mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Normed.Algebra.UnitizationL1

Goal state before:
pp: case h₂
𝕜 : Type u_1
A : Type u_2
inst✝⁴ : NormedField 𝕜
inst✝³ : NonUnitalNormedRing A
inst✝² : NormedSpace 𝕜 A
inst✝¹ : IsScalarTower 𝕜 A A
inst✝ : SMulCommClass 𝕜 A A
x y : WithLp 1 (Unitization 𝕜 A)
⊢ ‖x.ofLp.toProd.1 • y.ofLp.toProd.2‖ + ‖y.ofLp.toProd.1 • x.ofLp.toProd.2 + x.ofLp.toProd.2 * y.ofLp.toProd.2‖ ≤
    ‖x.ofLp.toProd.1‖ * ‖y.ofLp.toProd.2‖ +
      (‖x.ofLp.toProd.2‖ * ‖y.ofLp.toProd.1‖ + ‖x.ofLp.toProd.2‖ * ‖y.ofLp.toProd.2‖)

Tactic:
gcongr

Goal state after:
case h₁
𝕜 : Type u_1
A : Type u_2
inst✝⁴ : NormedField 𝕜
inst✝³ : NonUnitalNormedRing A
inst✝² : NormedSpace 𝕜 A
inst✝¹ : IsScalarTower 𝕜 A A
inst✝ : SMulCommClass 𝕜 A A
x y : WithLp 1 (Unitization 𝕜 A)
⊢ ‖x.ofLp.toProd.1 • y.ofLp.toProd.2‖ ≤ ‖x.ofLp.toProd.1‖ * ‖y.ofLp.toProd.2‖

case h₂
𝕜 : Type u_1
A : Type u_2
inst✝⁴ : NormedField 𝕜
inst✝³ : NonUnitalNormedRing A
inst✝² : NormedSpace 𝕜 A
inst✝¹ : IsScalarTower 𝕜 A A
inst✝ : SMulCommClass 𝕜 A A
x y : WithLp 1 (Unitization 𝕜 A)
⊢ ‖y.ofLp.toProd.1 • x.ofLp.toProd.2 + x.ofLp.toProd.2 * y.ofLp.toProd.2‖ ≤
    ‖x.ofLp.toProd.2‖ * ‖y.ofLp.toProd.1‖ + ‖x.ofLp.toProd.2‖ * ‖y.ofLp.toProd.2‖
Module: Mathlib.Analysis.Normed.Algebra.UnitizationL1

Goal state before:
pp: case h₂
𝕜 : Type u_1
A : Type u_2
inst✝⁴ : NormedField 𝕜
inst✝³ : NonUnitalNormedRing A
inst✝² : NormedSpace 𝕜 A
inst✝¹ : IsScalarTower 𝕜 A A
inst✝ : SMulCommClass 𝕜 A A
x y : WithLp 1 (Unitization 𝕜 A)
⊢ ‖y.ofLp.toProd.1 • x.ofLp.toProd.2‖ + ‖x.ofLp.toProd.2 * y.ofLp.toProd.2‖ ≤
    ‖x.ofLp.toProd.2‖ * ‖y.ofLp.toProd.1‖ + ‖x.ofLp.toProd.2‖ * ‖y.ofLp.toProd.2‖

Tactic:
gcongr

Goal state after:
case h₁
𝕜 : Type u_1
A : Type u_2
inst✝⁴ : NormedField 𝕜
inst✝³ : NonUnitalNormedRing A
inst✝² : NormedSpace 𝕜 A
inst✝¹ : IsScalarTower 𝕜 A A
inst✝ : SMulCommClass 𝕜 A A
x y : WithLp 1 (Unitization 𝕜 A)
⊢ ‖y.ofLp.toProd.1 • x.ofLp.toProd.2‖ ≤ ‖x.ofLp.toProd.2‖ * ‖y.ofLp.toProd.1‖

case h₂
𝕜 : Type u_1
A : Type u_2
inst✝⁴ : NormedField 𝕜
inst✝³ : NonUnitalNormedRing A
inst✝² : NormedSpace 𝕜 A
inst✝¹ : IsScalarTower 𝕜 A A
inst✝ : SMulCommClass 𝕜 A A
x y : WithLp 1 (Unitization 𝕜 A)
⊢ ‖x.ofLp.toProd.2 * y.ofLp.toProd.2‖ ≤ ‖x.ofLp.toProd.2‖ * ‖y.ofLp.toProd.2‖
Module: Mathlib.Analysis.CStarAlgebra.Unitization

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁹ : DenselyNormedField 𝕜
inst✝⁸ : NonUnitalNormedRing E
inst✝⁷ : StarRing E
inst✝⁶ : CStarRing E
inst✝⁵ : NormedSpace 𝕜 E
inst✝⁴ : IsScalarTower 𝕜 E E
inst✝³ : SMulCommClass 𝕜 E E
inst✝² : StarRing 𝕜
inst✝¹ : StarModule 𝕜 E
inst✝ : CStarRing 𝕜
x : Unitization 𝕜 E
h₁ : ∀ (x : Unitization 𝕜 E), ‖((splitMul 𝕜 E) x).2‖ ≤ ‖((splitMul 𝕜 E) x⋆).2‖
⊢ ‖((splitMul 𝕜 E) x⋆).2‖ * ‖((splitMul 𝕜 E) x).2‖ ≤ ‖((splitMul 𝕜 E) x).2‖ * ‖((splitMul 𝕜 E) x).2‖

Tactic:
gcongr

Goal state after:
case hbc
𝕜 : Type u_1
E : Type u_2
inst✝⁹ : DenselyNormedField 𝕜
inst✝⁸ : NonUnitalNormedRing E
inst✝⁷ : StarRing E
inst✝⁶ : CStarRing E
inst✝⁵ : NormedSpace 𝕜 E
inst✝⁴ : IsScalarTower 𝕜 E E
inst✝³ : SMulCommClass 𝕜 E E
inst✝² : StarRing 𝕜
inst✝¹ : StarModule 𝕜 E
inst✝ : CStarRing 𝕜
x : Unitization 𝕜 E
h₁ : ∀ (x : Unitization 𝕜 E), ‖((splitMul 𝕜 E) x).2‖ ≤ ‖((splitMul 𝕜 E) x⋆).2‖
⊢ ‖((splitMul 𝕜 E) x⋆).2‖ ≤ ‖((splitMul 𝕜 E) x).2‖
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unique

Goal state before:
pp: case refine_1
F : Type u_1
R : Type u_2
S : Type u_3
A : Type u_4
B : Type u_5
p : A → Prop
q : B → Prop
inst✝²⁴ : CommSemiring R
inst✝²³ : StarRing R
inst✝²² : MetricSpace R
inst✝²¹ : IsTopologicalSemiring R
inst✝²⁰ : ContinuousStar R
inst✝¹⁹ : Ring A
inst✝¹⁸ : StarRing A
inst✝¹⁷ : TopologicalSpace A
inst✝¹⁶ : Algebra R A
inst✝¹⁵ : Ring B
inst✝¹⁴ : StarRing B
inst✝¹³ : TopologicalSpace B
inst✝¹² : Algebra R B
inst✝¹¹ : CommSemiring S
inst✝¹⁰ : Algebra R S
inst✝⁹ : Algebra S A
inst✝⁸ : Algebra S B
inst✝⁷ : IsScalarTower R S A
inst✝⁶ : IsScalarTower R S B
inst✝⁵ : ContinuousFunctionalCalculus R A p
inst✝⁴ : ContinuousFunctionalCalculus R B q
inst✝³ : ContinuousMap.UniqueHom R B
inst✝² : FunLike F A B
inst✝¹ : AlgHomClass F S A B
inst✝ : StarHomClass F A B
φ : F
f : R → R
a : A
hf : ContinuousOn f (spectrum R a)
hφ : Continuous[inst✝¹⁷, inst✝¹³] ⇑φ
ha : p a
hφa : q (φ a)
ψ : A →⋆ₐ[R] B := StarAlgHom.restrictScalars R ↑φ
this : Continuous[inst✝¹⁷, inst✝¹³] ⇑ψ
h_spec : spectrum R (ψ a) ⊆ spectrum R a
hψa : q (ψ a)
ι : C(↑(spectrum R (ψ a)), ↑(spectrum R a)) := { toFun := Set.inclusion h_spec, continuous_toFun := ⋯ }
⊢ Continuous[_, inst✝¹³] ⇑(ψ.comp (cfcHom ha))

pp: case refine_2
F : Type u_1
R : Type u_2
S : Type u_3
A : Type u_4
B : Type u_5
p : A → Prop
q : B → Prop
inst✝²⁴ : CommSemiring R
inst✝²³ : StarRing R
inst✝²² : MetricSpace R
inst✝²¹ : IsTopologicalSemiring R
inst✝²⁰ : ContinuousStar R
inst✝¹⁹ : Ring A
inst✝¹⁸ : StarRing A
inst✝¹⁷ : TopologicalSpace A
inst✝¹⁶ : Algebra R A
inst✝¹⁵ : Ring B
inst✝¹⁴ : StarRing B
inst✝¹³ : TopologicalSpace B
inst✝¹² : Algebra R B
inst✝¹¹ : CommSemiring S
inst✝¹⁰ : Algebra R S
inst✝⁹ : Algebra S A
inst✝⁸ : Algebra S B
inst✝⁷ : IsScalarTower R S A
inst✝⁶ : IsScalarTower R S B
inst✝⁵ : ContinuousFunctionalCalculus R A p
inst✝⁴ : ContinuousFunctionalCalculus R B q
inst✝³ : ContinuousMap.UniqueHom R B
inst✝² : FunLike F A B
inst✝¹ : AlgHomClass F S A B
inst✝ : StarHomClass F A B
φ : F
f : R → R
a : A
hf : ContinuousOn f (spectrum R a)
hφ : Continuous[inst✝¹⁷, inst✝¹³] ⇑φ
ha : p a
hφa : q (φ a)
ψ : A →⋆ₐ[R] B := StarAlgHom.restrictScalars R ↑φ
this : Continuous[inst✝¹⁷, inst✝¹³] ⇑ψ
h_spec : spectrum R (ψ a) ⊆ spectrum R a
hψa : q (ψ a)
ι : C(↑(spectrum R (ψ a)), ↑(spectrum R a)) := { toFun := Set.inclusion h_spec, continuous_toFun := ⋯ }
⊢ Continuous[_, inst✝¹³] ⇑((cfcHom hψa).comp (ContinuousMap.compStarAlgHom' R R ι))

Tactic:
all_goals
    dsimp [ContinuousMap.compStarAlgHom']
    fun_prop

Goal state after:
Module: Mathlib.Topology.Semicontinuity.Hemicontinuity

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace α
f : α → Set β
s : Set α
x : α
inst✝ : TopologicalSpace β
t₁ t₂ : Set β
ht : t₁ ⊆ t₂
h : ∀ᶠ (x' : α) in 𝓝[s] x, t₁ ∈ 𝓝ˢ (f x')
x' : α
⊢ t₁ ∈ 𝓝ˢ (f x') → t₂ ∈ 𝓝ˢ (f x')

Tactic:
gcongr

Goal state after:
Module: Mathlib.Topology.Semicontinuity.Hemicontinuity

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace α
f : α → Set β
s✝ : Set α
x : α
inst✝ : TopologicalSpace β
s t : Set β
hst : s ⊆ t
⊢ f ⁻¹' Iic s ∈ 𝓝[s✝] x → f ⁻¹' Iic t ∈ 𝓝[s✝] x

Tactic:
gcongr

Goal state after:
Module: Mathlib.Topology.Semicontinuity.Hemicontinuity

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace α
f : α → Set β
s✝ : Set α
x : α
inst✝ : TopologicalSpace β
s t : Set β
hst : s ⊆ t
⊢ (∀ᶠ (x' : α) in 𝓝[s✝] x, s ∈ 𝓝ˢ (f x')) → ∀ᶠ (x' : α) in 𝓝[s✝] x, t ∈ 𝓝ˢ (f x')

Tactic:
gcongr

Goal state after:
Module: Mathlib.Topology.Semicontinuity.Hemicontinuity

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace α
f : α → Set β
inst✝ : TopologicalSpace β
x : α
s t : Set β
hst : s ⊆ t
⊢ f ⁻¹' Iic s ∈ 𝓝 x → f ⁻¹' Iic t ∈ 𝓝 x

Tactic:
gcongr

Goal state after:
Module: Mathlib.Topology.Semicontinuity.Hemicontinuity

Goal state before:
pp: α : Type u_3
β : Type u_4
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
f : α → Set β
ι : Type u_5
inst✝ : RegularSpace β
x₀ : α
l : Filter ι
hf : UpperHemicontinuousAt f x₀
hf_closed : IsClosed[inst✝¹] (f x₀)
x : ι → α
hx : Tendsto x l (𝓝 x₀)
y : ι → β
hy : ∃ᶠ (n : ι) in l, y n ∈ f (x n)
y₀ : β
hy₀ : Tendsto y l (𝓝 y₀)
this : y₀ ∉ f x₀
s : Set β
hs : s ∈ 𝓝ˢ (f x₀)
t : Set β
ht : t ∈ 𝓝 y₀
hst : Disjoint s t
n : ι
hn : n ∈ x ⁻¹' {x | s ∈ 𝓝ˢ (f x)}
hyn : y n ∈ f (x n)
⊢ y n ∈ s

Tactic:
simp only [← subset_interior_iff_mem_nhdsSet, preimage_ofPred_eq, mem_ofPred_eq] at hn

Goal state after:
α : Type u_3
β : Type u_4
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
f : α → Set β
ι : Type u_5
inst✝ : RegularSpace β
x₀ : α
l : Filter ι
hf : UpperHemicontinuousAt f x₀
hf_closed : IsClosed[inst✝¹] (f x₀)
x : ι → α
hx : Tendsto x l (𝓝 x₀)
y : ι → β
hy : ∃ᶠ (n : ι) in l, y n ∈ f (x n)
y₀ : β
hy₀ : Tendsto y l (𝓝 y₀)
this : y₀ ∉ f x₀
s : Set β
hs : s ∈ 𝓝ˢ (f x₀)
t : Set β
ht : t ∈ 𝓝 y₀
hst : Disjoint s t
n : ι
hyn : y n ∈ f (x n)
hn : f (x n) ⊆ interior s
⊢ y n ∈ s
Module: Mathlib.Topology.UniformSpace.Closeds

Goal state before:
pp: α : Type u_1
U V : SetRel α α
h : U ⊆ V
⊢ {x | x.1 ⊆ U.preimage x.2 ∧ x.2 ⊆ U.image x.1} ⊆ {x | x.1 ⊆ V.preimage x.2 ∧ x.2 ⊆ V.image x.1}

Tactic:
gcongr

Goal state after:
Module: Mathlib.Analysis.Normed.Algebra.Spectrum

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : CompleteSpace A
a : A
z : 𝕜
hz : z ∈ spectrum 𝕜 a
this : NormedAlgebra ℚ A
hexpmul : exp a = exp (a - ↑ₐ z) * ↑ₐ (exp z)
b : A := ∑' (n : ℕ), (↑(n + 1).factorial)⁻¹ • (a - ↑ₐ z) ^ n
n : ℕ
hn : n ≠ 0
⊢ ‖(a - ↑ₐ z) ^ n‖ / ↑(n + 1).factorial ≤ ‖a - ↑ₐ z‖ ^ n / ↑n.factorial

Tactic:
gcongr

Goal state after:
case hac
𝕜 : Type u_1
A : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : CompleteSpace A
a : A
z : 𝕜
hz : z ∈ spectrum 𝕜 a
this : NormedAlgebra ℚ A
hexpmul : exp a = exp (a - ↑ₐ z) * ↑ₐ (exp z)
b : A := ⋯
n : ℕ
hn : n ≠ 0
⊢ ‖(a - ↑ₐ z) ^ n‖ ≤ ‖a - ↑ₐ z‖ ^ n

case h
𝕜 : Type u_1
A : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : CompleteSpace A
a : A
z : 𝕜
hz : z ∈ spectrum 𝕜 a
this : NormedAlgebra ℚ A
hexpmul : exp a = exp (a - ↑ₐ z) * ↑ₐ (exp z)
b : A := ⋯
n : ℕ
hn : n ≠ 0
⊢ n ≤ n + 1
Module: Mathlib.Topology.Sets.VietorisTopology

Goal state before:
pp: case neg
α : Type u_1
inst✝ : TopologicalSpace α
K : Set α
hK : IsCompact K
s : Set (Set α)
hsK : s ⊆ 𝒫 K
hs : ∀ L ∈ s, IsCompact L
S : Set (Set (Set α))
hS : S ⊆ powerset '' {U | IsOpen[inst✝] U} ∪ (fun V ↦ {s | (s ∩ V).Nonempty}) '' {V | IsOpen[inst✝] V}
hKS : {t | t ⊆ K ∧ ∀ L ∈ s, (t ∩ L).Nonempty} ⊆ ⋃₀ S
u : Set (Set α) := {U | IsOpen[inst✝] U ∧ {s | (s ∩ U).Nonempty} ∈ S}
hsu : ∀ L ∈ s, (K \ ⋃₀ u ∩ L).Nonempty
w✝ : Set (Set α)
hUS : w✝ ∈ S
hUu : K \ ⋃₀ u ∈ w✝
⊢ ∃ Q ⊆ S, Q.Finite ∧ {t | t ⊆ K ∧ ∀ L ∈ s, (t ∩ L).Nonempty} ⊆ ⋃₀ Q

Tactic:
rcases hS hUS with ⟨U, hU, rfl⟩ | ⟨U, hU, rfl⟩

Goal state after:
case neg.inl
α : Type u_1
inst✝ : TopologicalSpace α
K : Set α
hK : IsCompact K
s : Set (Set α)
hsK : s ⊆ 𝒫 K
hs : ∀ L ∈ s, IsCompact L
S : Set (Set (Set α))
hS : S ⊆ powerset '' {U | IsOpen[inst✝] U} ∪ (fun V ↦ {s | (s ∩ V).Nonempty}) '' {V | IsOpen[inst✝] V}
hKS : {t | t ⊆ K ∧ ∀ L ∈ s, (t ∩ L).Nonempty} ⊆ ⋃₀ S
u : Set (Set α) := {U | IsOpen[inst✝] U ∧ {s | (s ∩ U).Nonempty} ∈ S}
hsu : ∀ L ∈ s, (K \ ⋃₀ u ∩ L).Nonempty
U : Set α
hU : U ∈ {U | IsOpen[inst✝] U}
hUS : 𝒫 U ∈ S
hUu : K \ ⋃₀ u ∈ 𝒫 U
⊢ ∃ Q ⊆ S, Q.Finite ∧ {t | t ⊆ K ∧ ∀ L ∈ s, (t ∩ L).Nonempty} ⊆ ⋃₀ Q

case neg.inr
α : Type u_1
inst✝ : TopologicalSpace α
K : Set α
hK : IsCompact K
s : Set (Set α)
hsK : s ⊆ 𝒫 K
hs : ∀ L ∈ s, IsCompact L
S : Set (Set (Set α))
hS : S ⊆ powerset '' {U | IsOpen[inst✝] U} ∪ (fun V ↦ {s | (s ∩ V).Nonempty}) '' {V | IsOpen[inst✝] V}
hKS : {t | t ⊆ K ∧ ∀ L ∈ s, (t ∩ L).Nonempty} ⊆ ⋃₀ S
u : Set (Set α) := {U | IsOpen[inst✝] U ∧ {s | (s ∩ U).Nonempty} ∈ S}
hsu : ∀ L ∈ s, (K \ ⋃₀ u ∩ L).Nonempty
U : Set α
hU : U ∈ {V | IsOpen[inst✝] V}
hUS : (fun V ↦ {s | (s ∩ V).Nonempty}) U ∈ S
hUu : K \ ⋃₀ u ∈ (fun V ↦ {s | (s ∩ V).Nonempty}) U
⊢ ∃ Q ⊆ S, Q.Finite ∧ {t | t ⊆ K ∧ ∀ L ∈ s, (t ∩ L).Nonempty} ⊆ ⋃₀ Q
Module: Mathlib.Analysis.Normed.Algebra.Spectrum

Goal state before:
pp: 𝕜 : Type u_3
A : Type u_4
SA : Type u_5
inst✝⁵ : NormedRing A
inst✝⁴ : CompleteSpace A
inst✝³ : SetLike SA A
inst✝² : SubringClass SA A
inst✝¹ : NormedField 𝕜
inst✝ : NormedAlgebra 𝕜 A
instSMulMem : SMulMemClass SA 𝕜 A
S : SA
hS : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ↑S
x : ↥S
⊢ frontier (σ 𝕜 x) ⊆ σ 𝕜 ↑x

Tactic:
have : CompleteSpace S := hS.completeSpace_coe
  intro μ hμ
  by_contra h
  rw [spectrum.notMem_iff] at h
  rw [← frontier_compl, (spectrum.isClosed _).isOpen_compl.frontier_eq, Set.mem_sdiff] at hμ
  obtain ⟨hμ₁, hμ₂⟩ := hμ
  rw [mem_closure_iff_clusterPt] at hμ₁
  apply hμ₂
  rw [mem_compl_iff, spectrum.notMem_iff]
  refine Subalgebra.isUnit_of_isUnit_val_of_eventually S h ?_ ?_ <| .map hμ₁ (algebraMap 𝕜 S · - x)
  · calc
      _ ≤ map _ (𝓝 μ) := map_mono (by simp)
      _ ≤ _ := by rw [← Filter.Tendsto, ← ContinuousAt]; fun_prop
  · rw [eventually_map]
    apply Eventually.filter_mono inf_le_right
    simp [spectrum.notMem_iff]

Goal state after:
Module: Mathlib.Topology.Sets.VietorisTopology

Goal state before:
pp: α : Type u_1
inst✝ : TopologicalSpace α
s : Set α
⊢ closure {K | (↑K).Finite ∧ ↑K ⊆ s} = {K | ↑K ⊆ closure[inst✝] s}

Tactic:
change closure (SetLike.coe ⁻¹' {K : Set α | K.Finite ∧ K ⊆ s}) =
    SetLike.coe ⁻¹' (closure s).powerset

Goal state after:
α : Type u_1
inst✝ : TopologicalSpace α
s : Set α
⊢ closure (SetLike.coe ⁻¹' {K | K.Finite ∧ K ⊆ s}) = SetLike.coe ⁻¹' 𝒫 closure[inst✝] s
Module: Mathlib.Analysis.Normed.Algebra.Spectrum

Goal state before:
pp: 𝕜₁ : Type u_3
𝕜₂ : Type u_4
A : Type u_5
inst✝⁶ : NormedField 𝕜₁
inst✝⁵ : NormedField 𝕜₂
inst✝⁴ : NormedRing A
inst✝³ : NormedAlgebra 𝕜₁ A
inst✝² : NormedAlgebra 𝕜₂ A
inst✝¹ : NormedAlgebra 𝕜₁ 𝕜₂
inst✝ : IsScalarTower 𝕜₁ 𝕜₂ A
f : 𝕜₂ → 𝕜₁
a : A
h : SpectrumRestricts a f
this : ∀ (x : 𝕜₁), ‖(algebraMap 𝕜₁ 𝕜₂) x‖₊ = ‖x‖₊
x : 𝕜₁
hx : x ∈ spectrum 𝕜₁ a
⊢ ↑‖x‖₊ ≤ ⨆ k ∈ spectrum 𝕜₂ a, ↑‖k‖₊

pp: 𝕜₁ : Type u_3
𝕜₂ : Type u_4
A : Type u_5
inst✝⁶ : NormedField 𝕜₁
inst✝⁵ : NormedField 𝕜₂
inst✝⁴ : NormedRing A
inst✝³ : NormedAlgebra 𝕜₁ A
inst✝² : NormedAlgebra 𝕜₂ A
inst✝¹ : NormedAlgebra 𝕜₁ 𝕜₂
inst✝ : IsScalarTower 𝕜₁ 𝕜₂ A
f : 𝕜₂ → 𝕜₁
a : A
h : SpectrumRestricts a f
this : ∀ (x : 𝕜₁), ‖(algebraMap 𝕜₁ 𝕜₂) x‖₊ = ‖x‖₊
x : 𝕜₂
hx : x ∈ spectrum 𝕜₂ a
⊢ ↑‖x‖₊ ≤ ⨆ k ∈ spectrum 𝕜₁ a, ↑‖k‖₊

Tactic:
· refine congr_arg ((↑) : ℝ≥0 → ℝ≥0∞) (this x) |>.symm.trans_le <| le_iSup₂ (α := ℝ≥0∞) _ ?_
    exact (spectrum.algebraMap_mem_iff _).mpr hx

Goal state after:
𝕜₁ : Type u_3
𝕜₂ : Type u_4
A : Type u_5
inst✝⁶ : NormedField 𝕜₁
inst✝⁵ : NormedField 𝕜₂
inst✝⁴ : NormedRing A
inst✝³ : NormedAlgebra 𝕜₁ A
inst✝² : NormedAlgebra 𝕜₂ A
inst✝¹ : NormedAlgebra 𝕜₁ 𝕜₂
inst✝ : IsScalarTower 𝕜₁ 𝕜₂ A
f : 𝕜₂ → 𝕜₁
a : A
h : SpectrumRestricts a f
this : ∀ (x : 𝕜₁), ‖(algebraMap 𝕜₁ 𝕜₂) x‖₊ = ‖x‖₊
x : 𝕜₂
hx : x ∈ spectrum 𝕜₂ a
⊢ ↑‖x‖₊ ≤ ⨆ k ∈ spectrum 𝕜₁ a, ↑‖k‖₊
Module: Mathlib.Analysis.Normed.Algebra.Spectrum

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
inst✝⁴ : NormedField 𝕜
inst✝³ : ProperSpace 𝕜
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : CompleteSpace A
a₀ a : A
ha : a ∈ Metric.closedBall a₀ 1
⊢ ‖a‖ * ‖1‖ ≤ (‖a₀‖ + 1) * ‖1‖

Tactic:
gcongr

Goal state after:
case hbc
𝕜 : Type u_1
A : Type u_2
inst✝⁴ : NormedField 𝕜
inst✝³ : ProperSpace 𝕜
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : CompleteSpace A
a₀ a : A
ha : a ∈ Metric.closedBall a₀ 1
⊢ ‖a‖ ≤ ‖a₀‖ + 1
Module: Mathlib.Analysis.Normed.Algebra.Spectrum

Goal state before:
pp: case hbc
𝕜 : Type u_1
A : Type u_2
inst✝⁴ : NormedField 𝕜
inst✝³ : ProperSpace 𝕜
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : CompleteSpace A
a₀ a : A
ha : a ∈ Metric.closedBall a₀ 1
⊢ ‖a₀‖ + ‖a - a₀‖ ≤ ‖a₀‖ + 1

Tactic:
gcongr

Goal state after:
case h₂
𝕜 : Type u_1
A : Type u_2
inst✝⁴ : NormedField 𝕜
inst✝³ : ProperSpace 𝕜
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : CompleteSpace A
a₀ a : A
ha : a ∈ Metric.closedBall a₀ 1
⊢ ‖a - a₀‖ ≤ 1
Module: Mathlib.Analysis.Complex.Convex

Goal state before:
pp: r : ℝ
⊢ IsPathConnected {0}ᶜ

Tactic:
by
      convert!
        (((convex_halfSpace_im_gt 0).isPathConnected ⟨.I, by simp⟩).union
              ((convex_halfSpace_re_gt 0).isPathConnected ⟨1, by simp⟩) ⟨1 + .I, by simp⟩).union
          (((convex_halfSpace_im_lt 0).isPathConnected ⟨-.I, by simp⟩).union
            ((convex_halfSpace_re_lt 0).isPathConnected ⟨-1, by simp⟩) ⟨-1 - .I, by simp⟩)
          ⟨1 - .I, by simp⟩ using 1
      ext x
      refine ⟨?_, by aesop⟩
      simp +contextual [Complex.ext_iff, -not_and, not_and_or, or_imp, ← ne_eq, ← lt_or_lt_iff_ne]

Goal state after:
Module: Mathlib.Topology.Sets.VietorisTopology

Goal state before:
pp: case inr.refine_1
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
f : α → β
inst✝ : RegularSpace α
U : Set α
hU : U ∈ {U | IsOpen[inst✝³] U}
K : Compacts α
x : α
hx₁ : x ∈ ↑K
hx₂ : x ∈ U
V W : Set α
hV : IsOpen[inst✝³] V
hW : IsOpen[inst✝³] W
hxV : {x} ⊆ V
hUW : Uᶜ ⊆ W
hVW : Disjoint V W
⊢ {a | ↑a ⊆ Uᶜ} ⊆ {K | ↑K ⊆ W}

Tactic:
gcongr

Goal state after:
Module: Mathlib.Topology.Sets.VietorisTopology

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
f : α → β
inst✝ : LocallyCompactSpace α
K : Compacts α
u : Set (Set α)
hu₁ : u.Finite
hu₂ : ∀ U ∈ u, IsOpen[inst✝³] U
hKu₁ : ↑K ⊆ ⋃₀ u
hKu₂ : ∀ U ∈ u, (↑K ∩ U).Nonempty
L : Set α
hL : IsCompact L
hLK : ↑K ⊆ interior L
hLu : L ⊆ ⋃₀ u
M : Set α → Set α
hM : ∀ U ∈ u, IsCompact (M U)
hML : ∀ U ∈ u, M U ⊆ L
hMU : ∀ U ∈ u, M U ⊆ U
hMK : ∀ U ∈ u, (↑K ∩ interior (M U)).Nonempty
⊢ {K' | ↑K' ⊆ L ∧ ∀ U ∈ u, (↑K' ∩ M U).Nonempty} ⊆ {K | ↑K ⊆ ⋃₀ u ∧ ∀ U ∈ u, (↑K ∩ U).Nonempty}

Tactic:
gcongr

Goal state after:
case h₂
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
f : α → β
inst✝ : LocallyCompactSpace α
K : Compacts α
u : Set (Set α)
hu₁ : u.Finite
hu₂ : ∀ U ∈ u, IsOpen[inst✝³] U
hKu₁ : ↑K ⊆ ⋃₀ u
hKu₂ : ∀ U ∈ u, (↑K ∩ U).Nonempty
L : Set α
hL : IsCompact L
hLK : ↑K ⊆ interior L
hLu : L ⊆ ⋃₀ u
M : Set α → Set α
hM : ∀ U ∈ u, IsCompact (M U)
hML : ∀ U ∈ u, M U ⊆ L
hMU : ∀ U ∈ u, M U ⊆ U
hMK : ∀ U ∈ u, (↑K ∩ interior (M U)).Nonempty
K' : Compacts α
a✝¹ : ↑K' ⊆ L
U : Set α
a✝ : U ∈ u
⊢ M U ⊆ U
Module: Mathlib.Topology.ContinuousMap.ZeroAtInfty

Goal state before:
pp: α : Type u
β : Type v
inst✝² : TopologicalSpace α
inst✝¹ : PseudoMetricSpace β
inst✝ : Zero β
⊢ Isometry toBCF

Tactic:
by tauto

Goal state after:
Module: Mathlib.Analysis.LocallyConvex.Polar

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : AddCommMonoid E
inst✝² : AddCommMonoid F
inst✝¹ : Module 𝕜 E
inst✝ : Module 𝕜 F
B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜
h : B.SeparatingRight
y : F
hy : y ∈ B.polar Set.univ
x : E
ε : ℝ
hε : 0 < ε
c : 𝕜
hc : 0 < ‖c‖
hcε : ‖c‖ < ε
⊢ ‖c‖ * ‖(B (c⁻¹ • x)) y‖ ≤ ε * 1

Tactic:
gcongr

Goal state after:
case h₂
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : AddCommMonoid E
inst✝² : AddCommMonoid F
inst✝¹ : Module 𝕜 E
inst✝ : Module 𝕜 F
B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜
h : B.SeparatingRight
y : F
hy : y ∈ B.polar Set.univ
x : E
ε : ℝ
hε : 0 < ε
c : 𝕜
hc : 0 < ‖c‖
hcε : ‖c‖ < ε
⊢ ‖(B (c⁻¹ • x)) y‖ ≤ 1
Module: Mathlib.Analysis.LocallyConvex.WeakDual

Goal state before:
pp: ι : Type u_4
𝕜 : Type u_5
E : Type u_6
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
f : ι → E →ₗ[𝕜] 𝕜
φ : E →ₗ[𝕜] 𝕜
t𝕜 : TopologicalSpace 𝕜 := inferInstance
t₁ : TopologicalSpace E := ⨅ i, induced (⇑(f i)) t𝕜
t₂ : Finset ι → TopologicalSpace E := fun s ↦ ⨅ i, induced (⇑(f ↑i)) t𝕜
t₁_group : IsTopologicalAddGroup E
t₂_group : ∀ (s : Finset ι), IsTopologicalAddGroup E
t₁_smul : ContinuousSMul 𝕜 E
t₂_smul : ∀ (s : Finset ι), ContinuousSMul 𝕜 E
s : Finset ι
| Continuous[t₂ s, _] ⇑((normSeminorm 𝕜 𝕜).comp φ)

Tactic:
rw [letI := t₂ s; Seminorm.continuous_iff one_pos, nhds_iInf, iInf_subtype]

Goal state after:
ι : Type u_4
𝕜 : Type u_5
E : Type u_6
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
f : ι → E →ₗ[𝕜] 𝕜
φ : E →ₗ[𝕜] 𝕜
t𝕜 : TopologicalSpace 𝕜 := inferInstance
t₁ : TopologicalSpace E := ⨅ i, induced (⇑(f i)) t𝕜
t₂ : Finset ι → TopologicalSpace E := fun s ↦ ⨅ i, induced (⇑(f ↑i)) t𝕜
t₁_group : IsTopologicalAddGroup E
t₂_group : ∀ (s : Finset ι), IsTopologicalAddGroup E
t₁_smul : ContinuousSMul 𝕜 E
t₂_smul : ∀ (s : Finset ι), ContinuousSMul 𝕜 E
s : Finset ι
| ((normSeminorm 𝕜 𝕜).comp φ).ball 0 1 ∈ ⨅ i, ⨅ (h : i ∈ s), 𝓝 0
Module: Mathlib.Topology.ContinuousMap.Units

Goal state before:
pp: X : Type u_1
𝕜 : Type u_4
inst✝² : TopologicalSpace X
inst✝¹ : NormedField 𝕜
inst✝ : CompleteSpace 𝕜
f : C(X, 𝕜)
⊢ spectrum 𝕜 f = Set.range ⇑f

Tactic:
rw [spectrum_eq_preimage_range, Algebra.algebraMap_self]

Goal state after:
X : Type u_1
𝕜 : Type u_4
inst✝² : TopologicalSpace X
inst✝¹ : NormedField 𝕜
inst✝ : CompleteSpace 𝕜
f : C(X, 𝕜)
⊢ ⇑(RingHom.id 𝕜) ⁻¹' Set.range ⇑f = Set.range ⇑f
Module: Mathlib.LinearAlgebra.AffineSpace.Ordered

Goal state before:
pp: k : Type u_1
E : Type u_2
inst✝⁷ : Ring k
inst✝⁶ : PartialOrder k
inst✝⁵ : IsOrderedRing k
inst✝⁴ : AddCommGroup E
inst✝³ : PartialOrder E
inst✝² : IsOrderedAddMonoid E
inst✝¹ : Module k E
inst✝ : IsStrictOrderedModule k E
a a' b : E
r : k
ha : a ≤ a'
hr : r ≤ 1
⊢ (1 - r) • a + r • b ≤ (1 - r) • a' + r • b

Tactic:
gcongr

Goal state after:
case ha
k : Type u_1
E : Type u_2
inst✝⁷ : Ring k
inst✝⁶ : PartialOrder k
inst✝⁵ : IsOrderedRing k
inst✝⁴ : AddCommGroup E
inst✝³ : PartialOrder E
inst✝² : IsOrderedAddMonoid E
inst✝¹ : Module k E
inst✝ : IsStrictOrderedModule k E
a a' b : E
r : k
ha : a ≤ a'
hr : r ≤ 1
⊢ 0 ≤ 1 - r
Module: Mathlib.LinearAlgebra.AffineSpace.Ordered

Goal state before:
pp: k : Type u_1
E : Type u_2
inst✝⁷ : Ring k
inst✝⁶ : PartialOrder k
inst✝⁵ : IsOrderedRing k
inst✝⁴ : AddCommGroup E
inst✝³ : PartialOrder E
inst✝² : IsOrderedAddMonoid E
inst✝¹ : Module k E
inst✝ : IsStrictOrderedModule k E
a a' b : E
r : k
ha : a < a'
hr : r < 1
⊢ (1 - r) • a + r • b < (1 - r) • a' + r • b

Tactic:
gcongr

Goal state after:
case ha
k : Type u_1
E : Type u_2
inst✝⁷ : Ring k
inst✝⁶ : PartialOrder k
inst✝⁵ : IsOrderedRing k
inst✝⁴ : AddCommGroup E
inst✝³ : PartialOrder E
inst✝² : IsOrderedAddMonoid E
inst✝¹ : Module k E
inst✝ : IsStrictOrderedModule k E
a a' b : E
r : k
ha : a < a'
hr : r < 1
⊢ 0 < 1 - r
Module: Mathlib.LinearAlgebra.AffineSpace.Ordered

Goal state before:
pp: k : Type u_1
E : Type u_2
inst✝⁶ : Ring k
inst✝⁵ : PartialOrder k
inst✝⁴ : AddCommGroup E
inst✝³ : PartialOrder E
inst✝² : IsOrderedAddMonoid E
inst✝¹ : Module k E
inst✝ : IsStrictOrderedModule k E
a b b' : E
r : k
hb : b ≤ b'
hr : 0 ≤ r
⊢ (1 - r) • a + r • b ≤ (1 - r) • a + r • b'

Tactic:
gcongr

Goal state after:
Module: Mathlib.LinearAlgebra.AffineSpace.Ordered

Goal state before:
pp: k : Type u_1
E : Type u_2
inst✝⁶ : Ring k
inst✝⁵ : PartialOrder k
inst✝⁴ : AddCommGroup E
inst✝³ : PartialOrder E
inst✝² : IsOrderedAddMonoid E
inst✝¹ : Module k E
inst✝ : IsStrictOrderedModule k E
a b b' : E
r : k
hb : b < b'
hr : 0 < r
⊢ (1 - r) • a + r • b < (1 - r) • a + r • b'

Tactic:
gcongr

Goal state after:
Module: Mathlib.Analysis.Normed.Module.WeakDual

Goal state before:
pp: case e'_4
𝕜 : Type u_1
E : Type u_3
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
⊢ instTopologicalSpaceWeakDual 𝕜 E =
    TopologicalSpace.induced (fun x' ↦ StrongDual.toWeakDual x') (instTopologicalSpaceWeakDual 𝕜 E)

Tactic:
exact induced_id.symm

Goal state after:
Module: Mathlib.Topology.UrysohnsLemma

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
P : Set X → Set X → Prop
h0 : 0 < 2⁻¹
h1234 : 2⁻¹ < 3 / 4
h1 : 3 / 4 < 1
x : X
x✝ : True
n : ℕ
ihn : ∀ (c : CU P), ∀ᶠ (x_1 : X) in 𝓝 x, dist (c.lim x_1) (c.lim x) ≤ (3 / 4) ^ n
c : CU P
hxl : x ∈ c.left.U
a✝ : X
hyl : a✝ ∈ c.left.U
hyd : dist (c.left.lim a✝) (c.left.lim x) ≤ (3 / 4) ^ n
⊢ 2⁻¹ * dist (c.left.lim a✝) (c.left.lim x) ≤ 3 / 4 * (3 / 4) ^ n

Tactic:
gcongr

Goal state after:
Module: Mathlib.Topology.UrysohnsLemma

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
P : Set X → Set X → Prop
h0 : 0 < 2⁻¹
h1234 : 2⁻¹ < 3 / 4
h1 : 3 / 4 < 1
x : X
x✝ : True
n : ℕ
c : CU P
y : X
hxl : x ∉ c.left.left.U
hyl : y ∉ c.left.left.U
r : ℝ := (3 / 4) ^ n
ihn : ∀ (c : CU P), ∀ᶠ (x_1 : X) in 𝓝 x, dist (c.lim x_1) (c.lim x) ≤ r
hydl : dist (c.left.right.lim y) (c.left.right.lim x) ≤ r
hydr : dist (c.right.lim y) (c.right.lim x) ≤ r
⊢ (dist (c.left.right.lim y) (c.left.right.lim x) / 2 + dist (c.right.lim y) (c.right.lim x)) / 2 ≤ (r / 2 + r) / 2

Tactic:
gcongr

Goal state after: