mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Measure.Stieltjes

Goal state before:
pp: case neg
R : Type u_1
inst✝⁷ : LinearOrder R
inst✝⁶ : TopologicalSpace R
f : StieltjesFunction R
inst✝⁵ : OrderTopology R
inst✝⁴ : CompactIccSpace R
inst✝³ : MeasurableSpace R
inst✝² : BorelSpace R
inst✝¹ : SecondCountableTopology R
inst✝ : DenselyOrdered R
a : R
ha : ¬IsBot a
b : R
hb : b < a
u : ℕ → R
u_mono : StrictMono u
u_lt_a : ∀ (n : ℕ), u n ∈ Ioo b a
u_lim : Tendsto u atTop (𝓝 a)
⊢ f.measure {a} = ofReal (↑f a - leftLim (↑f) a)

Tactic:
replace u_lt_a n : u n < a := (u_lt_a n).2

Goal state after:
case neg
R : Type u_1
inst✝⁷ : LinearOrder R
inst✝⁶ : TopologicalSpace R
f : StieltjesFunction R
inst✝⁵ : OrderTopology R
inst✝⁴ : CompactIccSpace R
inst✝³ : MeasurableSpace R
inst✝² : BorelSpace R
inst✝¹ : SecondCountableTopology R
inst✝ : DenselyOrdered R
a : R
ha : ¬IsBot a
b : R
hb : b < a
u : ℕ → R
u_mono : StrictMono u
u_lim : Tendsto u atTop (𝓝 a)
u_lt_a : ∀ (n : ℕ), u n < a
⊢ f.measure {a} = ofReal (↑f a - leftLim (↑f) a)
Module: Mathlib.MeasureTheory.Measure.Haar.Basic

Goal state before:
pp: G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
K₁ K₂ : Compacts G
V : Set G
hV : (interior V).Nonempty
h : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)
s : Finset G
h1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ g ∈ s, (fun h ↦ g * h) ⁻¹' V
h2s : s.card = index (K₁.carrier ∪ K₂.carrier) V
this : ∀ K ⊆ ⋃ g ∈ s, (fun x ↦ g * x) ⁻¹' V, index K V ≤ {g ∈ s | ((fun x ↦ g * x) ⁻¹' V ∩ K).Nonempty}.card
⊢ index K₁.carrier V + index K₂.carrier V ≤ s.card

Tactic:
refine
    le_trans
      (add_le_add (this K₁.1 <| Subset.trans subset_union_left h1s)
        (this K₂.1 <| Subset.trans subset_union_right h1s)) ?_

Goal state after:
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
K₁ K₂ : Compacts G
V : Set G
hV : (interior V).Nonempty
h : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)
s : Finset G
h1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ g ∈ s, (fun h ↦ g * h) ⁻¹' V
h2s : s.card = index (K₁.carrier ∪ K₂.carrier) V
this : ∀ K ⊆ ⋃ g ∈ s, (fun x ↦ g * x) ⁻¹' V, index K V ≤ {g ∈ s | ((fun x ↦ g * x) ⁻¹' V ∩ K).Nonempty}.card
⊢ {g ∈ s | ((fun x ↦ g * x) ⁻¹' V ∩ K₁.carrier).Nonempty}.card +
      {g ∈ s | ((fun x ↦ g * x) ⁻¹' V ∩ K₂.carrier).Nonempty}.card ≤
    s.card
Module: Mathlib.MeasureTheory.Group.FundamentalDomain

Goal state before:
pp: case refine_1
G : Type u_1
α : Type u_3
inst✝⁶ : Group G
inst✝⁵ : MulAction G α
inst✝⁴ : MeasurableSpace α
ν : Measure α
inst✝³ : SMulInvariantMeasure G α ν
inst✝² : Countable G
inst✝¹ : MeasurableConstSMul G α
i : SigmaFinite ν
i' : HasFundamentalDomain G α ν
μ : Measure (Quotient α_mod_G)
inst✝ : QuotientMeasureEqMeasurePreimage ν μ
A : ℕ → Set α
hA_meas : ∀ (i : ℕ), MeasurableSet (A i)
hA : ∀ (i : ℕ), ν (A i) < ∞
hA' : ⋃ i, A i = univ
n : ℕ
⊢ μ (Quotient.mk α_mod_G '' A n) < ∞

pp: case refine_2
G : Type u_1
α : Type u_3
inst✝⁶ : Group G
inst✝⁵ : MulAction G α
inst✝⁴ : MeasurableSpace α
ν : Measure α
inst✝³ : SMulInvariantMeasure G α ν
inst✝² : Countable G
inst✝¹ : MeasurableConstSMul G α
i : SigmaFinite ν
i' : HasFundamentalDomain G α ν
μ : Measure (Quotient α_mod_G)
inst✝ : QuotientMeasureEqMeasurePreimage ν μ
A : ℕ → Set α
hA_meas : ∀ (i : ℕ), MeasurableSet (A i)
hA : ∀ (i : ℕ), ν (A i) < ∞
hA' : ⋃ i, A i = univ
⊢ ⋃ i, Quotient.mk α_mod_G '' A i = univ

Tactic:
· obtain ⟨s, fund_dom_s⟩ := i'
    have : π ⁻¹' π '' (A n) = _ := MulAction.quotient_preimage_image_eq_union_mul (A n) (G := G)
    have measπAn : MeasurableSet (π '' A n) := by
      rw [measurableSet_quotient, Quotient.mk''_eq_mk, this]
      apply MeasurableSet.iUnion
      exact fun g ↦ MeasurableSet.const_smul (hA_meas n) g
    rw [fund_dom_s.projection_respects_measure_apply (μ := μ) measπAn, this, iUnion_inter]
    refine lt_of_le_of_lt ?_ (hA n)
    rw [fund_dom_s.measure_eq_tsum (A n)]
    exact measure_iUnion_le _

Goal state after:
case refine_2
G : Type u_1
α : Type u_3
inst✝⁶ : Group G
inst✝⁵ : MulAction G α
inst✝⁴ : MeasurableSpace α
ν : Measure α
inst✝³ : SMulInvariantMeasure G α ν
inst✝² : Countable G
inst✝¹ : MeasurableConstSMul G α
i : SigmaFinite ν
i' : HasFundamentalDomain G α ν
μ : Measure (Quotient α_mod_G)
inst✝ : QuotientMeasureEqMeasurePreimage ν μ
A : ℕ → Set α
hA_meas : ∀ (i : ℕ), MeasurableSet (A i)
hA : ∀ (i : ℕ), ν (A i) < ∞
hA' : ⋃ i, A i = univ
⊢ ⋃ i, Quotient.mk α_mod_G '' A i = univ
Module: Mathlib.MeasureTheory.Integral.Bochner.Set

Goal state before:
pp: case pos.hf
X : Type u_1
mX : MeasurableSpace X
μ : Measure X
f : X → ℝ
s : Set X
hs : ∀ x ∈ s, f x ≤ 1
h's : ∀ x ∈ sᶜ, f x ≤ 0
H : Integrable f μ
g : X → ℝ := ⋯
g_int : Integrable g μ
this : ENNReal.ofReal (∫ (x : X), f x ∂μ) ≤ ENNReal.ofReal (∫ (x : X), g x ∂μ)
x : X
⊢ g x ≤ ENNReal.toReal 1

Tactic:
by_cases H : x ∈ s

Goal state after:
case pos
X : Type u_1
mX : MeasurableSpace X
μ : Measure X
f : X → ℝ
s : Set X
hs : ∀ x ∈ s, f x ≤ 1
h's : ∀ x ∈ sᶜ, f x ≤ 0
H✝ : Integrable f μ
g : X → ℝ := fun x ↦ max (f x) 0
g_int : Integrable g μ
this : ENNReal.ofReal (∫ (x : X), f x ∂μ) ≤ ENNReal.ofReal (∫ (x : X), g x ∂μ)
x : X
H : x ∈ s
⊢ g x ≤ ENNReal.toReal 1

case neg
X : Type u_1
mX : MeasurableSpace X
μ : Measure X
f : X → ℝ
s : Set X
hs : ∀ x ∈ s, f x ≤ 1
h's : ∀ x ∈ sᶜ, f x ≤ 0
H✝ : Integrable f μ
g : X → ℝ := fun x ↦ max (f x) 0
g_int : Integrable g μ
this : ENNReal.ofReal (∫ (x : X), f x ∂μ) ≤ ENNReal.ofReal (∫ (x : X), g x ∂μ)
x : X
H : x ∉ s
⊢ g x ≤ ENNReal.toReal 1
Module: Mathlib.Analysis.InnerProductSpace.Defs

Goal state before:
pp: 𝕜 : Type u_1
F : Type u_3
inst✝² : RCLike 𝕜
inst✝¹ : AddCommGroup F
inst✝ : Module 𝕜 F
c : PreInnerProductSpace.Core 𝕜 F
x y : F
⊢ normSq (⟪x, y⟫ • x - ⟪x, x⟫ • y) = normSq x * (normSq x * normSq y - ‖⟪x, y⟫‖ ^ 2)

Tactic:
by
  rw [← @ofReal_inj 𝕜, ofReal_normSq_eq_inner_self]
  simp only [inner_sub_sub_self, inner_smul_left, inner_smul_right, conj_ofReal, mul_sub, ←
    ofReal_normSq_eq_inner_self x, ← ofReal_normSq_eq_inner_self y]
  rw [← mul_assoc, mul_conj, RCLike.conj_mul, mul_left_comm, ← inner_conj_symm y, mul_conj]
  push_cast
  ring

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
x y : E
⊢ ‖x + y‖ * ‖x + y‖ = ‖x‖ * ‖x‖ + 2 * re ⟪x, y⟫ + ‖y‖ * ‖y‖

Tactic:
repeat' rw [← sq (M := ℝ)]

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
x y : E
⊢ ‖x + y‖ ^ 2 = ‖x‖ ^ 2 + 2 * re ⟪x, y⟫ + ‖y‖ ^ 2
Module: Mathlib.Analysis.InnerProductSpace.Basic

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
x y : E
⊢ ‖x - y‖ * ‖x - y‖ = ‖x‖ * ‖x‖ - 2 * re ⟪x, y⟫ + ‖y‖ * ‖y‖

Tactic:
repeat' rw [← sq (M := ℝ)]

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
x y : E
⊢ ‖x - y‖ ^ 2 = ‖x‖ ^ 2 - 2 * re ⟪x, y⟫ + ‖y‖ ^ 2
Module: Mathlib.Analysis.InnerProductSpace.Orthonormal

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
ι : Type u_4
v : ι → E
hv : Orthonormal 𝕜 v
l₁ l₂ : ι →₀ 𝕜
⊢ ⟪(linearCombination 𝕜 v) l₁, (linearCombination 𝕜 v) l₂⟫ = l₁.sum fun i y ↦ (starRingEnd 𝕜) y * l₂ i

Tactic:
by
  simp [l₁.linearCombination_apply, Finsupp.sum_inner, hv.inner_right_finsupp, inner_smul_left]

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Projection.Minimal

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
K : Submodule 𝕜 E
h : IsComplete ↑K
this : InnerProductSpace ℝ E := rclikeToReal 𝕜 E
K' : Submodule ℝ E := restrictScalars ℝ K
⊢ ∀ (u : E), ∃ v ∈ K, ‖u - v‖ = ⨅ w, ‖u - ↑w‖

Tactic:
exact exists_norm_eq_iInf_of_complete_convex ⟨0, K'.zero_mem⟩ h K'.convex

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Projection.Minimal

Goal state before:
pp: F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
K : Submodule ℝ F
u v : F
hv : v ∈ K
h : ‖u - v‖ = ⨅ w, ‖u - ↑w‖
⊢ ∀ w ∈ K, ⟪u - v, w⟫_ℝ = 0

Tactic:
have h : ∀ w ∈ K, ⟪u - v, w - v⟫_ℝ ≤ 0 := by
        rwa [norm_eq_iInf_iff_real_inner_le_zero] at h
        exacts [K.convex, hv]

Goal state after:
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
K : Submodule ℝ F
u v : F
hv : v ∈ K
h✝ : ‖u - v‖ = ⨅ w, ‖u - ↑w‖
h : ∀ w ∈ K, ⟪u - v, w - v⟫_ℝ ≤ 0
⊢ ∀ w ∈ K, ⟪u - v, w⟫_ℝ = 0
Module: Mathlib.Analysis.InnerProductSpace.Projection.Minimal

Goal state before:
pp: F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
K : Submodule ℝ F
u v : F
hv : v ∈ K
h✝ : ‖u - v‖ = ⨅ w, ‖u - ↑w‖
h : ∀ w ∈ K, ⟪u - v, w - v⟫_ℝ ≤ 0
⊢ ∀ w ∈ K, ⟪u - v, w⟫_ℝ = 0

Tactic:
intro w hw

Goal state after:
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
K : Submodule ℝ F
u v : F
hv : v ∈ K
h✝ : ‖u - v‖ = ⨅ w, ‖u - ↑w‖
h : ∀ w ∈ K, ⟪u - v, w - v⟫_ℝ ≤ 0
w : F
hw : w ∈ K
⊢ ⟪u - v, w⟫_ℝ = 0
Module: Mathlib.Analysis.InnerProductSpace.LinearMap

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
x : E
h : 0 < ‖x‖
⊢ ‖x‖ * ‖x‖ = ‖⟪x, x⟫‖

Tactic:
by
        rw [← sq, inner_self_eq_norm_sq_to_K, norm_pow, norm_ofReal, abs_norm]

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Projection.Minimal

Goal state before:
pp: F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
K : Submodule ℝ F
u v : F
hv : v ∈ K
h : ∀ w ∈ K, ⟪u - v, w⟫_ℝ = 0
⊢ ∀ w ∈ K, ⟪u - v, w - v⟫_ℝ ≤ 0

Tactic:
intro w hw

Goal state after:
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
K : Submodule ℝ F
u v : F
hv : v ∈ K
h : ∀ w ∈ K, ⟪u - v, w⟫_ℝ = 0
w : F
hw : w ∈ K
⊢ ⟪u - v, w - v⟫_ℝ ≤ 0
Module: Mathlib.Analysis.InnerProductSpace.Projection.Minimal

Goal state before:
pp: case mp
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
K : Submodule 𝕜 E
u v : E
hv : v ∈ K
this : InnerProductSpace ℝ E := rclikeToReal 𝕜 E
K' : Submodule ℝ E := restrictScalars ℝ K
H : ‖u - v‖ = ⨅ w, ‖u - ↑w‖
A : ∀ w ∈ K, re ⟪u - v, w⟫_𝕜 = 0
⊢ ∀ w ∈ K, ⟪u - v, w⟫_𝕜 = 0

Tactic:
intro w hw

Goal state after:
case mp
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
K : Submodule 𝕜 E
u v : E
hv : v ∈ K
this : InnerProductSpace ℝ E := rclikeToReal 𝕜 E
K' : Submodule ℝ E := restrictScalars ℝ K
H : ‖u - v‖ = ⨅ w, ‖u - ↑w‖
A : ∀ w ∈ K, re ⟪u - v, w⟫_𝕜 = 0
w : E
hw : w ∈ K
⊢ ⟪u - v, w⟫_𝕜 = 0
Module: Mathlib.Analysis.InnerProductSpace.Projection.Minimal

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
K : Submodule 𝕜 E
u v : E
hv : v ∈ K
this : InnerProductSpace ℝ E := rclikeToReal 𝕜 E
K' : Submodule ℝ E := restrictScalars ℝ K
H : ∀ w ∈ K, ⟪u - v, w⟫_𝕜 = 0
⊢ ∀ w ∈ K', ⟪u - v, w⟫_ℝ = 0

Tactic:
intro w hw

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
K : Submodule 𝕜 E
u v : E
hv : v ∈ K
this : InnerProductSpace ℝ E := rclikeToReal 𝕜 E
K' : Submodule ℝ E := restrictScalars ℝ K
H : ∀ w ∈ K, ⟪u - v, w⟫_𝕜 = 0
w : E
hw : w ∈ K'
⊢ ⟪u - v, w⟫_ℝ = 0
Module: Mathlib.Topology.Algebra.Module.ClosedSubmodule

Goal state before:
pp: R : Type u_2
M : Type u_3
N : Type u_4
inst✝¹⁰ : Semiring R
inst✝⁹ : AddCommMonoid M
inst✝⁸ : TopologicalSpace M
inst✝⁷ : Module R M
inst✝⁶ : AddCommMonoid N
inst✝⁵ : TopologicalSpace N
inst✝⁴ : Module R N
inst✝³ : ContinuousAdd N
inst✝² : ContinuousConstSMul R N
inst✝¹ : ContinuousAdd M
inst✝ : ContinuousConstSMul R M
f : M ≃L[R] N
s t : ClosedSubmodule R M
⊢ (mapEquiv f) (s ⊔ t) = (mapEquiv f) s ⊔ (mapEquiv f) t

Tactic:
by
  ext x
  simp only [mapEquiv_apply, toSubmodule_sup, Submodule.carrier_eq_coe, Submodule.map_coe,
    LinearEquiv.coe_coe, ContinuousLinearEquiv.coe_toLinearEquiv, coe_toSubmodule,
    Submodule.coe_closure, Set.mem_image]
  have : f = f.toLinearEquiv.toLinearMap := by
    exact LinearMap.ext (congrFun rfl)
  rw [← this, ← Submodule.coe_closure, ← Submodule.map_sup, Submodule.map_coe]
  simp [← ContinuousLinearEquiv.image_closure]

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
S T : E →ₗ[𝕜] E
hS : S.IsSymmetric
hT : T.IsSymmetric
h : S.range ≤ T.range
v : E
hv : T v = 0
y : E
hy : T y = S (S v)
⊢ ⟪y, T v⟫ = 0

Tactic:
hv,

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
S T : E →ₗ[𝕜] E
hS : S.IsSymmetric
hT : T.IsSymmetric
h : S.range ≤ T.range
v : E
hv : T v = 0
y : E
hy : T y = S (S v)
⊢ ⟪y, 0⟫ = 0
Module: Mathlib.Analysis.InnerProductSpace.Projection.Basic

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
K : Submodule 𝕜 E
inst✝ : K.HasOrthogonalProjection
v : E
⊢ v - K.starProjection v ∈ Kᗮ

Tactic:
intro w hw

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
K : Submodule 𝕜 E
inst✝ : K.HasOrthogonalProjection
v w : E
hw : w ∈ K
⊢ ⟪w, v - K.starProjection v⟫ = 0
Module: Mathlib.Analysis.InnerProductSpace.Projection.FiniteDimensional

Goal state before:
pp: F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : InnerProductSpace ℝ F
inst✝ : FiniteDimensional ℝ F
n : ℕ
IH :
  ∀ (φ : F ≃ₗᵢ[ℝ] F),
    finrank ℝ ↥(↑(ContinuousLinearMap.id ℝ F - ↑↑φ)).kerᗮ ≤ n →
      ∃ l, l.length ≤ n ∧ φ = (List.map (fun v ↦ (ℝ ∙ v)ᗮ.reflection) l).prod
φ : F ≃ₗᵢ[ℝ] F
hn : finrank ℝ ↥(↑(ContinuousLinearMap.id ℝ F - ↑↑φ)).kerᗮ ≤ n + 1
W : Submodule ℝ F := (↑(ContinuousLinearMap.id ℝ F - ↑↑φ)).ker
hW : ∀ w ∈ W, φ w = w
hn' : ¬finrank ℝ ↥Wᗮ ≤ n
this : Nontrivial ↥Wᗮ
v : ↥Wᗮ
hv : v ≠ 0
⊢ φ ↑v ∈ Wᗮ

Tactic:
intro w hw

Goal state after:
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : InnerProductSpace ℝ F
inst✝ : FiniteDimensional ℝ F
n : ℕ
IH :
  ∀ (φ : F ≃ₗᵢ[ℝ] F),
    finrank ℝ ↥(↑(ContinuousLinearMap.id ℝ F - ↑↑φ)).kerᗮ ≤ n →
      ∃ l, l.length ≤ n ∧ φ = (List.map (fun v ↦ (ℝ ∙ v)ᗮ.reflection) l).prod
φ : F ≃ₗᵢ[ℝ] F
hn : finrank ℝ ↥(↑(ContinuousLinearMap.id ℝ F - ↑↑φ)).kerᗮ ≤ n + 1
W : Submodule ℝ F := (↑(ContinuousLinearMap.id ℝ F - ↑↑φ)).ker
hW : ∀ w ∈ W, φ w = w
hn' : ¬finrank ℝ ↥Wᗮ ≤ n
this : Nontrivial ↥Wᗮ
v : ↥Wᗮ
hv : v ≠ 0
w : F
hw : w ∈ W
⊢ ⟪w, φ ↑v⟫_ℝ = 0
Module: Mathlib.Analysis.InnerProductSpace.Projection.FiniteDimensional

Goal state before:
pp: F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : InnerProductSpace ℝ F
inst✝ : FiniteDimensional ℝ F
n : ℕ
IH :
  ∀ (φ : F ≃ₗᵢ[ℝ] F),
    finrank ℝ ↥(↑(ContinuousLinearMap.id ℝ F - ↑↑φ)).kerᗮ ≤ n →
      ∃ l, l.length ≤ n ∧ φ = (List.map (fun v ↦ (ℝ ∙ v)ᗮ.reflection) l).prod
φ : F ≃ₗᵢ[ℝ] F
hn : finrank ℝ ↥(↑(ContinuousLinearMap.id ℝ F - ↑↑φ)).kerᗮ ≤ n + 1
W : Submodule ℝ F := (↑(ContinuousLinearMap.id ℝ F - ↑↑φ)).ker
hW : ∀ w ∈ W, φ w = w
hn' : ¬finrank ℝ ↥Wᗮ ≤ n
this : Nontrivial ↥Wᗮ
v : ↥Wᗮ
hv : v ≠ 0
hφv : φ ↑v ∈ Wᗮ
hv' : ↑v ∉ W
x : F := ↑v - φ ↑v
ρ : F ≃ₗᵢ[ℝ] F := (ℝ ∙ x)ᗮ.reflection
V : Submodule ℝ F := (↑(ContinuousLinearMap.id ℝ F - ↑↑(φ.trans ρ))).ker
⊢ ∀ (w : F), ρ (φ w) = w → w ∈ V

Tactic:
intro w hw

Goal state after:
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : InnerProductSpace ℝ F
inst✝ : FiniteDimensional ℝ F
n : ℕ
IH :
  ∀ (φ : F ≃ₗᵢ[ℝ] F),
    finrank ℝ ↥(↑(ContinuousLinearMap.id ℝ F - ↑↑φ)).kerᗮ ≤ n →
      ∃ l, l.length ≤ n ∧ φ = (List.map (fun v ↦ (ℝ ∙ v)ᗮ.reflection) l).prod
φ : F ≃ₗᵢ[ℝ] F
hn : finrank ℝ ↥(↑(ContinuousLinearMap.id ℝ F - ↑↑φ)).kerᗮ ≤ n + 1
W : Submodule ℝ F := (↑(ContinuousLinearMap.id ℝ F - ↑↑φ)).ker
hW : ∀ w ∈ W, φ w = w
hn' : ¬finrank ℝ ↥Wᗮ ≤ n
this : Nontrivial ↥Wᗮ
v : ↥Wᗮ
hv : v ≠ 0
hφv : φ ↑v ∈ Wᗮ
hv' : ↑v ∉ W
x : F := ↑v - φ ↑v
ρ : F ≃ₗᵢ[ℝ] F := (ℝ ∙ x)ᗮ.reflection
V : Submodule ℝ F := (↑(ContinuousLinearMap.id ℝ F - ↑↑(φ.trans ρ))).ker
w : F
hw : ρ (φ w) = w
⊢ w ∈ V
Module: Mathlib.Analysis.InnerProductSpace.Projection.FiniteDimensional

Goal state before:
pp: F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : InnerProductSpace ℝ F
inst✝ : FiniteDimensional ℝ F
n : ℕ
IH :
  ∀ (φ : F ≃ₗᵢ[ℝ] F),
    finrank ℝ ↥(↑(ContinuousLinearMap.id ℝ F - ↑↑φ)).kerᗮ ≤ n →
      ∃ l, l.length ≤ n ∧ φ = (List.map (fun v ↦ (ℝ ∙ v)ᗮ.reflection) l).prod
φ : F ≃ₗᵢ[ℝ] F
hn : finrank ℝ ↥(↑(ContinuousLinearMap.id ℝ F - ↑↑φ)).kerᗮ ≤ n + 1
W : Submodule ℝ F := (↑(ContinuousLinearMap.id ℝ F - ↑↑φ)).ker
hW : ∀ w ∈ W, φ w = w
hn' : ¬finrank ℝ ↥Wᗮ ≤ n
this : Nontrivial ↥Wᗮ
v : ↥Wᗮ
hv : v ≠ 0
hφv : φ ↑v ∈ Wᗮ
hv' : ↑v ∉ W
x : F := ↑v - φ ↑v
ρ : F ≃ₗᵢ[ℝ] F := (ℝ ∙ x)ᗮ.reflection
V : Submodule ℝ F := (↑(ContinuousLinearMap.id ℝ F - ↑↑(φ.trans ρ))).ker
hV : ∀ (w : F), ρ (φ w) = w → w ∈ V
⊢ W ≤ V

Tactic:
intro w hw

Goal state after:
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : InnerProductSpace ℝ F
inst✝ : FiniteDimensional ℝ F
n : ℕ
IH :
  ∀ (φ : F ≃ₗᵢ[ℝ] F),
    finrank ℝ ↥(↑(ContinuousLinearMap.id ℝ F - ↑↑φ)).kerᗮ ≤ n →
      ∃ l, l.length ≤ n ∧ φ = (List.map (fun v ↦ (ℝ ∙ v)ᗮ.reflection) l).prod
φ : F ≃ₗᵢ[ℝ] F
hn : finrank ℝ ↥(↑(ContinuousLinearMap.id ℝ F - ↑↑φ)).kerᗮ ≤ n + 1
W : Submodule ℝ F := (↑(ContinuousLinearMap.id ℝ F - ↑↑φ)).ker
hW : ∀ w ∈ W, φ w = w
hn' : ¬finrank ℝ ↥Wᗮ ≤ n
this : Nontrivial ↥Wᗮ
v : ↥Wᗮ
hv : v ≠ 0
hφv : φ ↑v ∈ Wᗮ
hv' : ↑v ∉ W
x : F := ↑v - φ ↑v
ρ : F ≃ₗᵢ[ℝ] F := (ℝ ∙ x)ᗮ.reflection
V : Submodule ℝ F := (↑(ContinuousLinearMap.id ℝ F - ↑↑(φ.trans ρ))).ker
hV : ∀ (w : F), ρ (φ w) = w → w ∈ V
w : F
hw : w ∈ W
⊢ w ∈ V
Module: Mathlib.Analysis.InnerProductSpace.Projection.FiniteDimensional

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
v : Set E
hv : Orthonormal 𝕜 Subtype.val
x : E
hx' : x ∈ (span 𝕜 v)ᗮ
hx : x ≠ 0
e : E := (↑‖x‖)⁻¹ • x
he : ‖e‖ = 1
he' : e ∈ (span 𝕜 v)ᗮ
he'' : e ∉ v
a : E
⊢ a ∈ v → ⟪a, e⟫_𝕜 = 0

Tactic:
ha

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
v : Set E
hv : Orthonormal 𝕜 Subtype.val
x : E
hx' : x ∈ (span 𝕜 v)ᗮ
hx : x ≠ 0
e : E := (↑‖x‖)⁻¹ • x
he : ‖e‖ = 1
he' : e ∈ (span 𝕜 v)ᗮ
he'' : e ∉ v
a : E
ha : a ∈ v
⊢ ⟪a, e⟫_𝕜 = 0
Module: Mathlib.Analysis.Normed.Operator.Banach

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_3
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : CompleteSpace E
F : Type u_5
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : CompleteSpace F
f : E →L[𝕜] F
G : Submodule 𝕜 F
h : IsCompl (↑f).range G
hG : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ↑G
hker : (↑f).ker = ⊥
this : CompleteSpace ↥G
g : (E × ↥G) ≃L[𝕜] F := ⋯
⊢ IsClosed[instTopologicalSpaceProd] ↑(⊤.prod ⊥)

Tactic:
exact isClosed_univ.prod isClosed_singleton

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.PiLp

Goal state before:
pp: ι : Type u_2
β : ι → Type u_4
inst✝¹ : Fintype ι
inst✝ : (i : ι) → PseudoEMetricSpace (β i)
x y : WithLp ∞ ((i : ι) → β i)
⊢ edist x y ≤ edist x.ofLp y.ofLp

Tactic:
by simpa only [ENNReal.div_top, ENNReal.toReal_zero, NNReal.rpow_zero, ENNReal.coe_one,
      one_mul] using antilipschitzWith_ofLp ∞ β x y

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

Goal state before:
pp: 𝕜 : Type u_1
𝕜' : Type u_2
inst✝¹¹ : NontriviallyNormedField 𝕜
inst✝¹⁰ : NontriviallyNormedField 𝕜'
E : Type u_3
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
σ : 𝕜 →+* 𝕜'
σ' : 𝕜' →+* 𝕜
inst✝⁷ : RingHomInvPair σ σ'
F : Type u_4
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜' F
inst✝⁴ : CompleteSpace E
inst✝³ : CompleteSpace F
inst✝² : RingHomInvPair σ' σ
inst✝¹ : RingHomIsometric σ
inst✝ : RingHomIsometric σ'
f : E →SL[σ] F
⊢ Bijective ⇑f ↔ (↑f).range.topologicalClosure = ⊤ ∧ ∃ c, AntilipschitzWith c ⇑f

Tactic:
refine ⟨fun h ↦ ⟨?eq_top, ?anti⟩, fun ⟨hd, c, hf⟩ ↦ ⟨hf.injective, ?surj⟩⟩

Goal state after:
case eq_top
𝕜 : Type u_1
𝕜' : Type u_2
inst✝¹¹ : NontriviallyNormedField 𝕜
inst✝¹⁰ : NontriviallyNormedField 𝕜'
E : Type u_3
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
σ : 𝕜 →+* 𝕜'
σ' : 𝕜' →+* 𝕜
inst✝⁷ : RingHomInvPair σ σ'
F : Type u_4
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜' F
inst✝⁴ : CompleteSpace E
inst✝³ : CompleteSpace F
inst✝² : RingHomInvPair σ' σ
inst✝¹ : RingHomIsometric σ
inst✝ : RingHomIsometric σ'
f : E →SL[σ] F
h : Bijective ⇑f
⊢ (↑f).range.topologicalClosure = ⊤

case anti
𝕜 : Type u_1
𝕜' : Type u_2
inst✝¹¹ : NontriviallyNormedField 𝕜
inst✝¹⁰ : NontriviallyNormedField 𝕜'
E : Type u_3
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
σ : 𝕜 →+* 𝕜'
σ' : 𝕜' →+* 𝕜
inst✝⁷ : RingHomInvPair σ σ'
F : Type u_4
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜' F
inst✝⁴ : CompleteSpace E
inst✝³ : CompleteSpace F
inst✝² : RingHomInvPair σ' σ
inst✝¹ : RingHomIsometric σ
inst✝ : RingHomIsometric σ'
f : E →SL[σ] F
h : Bijective ⇑f
⊢ ∃ c, AntilipschitzWith c ⇑f

case surj
𝕜 : Type u_1
𝕜' : Type u_2
inst✝¹¹ : NontriviallyNormedField 𝕜
inst✝¹⁰ : NontriviallyNormedField 𝕜'
E : Type u_3
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
σ : 𝕜 →+* 𝕜'
σ' : 𝕜' →+* 𝕜
inst✝⁷ : RingHomInvPair σ σ'
F : Type u_4
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜' F
inst✝⁴ : CompleteSpace E
inst✝³ : CompleteSpace F
inst✝² : RingHomInvPair σ' σ
inst✝¹ : RingHomIsometric σ
inst✝ : RingHomIsometric σ'
f : E →SL[σ] F
x✝ : (↑f).range.topologicalClosure = ⊤ ∧ ∃ c, AntilipschitzWith c ⇑f
hd : (↑f).range.topologicalClosure = ⊤
c : ℝ≥0
hf : AntilipschitzWith c ⇑f
⊢ Surjective ⇑f
Module: Mathlib.MeasureTheory.Measure.Lebesgue.Basic

Goal state before:
pp: p : ℝ → Prop
a : ℝ
h : ∀ᶠ (x : ℝ) in 𝓝 a, p x
⊢ 0 < volume {x | p x}

Tactic:
by
  rcases h.exists_Ioo_subset with ⟨l, u, hx, hs⟩
  grw [← hs]
  simpa [-mem_Ioo] using hx.1.trans hx.2

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Haar.OfBasis

Goal state before:
pp: ι : Type u_1
ι' : Type u_2
E : Type u_3
inst✝³ : Fintype ι
inst✝² : Fintype ι'
inst✝¹ : AddCommGroup E
inst✝ : Module ℝ E
v : ι → E
e : ι' ≃ ι
K : (ι' → ℝ) ≃ (ι → ℝ) := Equiv.piCongrLeft' (fun _a ↦ ℝ) e
this✝ : Icc 0 1 = ⇑K '' Icc 0 1
x : E
this : ∀ (z : ι' → ℝ), ∑ x, z (e.symm x) • v x = ∑ i, z i • v (e i)
⊢ x ∈ (fun a ↦ ∑ x, a x • v (e x)) '' Icc 0 1 ↔ x ∈ (fun a ↦ ∑ i, K a i • v i) '' Icc 0 1

Tactic:
mem_image,

Goal state after:
ι : Type u_1
ι' : Type u_2
E : Type u_3
inst✝³ : Fintype ι
inst✝² : Fintype ι'
inst✝¹ : AddCommGroup E
inst✝ : Module ℝ E
v : ι → E
e : ι' ≃ ι
K : (ι' → ℝ) ≃ (ι → ℝ) := Equiv.piCongrLeft' (fun _a ↦ ℝ) e
this✝ : Icc 0 1 = ⇑K '' Icc 0 1
x : E
this : ∀ (z : ι' → ℝ), ∑ x, z (e.symm x) • v x = ∑ i, z i • v (e i)
⊢ (∃ x_1 ∈ Icc 0 1, ∑ x, x_1 x • v (e x) = x) ↔ ∃ x_1 ∈ Icc 0 1, ∑ i, K x_1 i • v i = x
Module: Mathlib.MeasureTheory.Measure.Haar.OfBasis

Goal state before:
pp: ι : Type u_1
E : Type u_3
inst✝² : Fintype ι
inst✝¹ : AddCommGroup E
inst✝ : Module ℝ E
v : ι → E
⊢ parallelepiped v = ∑ i, segment ℝ 0 (v i)

Tactic:
ext
  simp only [mem_parallelepiped_iff, Set.mem_finsetSum, Finset.mem_univ, forall_true_left,
    segment_eq_image, smul_zero, zero_add, ← Set.pi_univ_Icc, Set.mem_univ_pi]
  constructor
  · rintro ⟨t, ht, rfl⟩
    exact ⟨t • v, fun {i} => ⟨t i, ht _, by simp⟩, rfl⟩
  rintro ⟨g, hg, rfl⟩
  choose t ht hg using @hg
  refine ⟨@t, @ht, ?_⟩
  simp_rw [hg]

Goal state after:
Module: Mathlib.Algebra.Module.ZLattice.Basic

Goal state before:
pp: E : Type u_1
ι : Type u_2
K : Type u_3
inst✝⁶ : NormedField K
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace K E
b : Basis ι K E
inst✝³ : LinearOrder K
inst✝² : IsStrictOrderedRing K
inst✝¹ : FloorRing K
inst✝ : Fintype ι
m : E
i : ι
⊢ (b.repr m) i - (b.repr ↑(floor b m)) i = Int.fract ((b.repr m) i)

Tactic:
repr_floor_apply,

Goal state after:
E : Type u_1
ι : Type u_2
K : Type u_3
inst✝⁶ : NormedField K
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace K E
b : Basis ι K E
inst✝³ : LinearOrder K
inst✝² : IsStrictOrderedRing K
inst✝¹ : FloorRing K
inst✝ : Fintype ι
m : E
i : ι
⊢ (b.repr m) i - ↑⌊(b.repr m) i⌋ = Int.fract ((b.repr m) i)