mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Normed.Module.Multilinear.Basic

Goal state before:
pp: 𝕜 : Type u
ι : Type v
E : ι → Type wE
G : Type wG
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝³ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝² : SeminormedAddCommGroup G
inst✝¹ : NormedSpace 𝕜 G
inst✝ : Fintype ι
f : MultilinearMap 𝕜 E G
C : ℝ
D : ℝ := max C 1
D_pos : 0 ≤ D
H : ∀ (m : (i : ι) → E i), ‖f m‖ ≤ D * ∏ i, ‖m i‖
m m' : (i : ι) → E i
h' : dist m' m < 1
this : max ‖m'‖ ‖m‖ ≤ ‖m‖ + 1
⊢ D * ↑(Fintype.card ι) * max ‖m'‖ ‖m‖ ^ (Fintype.card ι - 1) * ‖m' - m‖ ≤
    D * ↑(Fintype.card ι) * (‖m‖ + 1) ^ (Fintype.card ι - 1) * ‖m' - m‖

Tactic:
gcongr

Goal state after:
Module: Mathlib.Analysis.Normed.Module.Multilinear.Basic

Goal state before:
pp: 𝕜 : Type u
ι : Type v
E : ι → Type wE
G : Type wG
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝³ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝² : SeminormedAddCommGroup G
inst✝¹ : NormedSpace 𝕜 G
inst✝ : Fintype ι
f : ContinuousMultilinearMap 𝕜 E G
⊢ IsLeast {c | 0 ≤ c ∧ ∀ (m : (i : ι) → E i), ‖f m‖ ≤ c * ∏ i, ‖m i‖} ‖f‖

Tactic:
by
  refine IsClosed.isLeast_csInf ?_ bounds_nonempty bounds_bddBelow
  simp only [Set.ofPred_and, Set.ofPred_forall]
  exact isClosed_Ici.inter (isClosed_iInter fun m ↦ isClosed_le continuous_const (by fun_prop))

Goal state after:
Module: Mathlib.Analysis.Normed.Module.Multilinear.Basic

Goal state before:
pp: 𝕜 : Type u
ι : Type v
E : ι → Type wE
G : Type wG
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝³ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝² : SeminormedAddCommGroup G
inst✝¹ : NormedSpace 𝕜 G
inst✝ : Fintype ι
f : ContinuousMultilinearMap 𝕜 E G
m : (i : ι) → E i
C : ℝ
b : ι → ℝ
hC : ‖f‖ ≤ C
hm : ∀ (i : ι), ‖m i‖ ≤ b i
⊢ ‖f‖ * ∏ i, ‖m i‖ ≤ C * ∏ i, b i

Tactic:
gcongr

Goal state after:
case b0
𝕜 : Type u
ι : Type v
E : ι → Type wE
G : Type wG
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝³ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝² : SeminormedAddCommGroup G
inst✝¹ : NormedSpace 𝕜 G
inst✝ : Fintype ι
f : ContinuousMultilinearMap 𝕜 E G
m : (i : ι) → E i
C : ℝ
b : ι → ℝ
hC : ‖f‖ ≤ C
hm : ∀ (i : ι), ‖m i‖ ≤ b i
⊢ 0 ≤ C

case h1
𝕜 : Type u
ι : Type v
E : ι → Type wE
G : Type wG
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝³ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝² : SeminormedAddCommGroup G
inst✝¹ : NormedSpace 𝕜 G
inst✝ : Fintype ι
f : ContinuousMultilinearMap 𝕜 E G
m : (i : ι) → E i
C : ℝ
b : ι → ℝ
hC : ‖f‖ ≤ C
hm : ∀ (i : ι), ‖m i‖ ≤ b i
i : ι
a✝ : i ∈ univ
⊢ ‖m i‖ ≤ b i
Module: Mathlib.Analysis.Normed.Module.Multilinear.Basic

Goal state before:
pp: 𝕜 : Type u
ι : Type v
E : ι → Type wE
G : Type wG
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝³ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝² : SeminormedAddCommGroup G
inst✝¹ : NormedSpace 𝕜 G
inst✝ : Fintype ι
A : ∀ (f : ContinuousMultilinearMap 𝕜 E G × ContinuousMultilinearMap 𝕜 E G), ‖-f.1 + f.2‖ = ‖f.1 - f.2‖
x✝¹ : Set ((i : ι) → E i) × ℝ
s : Set ((i : ι) → E i)
r : ℝ
x✝ : Bornology.IsVonNBounded 𝕜 (s, r).1 ∧ 0 < (s, r).2
hs : Bornology.IsVonNBounded 𝕜 (s, r).1
hr : 0 < (s, r).2
ε : ℝ
hε : ∀ x ∈ (s, r).1, ‖x‖ ≤ ε
δ : ℝ
hδ₀ : 0 < δ
hδ : ε ^ Fintype.card ι * δ < (s, r).2
f : ContinuousMultilinearMap 𝕜 E G
x : (i : ι) → E i
hf : ‖f‖ ≤ δ
hx : ‖x‖ ≤ ε
this : 0 ≤ ε
⊢ ‖f‖ * ε ^ Fintype.card ι ≤ δ * ε ^ Fintype.card ι

Tactic:
gcongr

Goal state after:
Module: Mathlib.Topology.Algebra.Module.FiniteDimension

Goal state before:
pp: 𝕜 : Type u
hnorm : NontriviallyNormedField 𝕜
E : Type v
inst✝⁷ : AddCommGroup E
inst✝⁶ : Module 𝕜 E
inst✝⁵ : TopologicalSpace E
inst✝⁴ : IsTopologicalAddGroup E
inst✝³ : ContinuousSMul 𝕜 E
inst✝² : CompleteSpace 𝕜
inst✝¹ : T2Space E
ι : Type v
inst✝ : Finite ι
ξ : Basis ι 𝕜 E
⊢ Continuous[inst✝⁵, Pi.topologicalSpace] ⇑ξ.equivFun

Tactic:
by
  have := Fintype.ofFinite ι
  let : UniformSpace E := IsTopologicalAddGroup.rightUniformSpace E
  let : IsUniformAddGroup E := isUniformAddGroup_of_addCommGroup
  suffices ∀ n, Fintype.card ι = n → Continuous ξ.equivFun by exact this _ rfl
  intro n hn
  induction n generalizing ι E with
  | zero =>
    rw [Fintype.card_eq_zero_iff] at hn
    exact continuous_of_const fun x y => funext hn.elim
  | succ n IH =>
    have : FiniteDimensional 𝕜 E := ξ.finiteDimensional_of_finite
    -- first step: thanks to the induction hypothesis, any n-dimensional subspace is equivalent
    -- to a standard space of dimension n, hence it is complete and therefore closed.
    have H₁ : ∀ s : Submodule 𝕜 E, finrank 𝕜 s = n → IsClosed (s : Set E) := by
      intro s s_dim
      let : IsUniformAddGroup s := s.toAddSubgroup.isUniformAddGroup
      let b := Basis.ofVectorSpace 𝕜 s
      have U : IsUniformEmbedding b.equivFun.symm.toEquiv := by
        have : Fintype.card (Basis.ofVectorSpaceIndex 𝕜 s) = n := by
          rw [← s_dim]
          exact (finrank_eq_card_basis b).symm
        have : Continuous b.equivFun := IH b inferInstance this
        exact
          b.equivFun.symm.isUniformEmbedding b.equivFun.symm.toLinearMap.continuous_on_pi this
      have : IsComplete (s : Set E) :=
        completeSpace_coe_iff_isComplete.1 ((completeSpace_congr U).1 inferInstance)
      exact this.isClosed
    -- second step: any linear form is continuous, as its kernel is closed by the first step
    have H₂ : ∀ f : E →ₗ[𝕜] 𝕜, Continuous f := by
      intro f
      by_cases H : finrank 𝕜 (LinearMap.range f) = 0
      · rw [Submodule.finrank_eq_zero, LinearMap.range_eq_bot] at H
        rw [H]
        exact continuous_zero
      · have : finrank 𝕜 (LinearMap.ker f) = n := by
          have Z := f.finrank_range_add_finrank_ker
          rw [finrank_eq_card_basis ξ, hn] at Z
          have : finrank 𝕜 (LinearMap.range f) = 1 :=
            le_antisymm (finrank_self 𝕜 ▸ (LinearMap.range f).finrank_le) (zero_lt_iff.mpr H)
          rw [this, add_comm, Nat.add_one] at Z
          exact Nat.succ.inj Z
        have : IsClosed (LinearMap.ker f : Set E) := H₁ _ this
        exact LinearMap.continuous_of_isClosed_ker f this
    rw [continuous_pi_iff]
    intro i
    change Continuous (ξ.coord i)
    exact H₂ (ξ.coord i)

Goal state after:
Module: Mathlib.MeasureTheory.Function.LocallyIntegrable

Goal state before:
pp: X : Type u_1
ε : Type u_3
inst✝⁴ : MeasurableSpace X
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace ε
inst✝¹ : ContinuousENorm ε
f : X → ε
μ ν : Measure X
s : Set X
inst✝ : OpensMeasurableSpace X
hf✝ : LocallyIntegrableOn f s μ
h : ν.restrict s ≤ μ.restrict s
x : X
hx : x ∈ s
t : Set X
ht : t ∈ 𝓝[s] x
hf : IntegrableOn f t μ
u : Set X
hu : IsOpen[inst✝³] u
hxu : x ∈ u
hut : u ∩ s ⊆ t
⊢ (ν.restrict s).restrict u ≤ (μ.restrict s).restrict u

Tactic:
gcongr

Goal state after:
Module: Mathlib.MeasureTheory.Function.LocallyIntegrable

Goal state before:
pp: X : Type u_1
ε : Type u_3
inst✝³ : MeasurableSpace X
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace ε
inst✝ : ContinuousENorm ε
f g : X → ε
μ : Measure X
s : Set X
h : f =ᵐ[μ.restrict s] g
hf : LocallyIntegrableOn f s μ
x : X
hx : x ∈ s
t : Set X
hxt : t ∈ 𝓝[s] x
hft : IntegrableOn f t μ
⊢ ae (μ.restrict (s ∩ t)) ≤ ae (μ.restrict s)

Tactic:
gcongr

Goal state after:
case hs
X : Type u_1
ε : Type u_3
inst✝³ : MeasurableSpace X
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace ε
inst✝ : ContinuousENorm ε
f g : X → ε
μ : Measure X
s : Set X
h : f =ᵐ[μ.restrict s] g
hf : LocallyIntegrableOn f s μ
x : X
hx : x ∈ s
t : Set X
hxt : t ∈ 𝓝[s] x
hft : IntegrableOn f t μ
⊢ s ∩ t ⊆ s
Module: Mathlib.MeasureTheory.Integral.IntegrableOn

Goal state before:
pp: case neg
α : Type u_1
ε' : Type u_4
mα : MeasurableSpace α
inst✝⁴ : PartialOrder α
inst✝³ : MeasurableSingletonClass α
inst✝² : TopologicalSpace ε'
inst✝¹ : ESeminormedAddMonoid ε'
inst✝ : PseudoMetrizableSpace ε'
f : α → ε'
μ : Measure α
a b : α
hb : μ {b} ≠ ∞
hb' : ‖f b‖ₑ ≠ ∞
hab : ¬a < b
⊢ IntegrableOn f (Ioc a b) μ ↔ IntegrableOn f (Ioo a b) μ

Tactic:
· rw [Ioo_eq_empty hab, Ioc_eq_empty hab]

Goal state after:
Module: Mathlib.Analysis.Normed.Module.Multilinear.Basic

Goal state before:
pp: case refine_1
𝕜 : Type u
ι : Type v
E : ι → Type wE
G : Type wG
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝³ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝² : SeminormedAddCommGroup G
inst✝¹ : NormedSpace 𝕜 G
inst✝ : Fintype ι
f : ContinuousMultilinearMap 𝕜 E 𝕜
z : G
m : (i : ι) → E i
⊢ ‖LinearMap.id (f.toMultilinearMap m)‖₊ * ‖z‖₊ ≤ (‖f‖₊ * ∏ i, ‖m i‖₊) * ‖z‖₊

Tactic:
gcongr

Goal state after:
case h₁
𝕜 : Type u
ι : Type v
E : ι → Type wE
G : Type wG
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝³ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝² : SeminormedAddCommGroup G
inst✝¹ : NormedSpace 𝕜 G
inst✝ : Fintype ι
f : ContinuousMultilinearMap 𝕜 E 𝕜
z : G
m : (i : ι) → E i
⊢ ‖LinearMap.id (f.toMultilinearMap m)‖₊ ≤ ‖f‖₊ * ∏ i, ‖m i‖₊
Module: Mathlib.Topology.Algebra.Module.FiniteDimension

Goal state before:
pp: 𝕜 : Type u_4
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : CompleteSpace 𝕜
Eᵤ : Type u_6
inst✝⁵ : AddCommGroup Eᵤ
inst✝⁴ : Module 𝕜 Eᵤ
inst✝³ : UniformSpace Eᵤ
inst✝² : T2Space Eᵤ
inst✝¹ : IsUniformAddGroup Eᵤ
inst✝ : ContinuousSMul 𝕜 Eᵤ
U : Set Eᵤ
hU_nhds : U ∈ 𝓝 0
hU_tb : TotallyBounded U
c : 𝕜
hc0 : 0 < ‖c‖
hc1 : ‖c‖ < 1
hc_ne : c ≠ 0
F : Set Eᵤ
hF_finite : F.Finite
hF_cover : U ⊆ ⋃ y ∈ F, y +ᵥ c • U
M : Submodule 𝕜 Eᵤ := Submodule.span 𝕜 F
this : FiniteDimensional 𝕜 ↥M := Finite.span_of_finite 𝕜 hF_finite
h_cover : U ⊆ ↑M + c • U
n : ℕ
ih : U ⊆ ↑M + c ^ n • U
⊢ ↑M + c ^ n • U ⊆ ↑M + c ^ n • (↑M + c • U)

Tactic:
gcongr

Goal state after:
Module: Mathlib.MeasureTheory.Function.L1Space.AEEqFun

Goal state before:
pp: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
f : ↥(Lp β 1 μ)
⊢ ENNReal.ofReal ‖f‖ = ∫⁻ (x : α), ‖↑↑f x‖ₑ ∂μ

Tactic:
rw [norm_def, ENNReal.ofReal_toReal]
  exact ne_of_lt (hasFiniteIntegral_coeFn f)

Goal state after:
Module: Mathlib.Analysis.Normed.Module.Multilinear.Basic

Goal state before:
pp: 𝕜 : Type u
ι : Type v
ι' : Type v'
E : ι → Type wE
E₁ : ι → Type wE₁
E' : ι' → Type wE'
G : Type wG
G' : Type wG'
inst✝¹⁰ : Fintype ι'
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝⁷ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝⁶ : (i : ι) → SeminormedAddCommGroup (E₁ i)
inst✝⁵ : (i : ι) → NormedSpace 𝕜 (E₁ i)
inst✝⁴ : SeminormedAddCommGroup G
inst✝³ : NormedSpace 𝕜 G
inst✝² : SeminormedAddCommGroup G'
inst✝¹ : NormedSpace 𝕜 G'
inst✝ : Fintype ι
f : ContinuousMultilinearMap 𝕜 E (G →L[𝕜] G')
x : G
m : (i : ι) → E i
⊢ ‖f m‖ * ‖x‖ ≤ (‖f‖ * ∏ i, ‖m i‖) * ‖x‖

Tactic:
gcongr

Goal state after:
case hbc
𝕜 : Type u
ι : Type v
ι' : Type v'
E : ι → Type wE
E₁ : ι → Type wE₁
E' : ι' → Type wE'
G : Type wG
G' : Type wG'
inst✝¹⁰ : Fintype ι'
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝⁷ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝⁶ : (i : ι) → SeminormedAddCommGroup (E₁ i)
inst✝⁵ : (i : ι) → NormedSpace 𝕜 (E₁ i)
inst✝⁴ : SeminormedAddCommGroup G
inst✝³ : NormedSpace 𝕜 G
inst✝² : SeminormedAddCommGroup G'
inst✝¹ : NormedSpace 𝕜 G'
inst✝ : Fintype ι
f : ContinuousMultilinearMap 𝕜 E (G →L[𝕜] G')
x : G
m : (i : ι) → E i
⊢ ‖f m‖ ≤ ‖f‖ * ∏ i, ‖m i‖
Module: Mathlib.Analysis.Normed.Module.Multilinear.Basic

Goal state before:
pp: 𝕜 : Type u
ι : Type v
ι' : Type v'
E : ι → Type wE
E₁ : ι → Type wE₁
E' : ι' → Type wE'
G : Type wG
G' : Type wG'
inst✝¹² : Fintype ι'
inst✝¹¹ : NontriviallyNormedField 𝕜
inst✝¹⁰ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝⁹ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝⁸ : (i : ι) → SeminormedAddCommGroup (E₁ i)
inst✝⁷ : (i : ι) → NormedSpace 𝕜 (E₁ i)
inst✝⁶ : SeminormedAddCommGroup G
inst✝⁵ : NormedSpace 𝕜 G
inst✝⁴ : SeminormedAddCommGroup G'
inst✝³ : NormedSpace 𝕜 G'
inst✝² : Fintype ι
α : Type u_1
inst✝¹ : Fintype α
f : ContinuousMultilinearMap 𝕜 E₁ G
s : Set ι
e : α ≃ ↑s
inst✝ : DecidablePred fun x ↦ x ∈ s
x : (i : { a // a ∉ s }) → E₁ ↑i
m : α → (i : ι) → E₁ i
⊢ ‖f‖ *
      ((∏ i ∈ s.toFinsetᶜ, ‖if h : i ∈ s then m (e.symm ⟨i, ⋯⟩) i else x ⟨i, ⋯⟩‖) *
        ∏ i ∈ s.toFinset, ‖if h : i ∈ s then m (e.symm ⟨i, ⋯⟩) i else x ⟨i, ⋯⟩‖) ≤
    ‖f‖ * ((∏ i, ‖x i‖) * ∏ i, ‖m i‖)

Tactic:
gcongr

Goal state after:
case h₁
𝕜 : Type u
ι : Type v
ι' : Type v'
E : ι → Type wE
E₁ : ι → Type wE₁
E' : ι' → Type wE'
G : Type wG
G' : Type wG'
inst✝¹² : Fintype ι'
inst✝¹¹ : NontriviallyNormedField 𝕜
inst✝¹⁰ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝⁹ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝⁸ : (i : ι) → SeminormedAddCommGroup (E₁ i)
inst✝⁷ : (i : ι) → NormedSpace 𝕜 (E₁ i)
inst✝⁶ : SeminormedAddCommGroup G
inst✝⁵ : NormedSpace 𝕜 G
inst✝⁴ : SeminormedAddCommGroup G'
inst✝³ : NormedSpace 𝕜 G'
inst✝² : Fintype ι
α : Type u_1
inst✝¹ : Fintype α
f : ContinuousMultilinearMap 𝕜 E₁ G
s : Set ι
e : α ≃ ↑s
inst✝ : DecidablePred fun x ↦ x ∈ s
x : (i : { a // a ∉ s }) → E₁ ↑i
m : α → (i : ι) → E₁ i
⊢ ∏ i ∈ s.toFinsetᶜ, ‖if h : i ∈ s then m (e.symm ⟨i, ⋯⟩) i else x ⟨i, ⋯⟩‖ ≤ ∏ i, ‖x i‖

case h₂
𝕜 : Type u
ι : Type v
ι' : Type v'
E : ι → Type wE
E₁ : ι → Type wE₁
E' : ι' → Type wE'
G : Type wG
G' : Type wG'
inst✝¹² : Fintype ι'
inst✝¹¹ : NontriviallyNormedField 𝕜
inst✝¹⁰ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝⁹ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝⁸ : (i : ι) → SeminormedAddCommGroup (E₁ i)
inst✝⁷ : (i : ι) → NormedSpace 𝕜 (E₁ i)
inst✝⁶ : SeminormedAddCommGroup G
inst✝⁵ : NormedSpace 𝕜 G
inst✝⁴ : SeminormedAddCommGroup G'
inst✝³ : NormedSpace 𝕜 G'
inst✝² : Fintype ι
α : Type u_1
inst✝¹ : Fintype α
f : ContinuousMultilinearMap 𝕜 E₁ G
s : Set ι
e : α ≃ ↑s
inst✝ : DecidablePred fun x ↦ x ∈ s
x : (i : { a // a ∉ s }) → E₁ ↑i
m : α → (i : ι) → E₁ i
⊢ ∏ i ∈ s.toFinset, ‖if h : i ∈ s then m (e.symm ⟨i, ⋯⟩) i else x ⟨i, ⋯⟩‖ ≤ ∏ i, ‖m i‖
Module: Mathlib.Analysis.Normed.Module.Multilinear.Basic

Goal state before:
pp: 𝕜 : Type u
ι : Type v
E₁ : ι → Type wE₁
G : Type wG
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : (i : ι) → SeminormedAddCommGroup (E₁ i)
inst✝⁵ : (i : ι) → NormedSpace 𝕜 (E₁ i)
inst✝⁴ : SeminormedAddCommGroup G
inst✝³ : NormedSpace 𝕜 G
inst✝² : Fintype ι
α : Type u_1
inst✝¹ : Fintype α
f : ContinuousMultilinearMap 𝕜 E₁ G
s : Set ι
e : α ≃ ↑s
inst✝ : DecidablePred fun x ↦ x ∈ s
x : (i : ι) → E₁ i
⊢ ‖f.iteratedFDerivComponent e‖ * ∏ i, ‖x ↑i‖ ≤ ‖f‖ * ∏ _i, ‖x‖

Tactic:
gcongr

Goal state after:
case h₁
𝕜 : Type u
ι : Type v
E₁ : ι → Type wE₁
G : Type wG
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : (i : ι) → SeminormedAddCommGroup (E₁ i)
inst✝⁵ : (i : ι) → NormedSpace 𝕜 (E₁ i)
inst✝⁴ : SeminormedAddCommGroup G
inst✝³ : NormedSpace 𝕜 G
inst✝² : Fintype ι
α : Type u_1
inst✝¹ : Fintype α
f : ContinuousMultilinearMap 𝕜 E₁ G
s : Set ι
e : α ≃ ↑s
inst✝ : DecidablePred fun x ↦ x ∈ s
x : (i : ι) → E₁ i
⊢ ‖f.iteratedFDerivComponent e‖ ≤ ‖f‖

case h1
𝕜 : Type u
ι : Type v
E₁ : ι → Type wE₁
G : Type wG
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : (i : ι) → SeminormedAddCommGroup (E₁ i)
inst✝⁵ : (i : ι) → NormedSpace 𝕜 (E₁ i)
inst✝⁴ : SeminormedAddCommGroup G
inst✝³ : NormedSpace 𝕜 G
inst✝² : Fintype ι
α : Type u_1
inst✝¹ : Fintype α
f : ContinuousMultilinearMap 𝕜 E₁ G
s : Set ι
e : α ≃ ↑s
inst✝ : DecidablePred fun x ↦ x ∈ s
x : (i : ι) → E₁ i
i : { a // a ∉ s }
a✝ : i ∈ univ
⊢ ‖x ↑i‖ ≤ ‖x‖
Module: Mathlib.Analysis.Normed.Module.Multilinear.Basic

Goal state before:
pp: 𝕜 : Type u
ι : Type v
E₁ : ι → Type wE₁
G : Type wG
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : (i : ι) → SeminormedAddCommGroup (E₁ i)
inst✝⁵ : (i : ι) → NormedSpace 𝕜 (E₁ i)
inst✝⁴ : SeminormedAddCommGroup G
inst✝³ : NormedSpace 𝕜 G
inst✝² : Fintype ι
α : Type u_1
inst✝¹ : Fintype α
f : ContinuousMultilinearMap 𝕜 E₁ G
s : Set ι
e : α ≃ ↑s
inst✝ : DecidablePred fun x ↦ x ∈ s
x : (i : ι) → E₁ i
⊢ ‖f‖ * ‖x‖ ^ #univ = ‖f‖ * ‖x‖ ^ Fintype.card { a // a ∉ s }

Tactic:
card_univ

Goal state after:
𝕜 : Type u
ι : Type v
E₁ : ι → Type wE₁
G : Type wG
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : (i : ι) → SeminormedAddCommGroup (E₁ i)
inst✝⁵ : (i : ι) → NormedSpace 𝕜 (E₁ i)
inst✝⁴ : SeminormedAddCommGroup G
inst✝³ : NormedSpace 𝕜 G
inst✝² : Fintype ι
α : Type u_1
inst✝¹ : Fintype α
f : ContinuousMultilinearMap 𝕜 E₁ G
s : Set ι
e : α ≃ ↑s
inst✝ : DecidablePred fun x ↦ x ∈ s
x : (i : ι) → E₁ i
⊢ ‖f‖ * ‖x‖ ^ Fintype.card { a // a ∉ s } = ‖f‖ * ‖x‖ ^ Fintype.card { a // a ∉ s }
Module: Mathlib.Analysis.Normed.Module.Multilinear.Basic

Goal state before:

Tactic:
‖f.iteratedFDeriv k x‖
  _ ≤ ∑ e : Fin k ↪ ι, ‖iteratedFDerivComponent f e.toEquivRange (fun i ↦ x i)‖ := norm_sum_le _ _
  _ ≤ ∑ _ : Fin k ↪ ι, ‖f‖ * ‖x‖ ^ (Fintype.card ι - k) := by
    gcongr with e _
    simpa using norm_iteratedFDerivComponent_le f e.toEquivRange x
  _ = Nat.descFactorial (Fintype.card ι) k * ‖f‖ * ‖x‖ ^ (Fintype.card ι - k) := by
    simp [card_univ, mul_assoc]

Goal state after:
Module: Mathlib.MeasureTheory.Integral.FinMeasAdditive

Goal state before:
pp: case refine_2
α : Type u_1
m : MeasurableSpace α
μ : Measure α
β : Type u_7
inst✝ : AddCommMonoid β
T : Set α → β
T_empty : T ∅ = 0
h_add : FinMeasAdditive μ T
ι : Type u_8
S : ι → Set α
sι : Finset ι
hS_meas : ∀ (i : ι), MeasurableSet (S i)
a : ι
s : Finset ι
has : a ∉ s
h : (∀ i ∈ s, μ (S i) ≠ ∞) → (∀ i ∈ s, ∀ j ∈ s, i ≠ j → Disjoint (S i) (S j)) → T (⋃ i ∈ s, S i) = ∑ i ∈ s, T (S i)
hps : ∀ i ∈ insert a s, μ (S i) ≠ ∞
h_disj : ∀ i ∈ insert a s, ∀ j ∈ insert a s, i ≠ j → Disjoint (S i) (S j)
⊢ T (⋃ i ∈ insert a s, S i) = T (S a) + T (⋃ i ∈ s, S i)

Tactic:
rw [←
    h_add (S a) (⋃ i ∈ s, S i) (hS_meas a) (measurableSet_biUnion _ fun i _ => hS_meas i)
      (hps a (Finset.mem_insert_self a s))]

Goal state after:
case refine_2
α : Type u_1
m : MeasurableSpace α
μ : Measure α
β : Type u_7
inst✝ : AddCommMonoid β
T : Set α → β
T_empty : T ∅ = 0
h_add : FinMeasAdditive μ T
ι : Type u_8
S : ι → Set α
sι : Finset ι
hS_meas : ∀ (i : ι), MeasurableSet (S i)
a : ι
s : Finset ι
has : a ∉ s
h : (∀ i ∈ s, μ (S i) ≠ ∞) → (∀ i ∈ s, ∀ j ∈ s, i ≠ j → Disjoint (S i) (S j)) → T (⋃ i ∈ s, S i) = ∑ i ∈ s, T (S i)
hps : ∀ i ∈ insert a s, μ (S i) ≠ ∞
h_disj : ∀ i ∈ insert a s, ∀ j ∈ insert a s, i ≠ j → Disjoint (S i) (S j)
⊢ T (⋃ i ∈ insert a s, S i) = T (S a ∪ ⋃ i ∈ s, S i)

case refine_2.a
α : Type u_1
m : MeasurableSpace α
μ : Measure α
β : Type u_7
inst✝ : AddCommMonoid β
T : Set α → β
T_empty : T ∅ = 0
h_add : FinMeasAdditive μ T
ι : Type u_8
S : ι → Set α
sι : Finset ι
hS_meas : ∀ (i : ι), MeasurableSet (S i)
a : ι
s : Finset ι
has : a ∉ s
h : (∀ i ∈ s, μ (S i) ≠ ∞) → (∀ i ∈ s, ∀ j ∈ s, i ≠ j → Disjoint (S i) (S j)) → T (⋃ i ∈ s, S i) = ∑ i ∈ s, T (S i)
hps : ∀ i ∈ insert a s, μ (S i) ≠ ∞
h_disj : ∀ i ∈ insert a s, ∀ j ∈ insert a s, i ≠ j → Disjoint (S i) (S j)
⊢ μ (⋃ i ∈ s, S i) ≠ ∞

case refine_2.a
α : Type u_1
m : MeasurableSpace α
μ : Measure α
β : Type u_7
inst✝ : AddCommMonoid β
T : Set α → β
T_empty : T ∅ = 0
h_add : FinMeasAdditive μ T
ι : Type u_8
S : ι → Set α
sι : Finset ι
hS_meas : ∀ (i : ι), MeasurableSet (S i)
a : ι
s : Finset ι
has : a ∉ s
h : (∀ i ∈ s, μ (S i) ≠ ∞) → (∀ i ∈ s, ∀ j ∈ s, i ≠ j → Disjoint (S i) (S j)) → T (⋃ i ∈ s, S i) = ∑ i ∈ s, T (S i)
hps : ∀ i ∈ insert a s, μ (S i) ≠ ∞
h_disj : ∀ i ∈ insert a s, ∀ j ∈ insert a s, i ≠ j → Disjoint (S i) (S j)
⊢ Disjoint (S a) (⋃ i ∈ s, S i)
Module: Mathlib.MeasureTheory.Function.SimpleFuncDenseLp

Goal state before:
pp: α : Type u_1
E : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
μ : Measure α
p : ℝ≥0∞
hp_pos : p ≠ 0
hp_ne_top : p ≠ ∞
f : α →ₛ E
hf : MemLp (⇑f) p μ
y : E
hy_ne : y ≠ 0
h_fin : (map (fun x ↦ ‖x‖ₑ ^ p.toReal) f).FinMeasSupp μ
this : ∀ {b : E}, (fun x ↦ ‖x‖ₑ ^ p.toReal) b = 0 ↔ b = 0
⊢ f.FinMeasSupp μ

Tactic:
rwa [FinMeasSupp.map_iff this] at h_fin

Goal state after:
Module: Mathlib.MeasureTheory.Integral.FinMeasAdditive

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
β : Type u_7
inst✝ : SeminormedAddCommGroup β
T : Set α → β
C : ℝ
μ' : Measure α
h : μ ≤ μ'
hT : DominatedFinMeasAdditive μ T C
hC : 0 ≤ C
h' : ∀ (s : Set α), μ s = ∞ → μ' s = ∞
s : Set α
hs : MeasurableSet s
hμ's : μ' s < ∞
hμs : μ s < ∞
⊢ C * (μ s).toReal ≤ C * (μ' s).toReal

Tactic:
gcongr

Goal state after:
case hb
α : Type u_1
m : MeasurableSpace α
μ : Measure α
β : Type u_7
inst✝ : SeminormedAddCommGroup β
T : Set α → β
C : ℝ
μ' : Measure α
h : μ ≤ μ'
hT : DominatedFinMeasAdditive μ T C
hC : 0 ≤ C
h' : ∀ (s : Set α), μ s = ∞ → μ' s = ∞
s : Set α
hs : MeasurableSet s
hμ's : μ' s < ∞
hμs : μ s < ∞
⊢ μ' s ≠ ∞
Module: Mathlib.MeasureTheory.Integral.FinMeasAdditive

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
β : Type u_7
inst✝ : SeminormedAddCommGroup β
T T' : Set α → β
C C' : ℝ
μ ν : Measure α
hT : DominatedFinMeasAdditive μ T C
hT' : DominatedFinMeasAdditive ν T' C'
s : Set α
hs : MeasurableSet s
hsf : (μ + ν) s < ∞
hμs : μ s < ∞
hνs : ν s < ∞
⊢ C * μ.real s + C' * ν.real s ≤ max C C' * μ.real s + max C C' * ν.real s

Tactic:
gcongr

Goal state after:
case hbc
α : Type u_1
m : MeasurableSpace α
β : Type u_7
inst✝ : SeminormedAddCommGroup β
T T' : Set α → β
C C' : ℝ
μ ν : Measure α
hT : DominatedFinMeasAdditive μ T C
hT' : DominatedFinMeasAdditive ν T' C'
s : Set α
hs : MeasurableSet s
hsf : (μ + ν) s < ∞
hμs : μ s < ∞
hνs : ν s < ∞
⊢ C ≤ max C C'

case hbc
α : Type u_1
m : MeasurableSpace α
β : Type u_7
inst✝ : SeminormedAddCommGroup β
T T' : Set α → β
C C' : ℝ
μ ν : Measure α
hT : DominatedFinMeasAdditive μ T C
hT' : DominatedFinMeasAdditive ν T' C'
s : Set α
hs : MeasurableSet s
hsf : (μ + ν) s < ∞
hμs : μ s < ∞
hνs : ν s < ∞
⊢ C' ≤ max C C'
Module: Mathlib.Analysis.Normed.Operator.Extend

Goal state before:
pp: 𝕜 : Type u_1
𝕜₂ : Type u_2
E : Type u_3
Eₗ : Type u_4
F : Type u_5
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : NontriviallyNormedField 𝕜₂
σ₁₂ : 𝕜 →+* 𝕜₂
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedAddCommGroup Eₗ
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedSpace 𝕜 Eₗ
inst✝² : NormedSpace 𝕜₂ F
inst✝¹ : CompleteSpace F
f : E →SL[σ₁₂] F
e : E →L[𝕜] Eₗ
N : ℝ≥0
inst✝ : RingHomIsometric σ₁₂
h_dense : DenseRange ⇑e
h_e : ∀ (x : E), ‖x‖ ≤ ↑N * ‖e x‖
x : E
⊢ ‖f‖ * ‖x‖ ≤ ‖f‖ * (↑N * ‖e x‖)

Tactic:
gcongr

Goal state after:
case hbc
𝕜 : Type u_1
𝕜₂ : Type u_2
E : Type u_3
Eₗ : Type u_4
F : Type u_5
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : NontriviallyNormedField 𝕜₂
σ₁₂ : 𝕜 →+* 𝕜₂
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedAddCommGroup Eₗ
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedSpace 𝕜 Eₗ
inst✝² : NormedSpace 𝕜₂ F
inst✝¹ : CompleteSpace F
f : E →SL[σ₁₂] F
e : E →L[𝕜] Eₗ
N : ℝ≥0
inst✝ : RingHomIsometric σ₁₂
h_dense : DenseRange ⇑e
h_e : ∀ (x : E), ‖x‖ ≤ ↑N * ‖e x‖
x : E
⊢ ‖x‖ ≤ ↑N * ‖e x‖
Module: Mathlib.MeasureTheory.Integral.Bochner.L1

Goal state before:
pp: case pos.inr
α : Type u_1
F : Type u_3
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
inst✝² : PartialOrder F
inst✝¹ : IsOrderedAddMonoid F
inst✝ : IsOrderedModule ℝ F
ν : Measure α
f : α →ₛ F
hf : 0 ≤ᵐ[ν] ⇑f
hμν : μ ≤ ν
hfν : Integrable (⇑f) ν
x : α
hx✝ : 0 ≤ f x
hx : 0 < f x
⊢ (μ (⇑f ⁻¹' {f x})).toReal • f x ≤ (ν (⇑f ⁻¹' {f x})).toReal • f x

Tactic:
gcongr

Goal state after:
case hb
α : Type u_1
F : Type u_3
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
inst✝² : PartialOrder F
inst✝¹ : IsOrderedAddMonoid F
inst✝ : IsOrderedModule ℝ F
ν : Measure α
f : α →ₛ F
hf : 0 ≤ᵐ[ν] ⇑f
hμν : μ ≤ ν
hfν : Integrable (⇑f) ν
x : α
hx✝ : 0 ≤ f x
hx : 0 < f x
⊢ ν (⇑f ⁻¹' {f x}) ≠ ∞