mathlib-initiative/mathlib-tactics huggingface dataset:
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
y : X
hydl : dist (c.left.right.lim y) (c.left.right.lim x) ≤ (3 / 4) ^ n
hydr : dist (c.right.lim y) (c.right.lim x) ≤ (3 / 4) ^ n
hxl : x ∉ c.left.left.U
hyl : y ∉ c.left.left.U
⊢ (dist (c.left.right.lim y) (c.left.right.lim x) / 2 + dist (c.right.lim y) (c.right.lim x)) / 2 ≤
    (3 / 4) ^ n * (3 / 4)

Tactic:
set r := (3 / 4 : ℝ) ^ n

Goal state after:
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 * (3 / 4)
Module: Mathlib.Topology.ContinuousMap.Ideals

Goal state before:
pp: X : Type u_1
𝕜 : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : TopologicalSpace X
inst✝¹ : CompactSpace X
inst✝ : T2Space X
s : Set X
x : X
hx : x ∈ interior s
⊢ x ∈ setOfIdeal (idealOfSet 𝕜 s)

Tactic:
rw [← compl_compl (interior s), ← closure_compl] at hx

Goal state after:
X : Type u_1
𝕜 : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : TopologicalSpace X
inst✝¹ : CompactSpace X
inst✝ : T2Space X
s : Set X
x : X
hx : x ∈ (closure sᶜ)ᶜ
⊢ x ∈ setOfIdeal (idealOfSet 𝕜 s)
Module: Mathlib.Topology.ContinuousMap.Ideals

Goal state before:
pp: X : Type u_1
𝕜 : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : TopologicalSpace X
inst✝¹ : CompactSpace X
inst✝ : T2Space X
s : Opens X
⊢ (idealOfSet 𝕜 ↑s).IsMaximal ↔ IsCoatom s

Tactic:
rw [Ideal.isMaximal_def]
  refine (idealOpensGI X 𝕜).isCoatom_iff (fun I hI => ?_) s
  rw [← Ideal.isMaximal_def] at hI
  exact idealOfSet_ofIdeal_isClosed inferInstance

Goal state after:
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Basic

Goal state before:
pp: A : Type u_1
inst✝ : NonUnitalCStarAlgebra A
a : A
ha : IsStarNormal a
⊢ Isometry ⇑(cfcₙHom ha)

Tactic:
refine AddMonoidHomClass.isometry_of_norm _ fun f ↦ ?_

Goal state after:
A : Type u_1
inst✝ : NonUnitalCStarAlgebra A
a : A
ha : IsStarNormal a
f : C(↑(σₙ ℂ a), ℂ)₀
⊢ ‖(cfcₙHom ha) f‖ = ‖f‖
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Basic

Goal state before:
pp: A : Type u_1
inst✝⁷ : PartialOrder A
inst✝⁶ : Ring A
inst✝⁵ : StarRing A
inst✝⁴ : TopologicalSpace A
inst✝³ : StarOrderedRing A
inst✝² : Algebra ℝ A
inst✝¹ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝ : NonnegSpectrumClass ℝ A
a : A
x y : ℝ
ha : IsUnit a
ha' : 0 ∉ spectrum ℝ≥0 a
z : ℝ≥0
hz : z ∈ spectrum ℝ≥0 a
⊢ (fun x_1 ↦ x_1 ^ (x + y)) z = (fun x_1 ↦ x_1 ^ x * x_1 ^ y) z

Tactic:
have : z ≠ 0 := by aesop

Goal state after:
A : Type u_1
inst✝⁷ : PartialOrder A
inst✝⁶ : Ring A
inst✝⁵ : StarRing A
inst✝⁴ : TopologicalSpace A
inst✝³ : StarOrderedRing A
inst✝² : Algebra ℝ A
inst✝¹ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝ : NonnegSpectrumClass ℝ A
a : A
x y : ℝ
ha : IsUnit a
ha' : 0 ∉ spectrum ℝ≥0 a
z : ℝ≥0
hz : z ∈ spectrum ℝ≥0 a
this : z ≠ 0
⊢ (fun x_1 ↦ x_1 ^ (x + y)) z = (fun x_1 ↦ x_1 ^ x * x_1 ^ y) z
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Order

Goal state before:
pp: A : Type u_1
inst✝² : CStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a : A
ha : 0 ≤ a
b : Aˣ
hb : IsStrictlyPositive ↑b
hbab : 0 ≤ ↑b ^ (-(1 / 2)) * a * ↑b ^ (-(1 / 2))
⊢ a ≤ ↑b ↔ ‖sqrt a * ↑b ^ (-(1 / 2))‖ ≤ 1

Tactic:
conv_rhs =>
    rw [← sq_le_one_iff₀ (norm_nonneg _), sq, ← CStarRing.norm_star_mul_self, star_mul,
      IsSelfAdjoint.of_nonneg (sqrt_nonneg a), IsSelfAdjoint.of_nonneg rpow_nonneg,
      ← mul_assoc, mul_assoc _ _ (sqrt a), sqrt_mul_sqrt_self a,
      CStarAlgebra.norm_le_one_iff_of_nonneg _ hbab]

Goal state after:
A : Type u_1
inst✝² : CStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a : A
ha : 0 ≤ a
b : Aˣ
hb : IsStrictlyPositive ↑b
hbab : 0 ≤ ↑b ^ (-(1 / 2)) * a * ↑b ^ (-(1 / 2))
⊢ a ≤ ↑b ↔ ↑b ^ (-(1 / 2)) * a * ↑b ^ (-(1 / 2)) ≤ 1
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Order

Goal state before:
pp: A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
⊢ IsClosed {a | 0 ≤ a}

Tactic:
simp only [nonneg_iff_isSelfAdjoint_and_quasispectrumRestricts,
    and_congr_right (SpectrumRestricts.nnreal_iff_nnnorm · le_rfl), Set.ofPred_and]

Goal state after:
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
⊢ IsClosed ({a | IsSelfAdjoint a} ∩ {a | ‖(algebraMap ℝ (Unitization ℂ A)) ↑‖a‖₊ - a‖₊ ≤ ‖a‖₊})
Module: Mathlib.Analysis.CStarAlgebra.Module.Defs

Goal state before:
pp: A : Type u_1
E : Type u_2
inst✝⁷ : NonUnitalCStarAlgebra A
inst✝⁶ : PartialOrder A
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module ℂ E
inst✝³ : SMul A E
inst✝² : Norm E
inst✝¹ : CStarModule A E
inst✝ : StarOrderedRing A
x y : E
⊢ ‖⟪x, x⟫ + ⟪y, x⟫ + ⟪x, y⟫ + ⟪y, y⟫‖ ≤ ‖⟪x, x⟫ + ⟪y, x⟫‖ + ‖⟪x, y⟫‖ + ‖⟪y, y⟫‖

Tactic:
exact norm_add₃_le

Goal state after:
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Order

Goal state before:
pp: A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a✝ e✝ : A
he✝ : IsStarProjection e✝
ha✝ : 0 ≤ a✝
hae✝ : a✝ ≤ e✝
a e : Unitization ℂ A
he : IsStarProjection e
ha : 0 ≤ a
hae : a ≤ e
⊢ sqrt a * (1 - e) = 0

Tactic:
rw [← norm_eq_zero, ← sq_eq_zero_iff, ← norm_star_mul_mul_self_of_nonneg, norm_eq_zero]

Goal state after:
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a✝ e✝ : A
he✝ : IsStarProjection e✝
ha✝ : 0 ≤ a✝
hae✝ : a✝ ≤ e✝
a e : Unitization ℂ A
he : IsStarProjection e
ha : 0 ≤ a
hae : a ≤ e
⊢ star (1 - e) * a * (1 - e) = 0
Module: Mathlib.Analysis.CStarAlgebra.Module.Constructions

Goal state before:
pp: A : Type u_1
inst✝¹⁰ : NonUnitalCStarAlgebra A
inst✝⁹ : PartialOrder A
E : Type u_2
F : Type u_3
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : Module ℂ E
inst✝⁶ : SMul A E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : Module ℂ F
inst✝³ : SMul A F
inst✝² : CStarModule A E
inst✝¹ : CStarModule A F
inst✝ : StarOrderedRing A
x : C⋆ᵐᵒᵈ(A, E × F)
h : x = 0
⊢ ⟪x.1, x.1⟫_A + ⟪x.2, x.2⟫_A = 0

Tactic:
simp [h]

Goal state after:
Module: Mathlib.Analysis.CStarAlgebra.Module.Constructions

Goal state before:
pp: A : Type u_1
inst✝⁷ : NonUnitalCStarAlgebra A
inst✝⁶ : PartialOrder A
ι : Type u_2
E : ι → Type u_3
inst✝⁵ : Fintype ι
inst✝⁴ : (i : ι) → NormedAddCommGroup (E i)
inst✝³ : (i : ι) → Module ℂ (E i)
inst✝² : (i : ι) → SMul A (E i)
inst✝¹ : (i : ι) → CStarModule A (E i)
inst✝ : StarOrderedRing A
x : C⋆ᵐᵒᵈ(A, (i : ι) → E i)
h : x = 0
⊢ ∑ i, ⟪x i, x i⟫_A = 0

Tactic:
simp [h]

Goal state after:
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity

Goal state before:
pp: X : Type u_1
R : Type u_2
A : Type u_3
p : A → Prop
inst✝¹² : CommSemiring R
inst✝¹¹ : StarRing R
inst✝¹⁰ : MetricSpace R
inst✝⁹ : Nontrivial R
inst✝⁸ : IsTopologicalSemiring R
inst✝⁷ : ContinuousStar R
inst✝⁶ : NonUnitalRing A
inst✝⁵ : StarRing A
inst✝⁴ : TopologicalSpace A
inst✝³ : Module R A
inst✝² : SMulCommClass R A A
inst✝¹ : IsScalarTower R A A
inst✝ : NonUnitalContinuousFunctionalCalculus R A p
l : Filter X
F : X → R → R
f : R → R
a : A
h_tendsto : TendstoUniformlyOn F f l (quasispectrum R a)
hF : ∀ᶠ (x : X) in l, ContinuousOn (F x) (quasispectrum R a)
hF0 : ∀ᶠ (x : X) in l, F x 0 = 0
hl : l.NeBot
hf : ContinuousOn f (quasispectrum R a)
hf0 : f 0 = 0
ha : p a
s : Set X := {x | ContinuousOn (F x) (quasispectrum R a) ∧ F x 0 = 0}
hs : s ∈ l
x : { x // x ∈ s }
| ((fun x ↦ cfcₙ (F x) a) ∘ Subtype.val) x

Tactic:
rw [Function.comp_apply, cfcₙ_apply (hf := x.2.1) (hf0 := x.2.2)]

Goal state after:
X : Type u_1
R : Type u_2
A : Type u_3
p : A → Prop
inst✝¹² : CommSemiring R
inst✝¹¹ : StarRing R
inst✝¹⁰ : MetricSpace R
inst✝⁹ : Nontrivial R
inst✝⁸ : IsTopologicalSemiring R
inst✝⁷ : ContinuousStar R
inst✝⁶ : NonUnitalRing A
inst✝⁵ : StarRing A
inst✝⁴ : TopologicalSpace A
inst✝³ : Module R A
inst✝² : SMulCommClass R A A
inst✝¹ : IsScalarTower R A A
inst✝ : NonUnitalContinuousFunctionalCalculus R A p
l : Filter X
F : X → R → R
f : R → R
a : A
h_tendsto : TendstoUniformlyOn F f l (quasispectrum R a)
hF : ∀ᶠ (x : X) in l, ContinuousOn (F x) (quasispectrum R a)
hF0 : ∀ᶠ (x : X) in l, F x 0 = 0
hl : l.NeBot
hf : ContinuousOn f (quasispectrum R a)
hf0 : f 0 = 0
ha : p a
s : Set X := {x | ContinuousOn (F x) (quasispectrum R a) ∧ F x 0 = 0}
hs : s ∈ l
x : { x // x ∈ s }
| (cfcₙHom ha) { toFun := (quasispectrum R a).domRestrict (F ↑x), continuous_toFun := ⋯, map_zero' := ⋯ }
Module: Mathlib.Analysis.Matrix.Normed

Goal state before:
pp: l : Type u_2
m : Type u_3
α : Type u_5
inst✝² : Fintype l
inst✝¹ : Fintype m
inst✝ : NonUnitalSeminormedRing α
A : Matrix l m α
v : m → α
⊢ ‖A *ᵥ v‖₊ ≤ ‖A‖₊ * ‖v‖₊

Tactic:
by
  rw [← linfty_opNNNorm_replicateCol (ι := Fin 1) (A *ᵥ v),
    ← linfty_opNNNorm_replicateCol v (ι := Fin 1)]
  exact linfty_opNNNorm_mul A (replicateCol (Fin 1) v)

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

Goal state before:
pp: case pos
α : Type u_5
inst✝¹ : NormedDivisionRing α
inst✝ : NormedAlgebra ℝ α
a : α
h : a = 0
⊢ a * 1 = (algebraMap ℝ α) ‖a‖

Tactic:
simp [h]

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

Goal state before:
pp: m : Type u_3
α : Type u_5
ι : Type u_7
inst✝² : Fintype m
inst✝¹ : Unique ι
inst✝ : SeminormedAddCommGroup α
v : m → α
⊢ (∑ j, ‖replicateRow ι v default j‖ ^ 2) ^ (1 / 2) = ‖toLp 2 v‖

Tactic:
PiLp.norm_eq_of_L2,

Goal state after:
m : Type u_3
α : Type u_5
ι : Type u_7
inst✝² : Fintype m
inst✝¹ : Unique ι
inst✝ : SeminormedAddCommGroup α
v : m → α
⊢ (∑ j, ‖replicateRow ι v default j‖ ^ 2) ^ (1 / 2) = √(∑ i, ‖(toLp 2 v).ofLp i‖ ^ 2)
Module: Mathlib.Analysis.Matrix.Normed

Goal state before:
pp: n : Type u_4
α : Type u_5
inst✝² : Fintype n
inst✝¹ : SeminormedAddCommGroup α
inst✝ : DecidableEq n
v : n → α
s : Finset (n × n) := Finset.map { toFun := fun i ↦ (i, i), inj' := ⋯ } Finset.univ
⊢ (∑ x ∈ s, ‖diagonal v x.1 x.2‖₊ ^ 2) ^ (1 / 2) = NNReal.sqrt (∑ x, ‖v x‖₊ ^ 2)

pp: n : Type u_4
α : Type u_5
inst✝² : Fintype n
inst✝¹ : SeminormedAddCommGroup α
inst✝ : DecidableEq n
v : n → α
s : Finset (n × n) := Finset.map { toFun := fun i ↦ (i, i), inj' := ⋯ } Finset.univ
i : n × n
_hi : i ∈ Finset.univ
his : i ∉ s
⊢ ‖diagonal v i.1 i.2‖₊ ^ 2 = 0

Tactic:
· rw [Finset.sum_map, NNReal.sqrt_eq_rpow]
    dsimp
    simp_rw [diagonal_apply_eq, NNReal.rpow_two]

Goal state after:
n : Type u_4
α : Type u_5
inst✝² : Fintype n
inst✝¹ : SeminormedAddCommGroup α
inst✝ : DecidableEq n
v : n → α
s : Finset (n × n) := Finset.map { toFun := fun i ↦ (i, i), inj' := ⋯ } Finset.univ
i : n × n
_hi : i ∈ Finset.univ
his : i ∉ s
⊢ ‖diagonal v i.1 i.2‖₊ ^ 2 = 0
Module: Mathlib.Analysis.CStarAlgebra.PositiveLinearMap

Goal state before:
pp: A₁ : Type u_1
A₂ : Type u_2
inst✝⁵ : NonUnitalCStarAlgebra A₁
inst✝⁴ : NonUnitalCStarAlgebra A₂
inst✝³ : PartialOrder A₁
inst✝² : StarOrderedRing A₁
inst✝¹ : PartialOrder A₂
inst✝ : StarOrderedRing A₂
f : A₁ →ₚ[ℂ] A₂
hcontra : ∀ (C : ℝ≥0), ∃ a, 0 ≤ a ∧ ↑C * ‖a‖ < ‖f a‖
x : ℕ → A₁
hx_nonneg : ∀ (x_1 : ℕ), 0 ≤ x x_1
hx_norm : ∀ (x_1 : ℕ), ‖x x_1‖ = 1
hx : ∀ (x_1 : ℕ), 2 ^ (2 * x_1) < ‖f (x x_1)‖
⊢ Summable fun a ↦ ‖2 ^ (-↑a) • x a‖

Tactic:
have : (2 : ℝ)⁻¹ < 1 := by norm_num

Goal state after:
A₁ : Type u_1
A₂ : Type u_2
inst✝⁵ : NonUnitalCStarAlgebra A₁
inst✝⁴ : NonUnitalCStarAlgebra A₂
inst✝³ : PartialOrder A₁
inst✝² : StarOrderedRing A₁
inst✝¹ : PartialOrder A₂
inst✝ : StarOrderedRing A₂
f : A₁ →ₚ[ℂ] A₂
hcontra : ∀ (C : ℝ≥0), ∃ a, 0 ≤ a ∧ ↑C * ‖a‖ < ‖f a‖
x : ℕ → A₁
hx_nonneg : ∀ (x_1 : ℕ), 0 ≤ x x_1
hx_norm : ∀ (x_1 : ℕ), ‖x x_1‖ = 1
hx : ∀ (x_1 : ℕ), 2 ^ (2 * x_1) < ‖f (x x_1)‖
this : 2⁻¹ < 1
⊢ Summable fun a ↦ ‖2 ^ (-↑a) • x a‖
Module: Mathlib.Analysis.CStarAlgebra.CStarMatrix

Goal state before:
pp: m : Type u_1
n : Type u_2
A : Type u_5
M N : CStarMatrix m n A
h : M = N
⊢ ∀ (i : m) (j : n), M i j = N i j

Tactic:
simp [h]

Goal state after:
Module: Mathlib.Analysis.CStarAlgebra.CStarMatrix

Goal state before:
pp: n : Type u_2
A : Type u_5
inst✝ : Star A
f : CStarMatrix n n A
hf : IsSelfAdjoint f
i j : n
⊢ star (f i j) = f j i

Tactic:
← star_apply,

Goal state after:
n : Type u_2
A : Type u_5
inst✝ : Star A
f : CStarMatrix n n A
hf : IsSelfAdjoint f
i j : n
⊢ star f j i = f j i
Module: Mathlib.Analysis.CStarAlgebra.CStarMatrix

Goal state before:
pp: m : Type u_1
n : Type u_2
R✝ : Type u_3
S : Type u_4
A✝ : Type u_5
B : Type u_6
R : Type ?u.14
A : Type ?u.16
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
inst✝⁴ : Semiring R
inst✝³ : AddCommMonoid A
inst✝² : Mul A
inst✝¹ : Module R A
inst✝ : Star A
e : m ≃ n
M N : CStarMatrix m m A
i j : n
⊢ (↑(reindexₗ R A e e)).toFun (M * N) i j = ((↑(reindexₗ R A e e)).toFun M * (↑(reindexₗ R A e e)).toFun N) i j

Tactic:
simp only [mul_apply]

Goal state after:
m : Type u_1
n : Type u_2
R✝ : Type u_3
S : Type u_4
A✝ : Type u_5
B : Type u_6
R : Type ?u.14
A : Type ?u.16
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
inst✝⁴ : Semiring R
inst✝³ : AddCommMonoid A
inst✝² : Mul A
inst✝¹ : Module R A
inst✝ : Star A
e : m ≃ n
M N : CStarMatrix m m A
i j : n
⊢ (↑(reindexₗ R A e e)).toFun (M * N) i j =
    ∑ j_1, (↑(reindexₗ R A e e)).toFun M i j_1 * (↑(reindexₗ R A e e)).toFun N j_1 j
Module: Mathlib.Analysis.CStarAlgebra.CStarMatrix

Goal state before:
pp: m : Type u_1
n : Type u_2
A : Type u_5
inst✝¹ : Fintype m
inst✝ : NonUnitalCStarAlgebra A
M : CStarMatrix m n A
h : toCLM M = 0
i : m
j : n
⊢ ‖(toCLM M) ((WithCStarModule.equiv A (m → A)).symm (Pi.single i (star (M i j)))) j‖ = 0

Tactic:
simp [h]

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Dual

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
ι : Type u_3
x y : E
b : Basis ι 𝕜 E
h : ∀ (i : ι), ⟪x, b i⟫ = ⟪y, b i⟫
⊢ x = y

Tactic:
refine ext_inner_left_basis b fun i => ?_

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
ι : Type u_3
x y : E
b : Basis ι 𝕜 E
h : ∀ (i : ι), ⟪x, b i⟫ = ⟪y, b i⟫
i : ι
⊢ ⟪b i, x⟫ = ⟪b i, y⟫
Module: Mathlib.Analysis.CStarAlgebra.GelfandNaimarkSegal

Goal state before:
pp: case hp
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
f : A →ₚ[ℂ] ℂ
inst✝ : StarOrderedRing A
x✝¹ x✝ : A
⊢ IsClosed
    {a |
      (f.leftMulMapPreGNS (x✝¹ + x✝)).completion a =
        ((f.leftMulMapPreGNS x✝¹).completion + (f.leftMulMapPreGNS x✝).completion) a}

Tactic:
apply isClosed_eq <;> fun_prop

Goal state after:
Module: Mathlib.Analysis.CStarAlgebra.GelfandNaimarkSegal

Goal state before:
pp: case hp
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
f : A →ₚ[ℂ] ℂ
inst✝ : StarOrderedRing A
x✝¹ x✝ : A
⊢ IsClosed
    {a |
      (f.leftMulMapPreGNS (x✝¹ * x✝)).completion a =
        ((f.leftMulMapPreGNS x✝¹).completion * (f.leftMulMapPreGNS x✝).completion) a}

Tactic:
apply isClosed_eq <;> fun_prop

Goal state after:
Module: Mathlib.Analysis.CStarAlgebra.GelfandNaimarkSegal

Goal state before:
pp: case hp
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
f : A →ₚ[ℂ] ℂ
inst✝ : StarOrderedRing A
a : A
⊢ IsClosed {x | ⟪(f.leftMulMapPreGNS (star a)).completion x.1, x.2⟫_ℂ = ⟪x.1, (f.leftMulMapPreGNS a).completion x.2⟫_ℂ}

Tactic:
apply isClosed_eq <;> fun_prop

Goal state after:
Module: Mathlib.Analysis.CStarAlgebra.GelfandNaimarkSegal

Goal state before:
pp: case hp
A : Type u_2
inst✝² : CStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
f : A →ₚ[ℂ] ℂ
⊢ IsClosed {a | (f.gnsNonUnitalStarAlgHom 1) a = 1 a}

Tactic:
apply isClosed_eq <;> fun_prop

Goal state after:
Module: Mathlib.Analysis.Complex.Circle

Goal state before:
pp: h : exp Real.pi = 1
⊢ ↑(exp Real.pi) = 1

Tactic:
simp [h]

Goal state after:
Module: Mathlib.Topology.OpenPartialHomeomorph.Constructions

Goal state before:
pp: case pos
X✝ : Type u_1
X'✝ : Type u_2
Y : Type u_3
Y' : Type u_4
Z✝ : Type u_5
Z' : Type u_6
inst✝⁹ : TopologicalSpace X✝
inst✝⁸ : TopologicalSpace X'✝
inst✝⁷ : TopologicalSpace Y
inst✝⁶ : TopologicalSpace Y'
inst✝⁵ : TopologicalSpace Z✝
inst✝⁴ : TopologicalSpace Z'
e✝ : OpenPartialHomeomorph X✝ Y
X : Type u_7
X' : Type u_8
Z : Type u_9
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace X'
inst✝¹ : TopologicalSpace Z
inst✝ : Nonempty Z
f : X → X'
e : OpenPartialHomeomorph X Z
hf : IsOpenEmbedding f
h✝ : Nonempty X
⊢ ContinuousOn (extend f ↑e fun x ↦ Classical.arbitrary Z) (f '' e.source)

pp: case neg
X✝ : Type u_1
X'✝ : Type u_2
Y : Type u_3
Y' : Type u_4
Z✝ : Type u_5
Z' : Type u_6
inst✝⁹ : TopologicalSpace X✝
inst✝⁸ : TopologicalSpace X'✝
inst✝⁷ : TopologicalSpace Y
inst✝⁶ : TopologicalSpace Y'
inst✝⁵ : TopologicalSpace Z✝
inst✝⁴ : TopologicalSpace Z'
e✝ : OpenPartialHomeomorph X✝ Y
X : Type u_7
X' : Type u_8
Z : Type u_9
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace X'
inst✝¹ : TopologicalSpace Z
inst✝ : Nonempty Z
f : X → X'
e : OpenPartialHomeomorph X Z
hf : IsOpenEmbedding f
h✝ : ¬Nonempty X
⊢ ContinuousOn (extend f ↑e fun x ↦ Classical.arbitrary Z) (f '' e.source)

Tactic:
swap

Goal state after:
case neg
X✝ : Type u_1
X'✝ : Type u_2
Y : Type u_3
Y' : Type u_4
Z✝ : Type u_5
Z' : Type u_6
inst✝⁹ : TopologicalSpace X✝
inst✝⁸ : TopologicalSpace X'✝
inst✝⁷ : TopologicalSpace Y
inst✝⁶ : TopologicalSpace Y'
inst✝⁵ : TopologicalSpace Z✝
inst✝⁴ : TopologicalSpace Z'
e✝ : OpenPartialHomeomorph X✝ Y
X : Type u_7
X' : Type u_8
Z : Type u_9
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace X'
inst✝¹ : TopologicalSpace Z
inst✝ : Nonempty Z
f : X → X'
e : OpenPartialHomeomorph X Z
hf : IsOpenEmbedding f
h✝ : ¬Nonempty X
⊢ ContinuousOn (extend f ↑e fun x ↦ Classical.arbitrary Z) (f '' e.source)

case pos
X✝ : Type u_1
X'✝ : Type u_2
Y : Type u_3
Y' : Type u_4
Z✝ : Type u_5
Z' : Type u_6
inst✝⁹ : TopologicalSpace X✝
inst✝⁸ : TopologicalSpace X'✝
inst✝⁷ : TopologicalSpace Y
inst✝⁶ : TopologicalSpace Y'
inst✝⁵ : TopologicalSpace Z✝
inst✝⁴ : TopologicalSpace Z'
e✝ : OpenPartialHomeomorph X✝ Y
X : Type u_7
X' : Type u_8
Z : Type u_9
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace X'
inst✝¹ : TopologicalSpace Z
inst✝ : Nonempty Z
f : X → X'
e : OpenPartialHomeomorph X Z
hf : IsOpenEmbedding f
h✝ : Nonempty X
⊢ ContinuousOn (extend f ↑e fun x ↦ Classical.arbitrary Z) (f '' e.source)
Module: Mathlib.Topology.OpenPartialHomeomorph.Constructions

Goal state before:
pp: case hf
X✝ : Type u_1
X'✝ : Type u_2
Y : Type u_3
Y' : Type u_4
Z✝ : Type u_5
Z' : Type u_6
inst✝⁹ : TopologicalSpace X✝
inst✝⁸ : TopologicalSpace X'✝
inst✝⁷ : TopologicalSpace Y
inst✝⁶ : TopologicalSpace Y'
inst✝⁵ : TopologicalSpace Z✝
inst✝⁴ : TopologicalSpace Z'
e✝ : OpenPartialHomeomorph X✝ Y
X : Type u_7
X' : Type u_8
Z : Type u_9
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace X'
inst✝¹ : TopologicalSpace Z
inst✝ : Nonempty Z
f : X → X'
e : OpenPartialHomeomorph X Z
hf : IsOpenEmbedding f
h✝ : Nonempty X
F : X' → Z := ⋯
F_eq : F = extend f ↑e fun x ↦ Classical.arbitrary Z
heq : EqOn F (↑e ∘ ↑(IsOpenEmbedding.toOpenPartialHomeomorph f hf).symm) (f '' e.source)
⊢ ContinuousOn (↑(IsOpenEmbedding.toOpenPartialHomeomorph f hf).symm) (f '' e.source)

pp: case h
X✝ : Type u_1
X'✝ : Type u_2
Y : Type u_3
Y' : Type u_4
Z✝ : Type u_5
Z' : Type u_6
inst✝⁹ : TopologicalSpace X✝
inst✝⁸ : TopologicalSpace X'✝
inst✝⁷ : TopologicalSpace Y
inst✝⁶ : TopologicalSpace Y'
inst✝⁵ : TopologicalSpace Z✝
inst✝⁴ : TopologicalSpace Z'
e✝ : OpenPartialHomeomorph X✝ Y
X : Type u_7
X' : Type u_8
Z : Type u_9
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace X'
inst✝¹ : TopologicalSpace Z
inst✝ : Nonempty Z
f : X → X'
e : OpenPartialHomeomorph X Z
hf : IsOpenEmbedding f
h✝ : Nonempty X
F : X' → Z := ⋯
F_eq : F = extend f ↑e fun x ↦ Classical.arbitrary Z
heq : EqOn F (↑e ∘ ↑(IsOpenEmbedding.toOpenPartialHomeomorph f hf).symm) (f '' e.source)
⊢ MapsTo (↑(IsOpenEmbedding.toOpenPartialHomeomorph f hf).symm) (f '' e.source) e.source

Tactic:
swap

Goal state after:
case h
X✝ : Type u_1
X'✝ : Type u_2
Y : Type u_3
Y' : Type u_4
Z✝ : Type u_5
Z' : Type u_6
inst✝⁹ : TopologicalSpace X✝
inst✝⁸ : TopologicalSpace X'✝
inst✝⁷ : TopologicalSpace Y
inst✝⁶ : TopologicalSpace Y'
inst✝⁵ : TopologicalSpace Z✝
inst✝⁴ : TopologicalSpace Z'
e✝ : OpenPartialHomeomorph X✝ Y
X : Type u_7
X' : Type u_8
Z : Type u_9
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace X'
inst✝¹ : TopologicalSpace Z
inst✝ : Nonempty Z
f : X → X'
e : OpenPartialHomeomorph X Z
hf : IsOpenEmbedding f
h✝ : Nonempty X
F : X' → Z := ⋯
F_eq : F = extend f ↑e fun x ↦ Classical.arbitrary Z
heq : EqOn F (↑e ∘ ↑(IsOpenEmbedding.toOpenPartialHomeomorph f hf).symm) (f '' e.source)
⊢ MapsTo (↑(IsOpenEmbedding.toOpenPartialHomeomorph f hf).symm) (f '' e.source) e.source

case hf
X✝ : Type u_1
X'✝ : Type u_2
Y : Type u_3
Y' : Type u_4
Z✝ : Type u_5
Z' : Type u_6
inst✝⁹ : TopologicalSpace X✝
inst✝⁸ : TopologicalSpace X'✝
inst✝⁷ : TopologicalSpace Y
inst✝⁶ : TopologicalSpace Y'
inst✝⁵ : TopologicalSpace Z✝
inst✝⁴ : TopologicalSpace Z'
e✝ : OpenPartialHomeomorph X✝ Y
X : Type u_7
X' : Type u_8
Z : Type u_9
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace X'
inst✝¹ : TopologicalSpace Z
inst✝ : Nonempty Z
f : X → X'
e : OpenPartialHomeomorph X Z
hf : IsOpenEmbedding f
h✝ : Nonempty X
F : X' → Z := ⋯
F_eq : F = extend f ↑e fun x ↦ Classical.arbitrary Z
heq : EqOn F (↑e ∘ ↑(IsOpenEmbedding.toOpenPartialHomeomorph f hf).symm) (f '' e.source)
⊢ ContinuousOn (↑(IsOpenEmbedding.toOpenPartialHomeomorph f hf).symm) (f '' e.source)
Module: Mathlib.Topology.OpenPartialHomeomorph.Constructions

Goal state before:
pp: X : Type u_7
X' : Type u_8
Z : Type u_9
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace X'
inst✝¹ : TopologicalSpace Z
inst✝ : Nonempty Z
f : X → X'
e : OpenPartialHomeomorph X Z
hf : IsOpenEmbedding f
⊢ (e.lift_openEmbedding hf).symm.target = f '' e.source

Tactic:
rw [OpenPartialHomeomorph.symm_target, e.lift_openEmbedding_source]

Goal state after:
Module: Mathlib.Topology.FiberBundle.Basic

Goal state before:
pp: case inr.inr
B : Type u_2
F : Type u_3
inst✝⁶ : TopologicalSpace B
inst✝⁵ : TopologicalSpace F
E : B → Type u_5
inst✝⁴ : TopologicalSpace (TotalSpace F E)
inst✝³ : (b : B) → TopologicalSpace (E b)
inst✝² : ConditionallyCompleteLinearOrder B
inst✝¹ : OrderTopology B
inst✝ : FiberBundle F E
a b : B
ea : Trivialization F TotalSpace.proj
hea : a ∈ ea.baseSet
hab : a ≤ b
s : Set B := {x | x ∈ Icc a b ∧ ∃ e, Icc a x ⊆ e.baseSet}
ha : a ∈ s
sne : s.Nonempty
hsb : b ∈ upperBounds s
sbd : BddAbove s
c : B := sSup s
hsc : IsLUB s c
hc : c ∈ Icc a b
ec : Trivialization F TotalSpace.proj
hec : Icc a c ⊆ ec.baseSet
hlt : c < b
d : B
hdcb : d ∈ Ioc c b
hd : ∃ e, Icc a d ⊆ e.baseSet
⊢ ∃ e, Icc a b ⊆ e.baseSet

Tactic:
exact ((hsc.1 ⟨⟨hc.1.trans hdcb.1.le, hdcb.2⟩, hd⟩).not_gt hdcb.1).elim

Goal state after:
Module: Mathlib.Topology.FiberBundle.Basic

Goal state before:
pp: ι : Type u_1
B : Type u_2
F : Type u_3
X : Type u_4
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace B
inst✝ : TopologicalSpace F
Z : FiberBundleCore ι B F
i✝ : ι
b : B
a : F
i : ι
⊢ ContinuousOn (↑(Z.localTrivAsPartialEquiv i)) (Z.localTrivAsPartialEquiv i).source

Tactic:
rw [continuousOn_open_iff (Z.open_source' i)]

Goal state after:
ι : Type u_1
B : Type u_2
F : Type u_3
X : Type u_4
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace B
inst✝ : TopologicalSpace F
Z : FiberBundleCore ι B F
i✝ : ι
b : B
a : F
i : ι
⊢ ∀ (t : Set (B × F)),
    IsOpen[instTopologicalSpaceProd] t →
      IsOpen ((Z.localTrivAsPartialEquiv i).source ∩ ↑(Z.localTrivAsPartialEquiv i) ⁻¹' t)
Module: Mathlib.Topology.FiberBundle.Trivialization

Goal state before:
pp: B : Type u_1
F : Type u_2
Z : Type u_4
inst✝⁴ : TopologicalSpace B
inst✝³ : TopologicalSpace F
proj : Z → B
inst✝² : TopologicalSpace Z
T : Trivialization F proj
ι : Type u_5
inst✝¹ : TopologicalSpace ι
inst✝ : LocallyCompactPair ι ↑T.baseSet
γ : C(ι, ↑T.baseSet)
i : ι
e : ↑T.source
h : proj ↑e = ↑(γ i)
⊢ proj ↑e ∈ T.baseSet

Tactic:
simp [h]

Goal state after: