂' F'
σ₁' : 𝕜₁' →+* 𝕜
σ₁₃' : 𝕜₁' →+* 𝕜₃
σ₂' : 𝕜₂' →+* 𝕜₂
σ₂₃' : 𝕜₂' →+* 𝕜₃
inst✝⁷ : RingHomCompTriple σ₁' σ₁₃ σ₁₃'
inst✝⁶ : RingHomCompTriple σ₂' σ₂₃ σ₂₃'
inst✝⁵ : RingHomIsometric σ₂₃
inst✝⁴ : RingHomIsometric σ₁₃'
inst✝³ : RingHomIsometric σ₂₃'
inst✝² : RingHomIsometric σ₁₃
inst✝¹ : RingHomIsometric σ₁'
inst✝ : RingHomIsometric σ₂'
c : StrongDual 𝕜 E
x : Fₗ
⊢ ‖({ toFun := smulRightₗ, map_add' := ⋯, map_smul' := ⋯ } c) x‖ ≤ 1 * ‖c‖ * ‖x‖

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.Module.Spaces.UniformConvergenceCLM

Goal state before:
pp: 𝕜₁ : Type u_1
𝕜₂ : Type u_2
inst✝¹⁵ : NormedField 𝕜₁
inst✝¹⁴ : NormedField 𝕜₂
σ : 𝕜₁ →+* 𝕜₂
E✝ : Type u_3
F✝ : Type u_4
G : Type u_5
inst✝¹³ : AddCommGroup E✝
inst✝¹² : Module 𝕜₁ E✝
inst✝¹¹ : TopologicalSpace E✝
inst✝¹⁰ : AddCommGroup F✝
inst✝⁹ : Module 𝕜₂ F✝
𝕜 : Type u_6
inst✝⁸ : NormedField 𝕜
E : Type u_7
ι : Type u_8
F : ι → Type u_9
inst✝⁷ : AddCommGroup E
inst✝⁶ : Module 𝕜 E
inst✝⁵ : TopologicalSpace E
inst✝⁴ : (i : ι) → AddCommGroup (F i)
inst✝³ : (i : ι) → Module 𝕜 (F i)
inst✝² : (i : ι) → TopologicalSpace (F i)
inst✝¹ : ∀ (i : ι), IsTopologicalAddGroup (F i)
inst✝ : ∀ (i : ι), ContinuousConstSMul 𝕜 (F i)
𝔖 : Set (Set E)
this✝ : (i : ι) → UniformSpace (F i) := fun i ↦ IsTopologicalAddGroup.rightUniformSpace (F i)
this : ∀ (i : ι), IsUniformAddGroup (F i)
⊢ Continuous[instTopologicalSpace (RingHom.id 𝕜) ((i : ι) → F i) 𝔖, Pi.topologicalSpace] fun f i ↦
    ContinuousLinearMap.proj i ∘SL f

Tactic:
by

Goal state after:
Module: Mathlib.Topology.UniformSpace.CompactConvergence

Goal state before:
pp: α : Type u₁
β : Type u₂
inst✝² : TopologicalSpace α
inst✝¹ : UniformSpace β
ι : Type u₃
δ : ι → Type u_1
inst✝ : (i : ι) → TopologicalSpace (δ i)
φ : (i : ι) → C(δ i, α)
h_proper : ∀ (i : ι), IsProperMap ⇑(φ i)
h_lf : LocallyFinite fun i ↦ range ⇑(φ i)
h_cover : ⋃ i, range ⇑(φ i) = univ
𝔖 : Set (Set α) := {K | IsCompact K}
𝔗 : (i : ι) → Set (Set (δ i)) := fun i ↦ {K | IsCompact K}
h_image : ∀ (i : ι), MapsTo (fun x ↦ ⇑(φ i) '' x) (𝔗 i) 𝔖
h_preimage : ∀ (i : ι), MapsTo (fun x ↦ ⇑(φ i) ⁻¹' x) 𝔖 (𝔗 i)
S : Set α
hS : S ∈ 𝔖
⊢ ∃ I, I.Finite ∧ S ⊆ ⋃ i ∈ I, range ⇑(φ i)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.Module.Spaces.UniformConvergenceCLM

Goal state before:
pp: 𝕜₁ : Type u_1
𝕜₂✝ : Type u_2
inst✝³⁶ : NormedField 𝕜₁
inst✝³⁵ : NormedField 𝕜₂✝
σ : 𝕜₁ →+* 𝕜₂✝
E✝ : Type u_3
F✝ : Type u_4
G✝ : Type u_5
inst✝³⁴ : AddCommGroup E✝
inst✝³³ : Module 𝕜₁ E✝
inst✝³² : TopologicalSpace E✝
inst✝³¹ : AddCommGroup F✝
inst✝³⁰ : Module 𝕜₂✝ F✝
𝕜 : Type u_6
𝕜₂ : Type u_7
𝕜₃ : Type u_8
𝕜₄ : Type u_9
E : Type u_10
F : Type u_11
G : Type u_12
H : Type u_13
inst✝²⁹ : AddCommGroup E
inst✝²⁸ : AddCommGroup F
inst✝²⁷ : AddCommGroup G
inst✝²⁶ : AddCommGroup H
inst✝²⁵ : NormedField 𝕜
inst✝²⁴ : NormedField 𝕜₂
inst✝²³ : NormedField 𝕜₃
inst✝²² : NormedField 𝕜₄
inst✝²¹ : Module 𝕜 E
inst✝²⁰ : Module 𝕜₂ F
inst✝¹⁹ : Module 𝕜₃ G
inst✝¹⁸ : Module 𝕜₄ H
inst✝¹⁷ : TopologicalSpace E
inst✝¹⁶ : TopologicalSpace F
inst✝¹⁵ : TopologicalSpace G
inst✝¹⁴ : TopologicalSpace H
inst✝¹³ : IsTopologicalAddGroup G
inst✝¹² : IsTopologicalAddGroup H
inst✝¹¹ : ContinuousConstSMul 𝕜₃ G
inst✝¹⁰ : ContinuousConstSMul 𝕜₄ H
σ₁₂ : 𝕜 →+* 𝕜₂
σ₂₁ : 𝕜₂ →+* 𝕜
σ₂₃ : 𝕜₂ →+* 𝕜₃
σ₁₃ : 𝕜 →+* 𝕜₃
σ₃₄ : 𝕜₃ →+* 𝕜₄
σ₄₃ : 𝕜₄ →+* 𝕜₃
σ₂₄ : 𝕜₂ →+* 𝕜₄
σ₁₄ : 𝕜 →+* 𝕜₄
inst✝⁹ : RingHomInvPair σ₁₂ σ₂₁
inst✝⁸ : RingHomInvPair σ₂₁ σ₁₂
inst✝⁷ : RingHomInvPair σ₃₄ σ₄₃
inst✝⁶ : RingHomInvPair σ₄₃ σ₃₄
inst✝⁵ : RingHomCompTriple σ₂₁ σ₁₄ σ₂₄
inst✝⁴ : RingHomCompTriple σ₂₄ σ₄₃ σ₂₃
inst✝³ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃
inst✝² : RingHomCompTriple σ₁₃ σ₃₄ σ₁₄
inst✝¹ : RingHomCompTriple σ₂₃ σ₃₄ σ₂₄
inst✝ : RingHomCompTriple σ₁₂ σ₂₄ σ₁₄
e₁₂ : E ≃SL[σ₁₂] F
e₄₃ : H ≃SL[σ₄₃] G
𝔖 : Set (Set E)
𝔗 : Set (Set F)
h : ∀ (t : Set F), t ∈ 𝔗 ↔ ⇑e₁₂ ⁻¹' t ∈ 𝔖
s : Set E
⊢ s ∈ 𝔖 → (fun x ↦ ⇑e₁₂ '' x) s ∈ 𝔗

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.Module.Spaces.UniformConvergenceCLM

Goal state before:
pp: 𝕜₁ : Type u_1
𝕜₂✝ : Type u_2
inst✝³⁶ : NormedField 𝕜₁
inst✝³⁵ : NormedField 𝕜₂✝
σ : 𝕜₁ →+* 𝕜₂✝
E✝ : Type u_3
F✝ : Type u_4
G✝ : Type u_5
inst✝³⁴ : AddCommGroup E✝
inst✝³³ : Module 𝕜₁ E✝
inst✝³² : TopologicalSpace E✝
inst✝³¹ : AddCommGroup F✝
inst✝³⁰ : Module 𝕜₂✝ F✝
𝕜 : Type u_6
𝕜₂ : Type u_7
𝕜₃ : Type u_8
𝕜₄ : Type u_9
E : Type u_10
F : Type u_11
G : Type u_12
H : Type u_13
inst✝²⁹ : AddCommGroup E
inst✝²⁸ : AddCommGroup F
inst✝²⁷ : AddCommGroup G
inst✝²⁶ : AddCommGroup H
inst✝²⁵ : NormedField 𝕜
inst✝²⁴ : NormedField 𝕜₂
inst✝²³ : NormedField 𝕜₃
inst✝²² : NormedField 𝕜₄
inst✝²¹ : Module 𝕜 E
inst✝²⁰ : Module 𝕜₂ F
inst✝¹⁹ : Module 𝕜₃ G
inst✝¹⁸ : Module 𝕜₄ H
inst✝¹⁷ : TopologicalSpace E
inst✝¹⁶ : TopologicalSpace F
inst✝¹⁵ : TopologicalSpace G
inst✝¹⁴ : TopologicalSpace H
inst✝¹³ : IsTopologicalAddGroup G
inst✝¹² : IsTopologicalAddGroup H
inst✝¹¹ : ContinuousConstSMul 𝕜₃ G
inst✝¹⁰ : ContinuousConstSMul 𝕜₄ H
σ₁₂ : 𝕜 →+* 𝕜₂
σ₂₁ : 𝕜₂ →+* 𝕜
σ₂₃ : 𝕜₂ →+* 𝕜₃
σ₁₃ : 𝕜 →+* 𝕜₃
σ₃₄ : 𝕜₃ →+* 𝕜₄
σ₄₃ : 𝕜₄ →+* 𝕜₃
σ₂₄ : 𝕜₂ →+* 𝕜₄
σ₁₄ : 𝕜 →+* 𝕜₄
inst✝⁹ : RingHomInvPair σ₁₂ σ₂₁
inst✝⁸ : RingHomInvPair σ₂₁ σ₁₂
inst✝⁷ : RingHomInvPair σ₃₄ σ₄₃
inst✝⁶ : RingHomInvPair σ₄₃ σ₃₄
inst✝⁵ : RingHomCompTriple σ₂₁ σ₁₄ σ₂₄
inst✝⁴ : RingHomCompTriple σ₂₄ σ₄₃ σ₂₃
inst✝³ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃
inst✝² : RingHomCompTriple σ₁₃ σ₃₄ σ₁₄
inst✝¹ : RingHomCompTriple σ₂₃ σ₃₄ σ₂₄
inst✝ : RingHomCompTriple σ₁₂ σ₂₄ σ₁₄
e₁₂ : E ≃SL[σ₁₂] F
e₄₃ : H ≃SL[σ₄₃] G
𝔖 : Set (Set E)
𝔗 : Set (Set F)
h : ∀ (t : Set F), t ∈ 𝔗 ↔ ⇑e₁₂ ⁻¹' t ∈ 𝔖
mapsto₁ : MapsTo (fun x ↦ ⇑e₁₂ '' x) 𝔖 𝔗
t : Set F
⊢ t ∈ 𝔗 → (fun x ↦ ⇑e₁₂.symm '' x) t ∈ 𝔖

Tactic:
by

Goal state after:
Module: Mathlib.Topology.UniformSpace.CompactConvergence

Goal state before:
pp: α : Type u₁
β : Type u₂
inst✝² : TopologicalSpace α
inst✝¹ : UniformSpace β
ι : Type u₃
δ : ι → Type u_1
inst✝ : (i : ι) → TopologicalSpace (δ i)
φ : (i : ι) → C(δ i, α)
h_proper : ∀ (i : ι), IsProperMap ⇑(φ i)
h_lf : LocallyFinite fun i ↦ range ⇑(φ i)
h_cover : ⋃ i, range ⇑(φ i) = univ
⊢ inferInstance = ⨅ i, UniformSpace.comap (fun x ↦ x.comp (φ i)) inferInstance

Tactic:
by

Goal state after:
Module: Mathlib.Topology.UniformSpace.CompactConvergence

Goal state before:
pp: α : Type u₁
β : Type u₂
inst✝³ : TopologicalSpace α
inst✝² : UniformSpace β
K : Set α
V : Set (β × β)
f : C(α, β)
ι : Type u₃
p : Filter ι
F : ι → C(α, β)
inst✝¹ : CompleteSpace β
inst✝ : CompactlyCoherentSpace α
⊢ CompleteSpace C(α, β)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.UniformSpace.CompactConvergence

Goal state before:
pp: α : Type u₁
β : Type u₂
inst✝² : TopologicalSpace α
inst✝¹ : UniformSpace β
inst✝ : CompleteSpace C(α, β)
f : α → β
s : Set α
l : Filter C(α, β)
hlc : Cauchy l
hlf : l ≤ 𝓟 {g | EqOn (⇑g) f s}
f' : C(α, β)
hf' : l ≤ 𝓝 f'
this : l.NeBot
x : α
hx : x ∈ s
⊢ Inseparable (f x) (f' x)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.UniformSpace.CompactConvergence

Goal state before:
pp: α : Type u₁
β : Type u₂
inst✝² : TopologicalSpace α
inst✝¹ : UniformSpace β
inst✝ : CompleteSpace C(α, β)
f : α → β
s : Set α
l : Filter C(α, β)
hlc : Cauchy l
hlf : l ≤ 𝓟 {g | EqOn (⇑g) f s}
f' : C(α, β)
hf' : l ≤ 𝓝 f'
this : l.NeBot
H₁ : ∀ x ∈ s, Inseparable (f x) (f' x)
x : α
⊢ Inseparable (s.piecewise f (⇑f') x) (f' x)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.UniformSpace.CompactConvergence

Goal state before:
pp: α : Type u₁
β : Type u₂
inst✝² : TopologicalSpace α
inst✝¹ : UniformSpace β
inst✝ : CompleteSpace C(α, β)
f : α → β
s : Set α
⊢ IsComplete {g | EqOn (⇑g) f s}

Tactic:
by

Goal state after:
Module: Mathlib.Probability.ConditionalProbability

Goal state before:
pp: Ω : Type u_1
m : MeasurableSpace Ω
μ : Measure Ω
s : Set Ω
hcs : μ s ≠ 0
hs : μ s ≠ ∞
⊢ μ[univ | s] = 1

Tactic:
by

Goal state after:
Module: Mathlib.Probability.ConditionalProbability

Goal state before:
pp: Ω : Type u_1
Ω' : Type u_2
α : Type u_3
m : MeasurableSpace Ω
m' : MeasurableSpace Ω'
μ : Measure Ω
s t : Set Ω
⊢ IsZeroOrProbabilityMeasure μ[|s]

Tactic:
by

Goal state after:
Module: Mathlib.Probability.ConditionalProbability

Goal state before:
pp: Ω : Type u_1
m : MeasurableSpace Ω
μ : Measure Ω
s : Set Ω
⊢ μ[|toMeasurable μ s] = μ[|s]

Tactic:
by

Goal state after:
Module: Mathlib.Probability.ConditionalProbability

Goal state before:
pp: Ω : Type u_1
m : MeasurableSpace Ω
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
⊢ μ ≪ μ[|univ]

Tactic:
by

Goal state after:
Module: Mathlib.Probability.ConditionalProbability

Goal state before:
pp: Ω : Type u_1
m : MeasurableSpace Ω
μ : Measure Ω
⊢ μ[|∅] = 0

Tactic:
by

Goal state after:
Module: Mathlib.Probability.ConditionalProbability

Goal state before:
pp: Ω : Type u_1
m : MeasurableSpace Ω
μ : Measure Ω
inst✝ : IsProbabilityMeasure μ
⊢ μ[|univ] = μ

Tactic:
by

Goal state after:
Module: Mathlib.Probability.ConditionalProbability

Goal state before:
pp: Ω : Type u_1
m : MeasurableSpace Ω
μ : Measure Ω
s : Set Ω
⊢ μ[|s] = 0 ↔ μ s = ∞ ∨ μ s = 0

Tactic:
by

Goal state after:
Module: Mathlib.Probability.ConditionalProbability

Goal state before:
pp: Ω : Type u_1
m : MeasurableSpace Ω
μ : Measure Ω
s : Set Ω
hμs : μ s = 0
⊢ μ[|s] = 0

Tactic:
by

Goal state after:
Module: Mathlib.Probability.ConditionalProbability

Goal state before:
pp: Ω : Type u_1
m : MeasurableSpace Ω
s : Set Ω
hms : MeasurableSet s
μ : Measure Ω
t : Set Ω
⊢ μ[t | s] = (μ s)⁻¹ * μ (s ∩ t)

Tactic:
by

Goal state after:
Module: Mathlib.Order.Filter.ENNReal

Goal state before:
pp: f : Filter ℝ
hf : ¬IsCobounded (fun x1 x2 ↦ x1 ≤ x2) f
⊢ f.limsSup = 0

Tactic:
by

Goal state after:
Module: Mathlib.Probability.ConditionalProbability

Goal state before:
pp: Ω : Type u_1
m : MeasurableSpace Ω
s t : Set Ω
ht : MeasurableSet t
μ : Measure Ω
⊢ μ[t | s] = (μ s)⁻¹ * μ (s ∩ t)

Tactic:
by

Goal state after:
Module: Mathlib.Order.Filter.ENNReal

Goal state before:
pp: f : Filter ℝ
hf : ¬IsBounded (fun x1 x2 ↦ x1 ≤ x2) f
⊢ f.limsSup = 0

Tactic:
by

Goal state after:
Module: Mathlib.Probability.ConditionalProbability

Goal state before:
pp: Ω : Type u_1
m : MeasurableSpace Ω
μ : Measure Ω
s : Set Ω
hs₀ : μ s ≠ 0
hs : μ s ≠ ∞
⊢ μ[s | s] = 1

Tactic:
by

Goal state after:
Module: Mathlib.Probability.ConditionalProbability

Goal state before:
pp: Ω : Type u_1
m : MeasurableSpace Ω
s : Set Ω
hms : MeasurableSet s
t : Set Ω
μ : Measure Ω
⊢ μ[s ∩ t | s] = μ[t | s]

Tactic:
by

Goal state after:
Module: Mathlib.Order.Filter.ENNReal

Goal state before:
pp: f : Filter ℝ
hf : ¬IsCobounded (fun x1 x2 ↦ x1 ≥ x2) f
⊢ f.limsInf = 0

Tactic:
by

Goal state after:
Module: Mathlib.Probability.ConditionalProbability

Goal state before:
pp: Ω : Type u_1
m : MeasurableSpace Ω
μ : Measure Ω
s t : Set Ω
hms : MeasurableSet s
hcst : μ[t | s] ≠ 0
⊢ 0 < μ (s ∩ t)

Tactic:
by

Goal state after:
Module: Mathlib.Probability.ConditionalProbability

Goal state before:
pp: Ω : Type u_1
m : MeasurableSpace Ω
μ : Measure Ω
s t : Set Ω
inst✝ : IsFiniteMeasure μ
hms : MeasurableSet s
hci : μ (s ∩ t) ≠ 0
⊢ 0 < μ[t | s]

Tactic:
by

Goal state after:
Module: Mathlib.Order.Filter.ENNReal

Goal state before:
pp: f : Filter ℝ
hf : ¬IsBounded (fun x1 x2 ↦ x1 ≥ x2) f
⊢ f.limsInf = 0

Tactic:
by

Goal state after:
Module: Mathlib.Order.Filter.ENNReal

Goal state before:
pp: ι : Type u_1
f : Filter ι
u : ι → ℝ≥0
b : ℝ
hb : ∀ᶠ (a : ι) in f, ↑(u a) ≤ b
⊢ 0 ≤ ↑b.toNNReal

Tactic:
by

Goal state after: