mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Normed.Lp.ProdLp

Goal state before:
pp: α : Type u_2
β : Type u_3
inst✝¹ : SeminormedAddCommGroup α
inst✝ : SeminormedAddCommGroup β
p : ℝ≥0∞
x : WithLp p (α × β)
⊢ toLp p (x.fst, 0) + toLp p (0, x.snd) = 1 x

Tactic:
AddMonoid.End.coe_one,

Goal state after:
α : Type u_2
β : Type u_3
inst✝¹ : SeminormedAddCommGroup α
inst✝ : SeminormedAddCommGroup β
p : ℝ≥0∞
x : WithLp p (α × β)
⊢ toLp p (x.fst, 0) + toLp p (0, x.snd) = id x
Module: Mathlib.Analysis.Normed.Lp.ProdLp

Goal state before:
pp: p : ℝ≥0∞
𝕜 : Type u_1
α✝ : Type u_2
β✝ : Type u_3
hp : Fact (1 ≤ p)
inst✝³ : SeminormedAddCommGroup α✝
inst✝² : SeminormedAddCommGroup β✝
α : Type u_4
β : Type u_5
inst✝¹ : SeminormedAddCommGroup α
inst✝ : SeminormedAddCommGroup β
x y : α × β
⊢ dist x y = ‖toLp p (-x + y)‖

Tactic:
by
    rw [dist_pseudoMetricSpaceToProd, SeminormedAddCommGroup.dist_eq, toLp_add, toLp_neg]

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

Goal state before:
pp: case refine_1
p : ℝ≥0∞
hp : Fact (1 ≤ p)
α : Type u_4
β : Type u_5
inst✝⁶ : SeminormedAddCommGroup α
inst✝⁵ : SeminormedAddCommGroup β
R : Type u_6
inst✝⁴ : SeminormedRing R
inst✝³ : Module R α
inst✝² : Module R β
inst✝¹ : IsBoundedSMul R α
inst✝ : IsBoundedSMul R β
this : PseudoMetricSpace (α × β) := pseudoMetricSpaceToProd p α β
x : R
y z : α × β
⊢ dist (x • y) (x • z) ≤ dist x 0 * dist y z

pp: case refine_2
p : ℝ≥0∞
hp : Fact (1 ≤ p)
α : Type u_4
β : Type u_5
inst✝⁶ : SeminormedAddCommGroup α
inst✝⁵ : SeminormedAddCommGroup β
R : Type u_6
inst✝⁴ : SeminormedRing R
inst✝³ : Module R α
inst✝² : Module R β
inst✝¹ : IsBoundedSMul R α
inst✝ : IsBoundedSMul R β
this : PseudoMetricSpace (α × β) := pseudoMetricSpaceToProd p α β
x y : R
z : α × β
⊢ dist (x • z) (y • z) ≤ dist x y * dist z 0

Tactic:
· simpa [dist_pseudoMetricSpaceToProd] using dist_smul_pair x (toLp p y) (toLp p z)

Goal state after:
case refine_2
p : ℝ≥0∞
hp : Fact (1 ≤ p)
α : Type u_4
β : Type u_5
inst✝⁶ : SeminormedAddCommGroup α
inst✝⁵ : SeminormedAddCommGroup β
R : Type u_6
inst✝⁴ : SeminormedRing R
inst✝³ : Module R α
inst✝² : Module R β
inst✝¹ : IsBoundedSMul R α
inst✝ : IsBoundedSMul R β
this : PseudoMetricSpace (α × β) := pseudoMetricSpaceToProd p α β
x y : R
z : α × β
⊢ dist (x • z) (y • z) ≤ dist x y * dist z 0
Module: Mathlib.Analysis.Normed.Lp.ProdLp

Goal state before:
pp: p : ℝ≥0∞
𝕜 : Type u_1
α✝ : Type u_2
β✝ : Type u_3
hp : Fact (1 ≤ p)
inst✝³ : SeminormedAddCommGroup α✝
inst✝² : SeminormedAddCommGroup β✝
α : Type u_4
β : Type u_5
inst✝¹ : NormedAddCommGroup α
inst✝ : NormedAddCommGroup β
x y : α × β
⊢ dist x y = ‖toLp p (-x + y)‖

Tactic:
by
    rw [dist_pseudoMetricSpaceToProd, SeminormedAddCommGroup.dist_eq, toLp_add, toLp_neg]

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

Goal state before:
pp: case inl
α : Type u_2
β : Type u_3
α' : Type u_5
β' : Type u_6
inst✝³ : PseudoEMetricSpace α
inst✝² : PseudoEMetricSpace β
inst✝¹ : PseudoEMetricSpace α'
inst✝ : PseudoEMetricSpace β'
f : α → α'
hf : Isometry f
g : β → β'
hg : Isometry g
hp : Fact (1 ≤ 0)
x1✝ x2✝ : WithLp 0 (α × β)
⊢ edist (WithLp.map 0 (Prod.map f g) x1✝) (WithLp.map 0 (Prod.map f g) x2✝) = edist x1✝ x2✝

Tactic:
absurd hp.elim; simp

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

Goal state before:
pp: case inl
𝕜 : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
α' : Type u_5
β' : Type u_6
inst✝⁴ : PseudoEMetricSpace α
inst✝³ : PseudoEMetricSpace β
inst✝² : PseudoEMetricSpace γ
inst✝¹ : PseudoEMetricSpace α'
inst✝ : PseudoEMetricSpace β'
hp : Fact (1 ≤ 0)
__spread✝⁻⁰ : WithLp 0 (α × β) ≃ WithLp 0 (β × α) := WithLp.congr 0 (Equiv.prodComm α β)
x✝¹ x✝ : WithLp 0 (α × β)
⊢ edist ((WithLp.congr 0 (Equiv.prodComm α β)).toFun x✝¹) ((WithLp.congr 0 (Equiv.prodComm α β)).toFun x✝) =
    edist x✝¹ x✝

Tactic:
absurd hp.elim; simp

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

Goal state before:
pp: case inl
𝕜 : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
α' : Type u_5
β' : Type u_6
inst✝⁴ : PseudoEMetricSpace α
inst✝³ : PseudoEMetricSpace β
inst✝² : PseudoEMetricSpace γ
inst✝¹ : PseudoEMetricSpace α'
inst✝ : PseudoEMetricSpace β'
hp : Fact (1 ≤ 0)
x✝¹ x✝ : WithLp 0 (WithLp 0 (α × β) × γ)
⊢ edist ((fun x ↦ WithLp.toLp 0 (x.fst.fst, WithLp.toLp 0 (x.fst.snd, x.snd))) x✝¹)
      ((fun x ↦ WithLp.toLp 0 (x.fst.fst, WithLp.toLp 0 (x.fst.snd, x.snd))) x✝) =
    edist x✝¹ x✝

Tactic:
absurd hp.elim; simp

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

Goal state before:
pp: case inl
𝕜 : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
α' : Type u_5
β' : Type u_6
inst✝⁵ : PseudoEMetricSpace α
inst✝⁴ : PseudoEMetricSpace β
inst✝³ : PseudoEMetricSpace γ
inst✝² : PseudoEMetricSpace α'
inst✝¹ : PseudoEMetricSpace β'
inst✝ : Unique β
hp : Fact (1 ≤ 0)
__spread✝⁻⁰ : WithLp 0 (α × β) ≃ α := (WithLp.equiv 0 (α × β)).trans (Equiv.prodUnique α β)
x y : WithLp 0 (α × β)
⊢ edist x.fst y.fst = edist x y

Tactic:
absurd hp.elim; simp

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar

Goal state before:
pp: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
f : E →ₗ[ℝ] E
hf : LinearMap.det f ≠ 0
ι : Type := Fin (finrank ℝ E)
this✝ : FiniteDimensional ℝ (ι → ℝ)
this : finrank ℝ E = finrank ℝ (ι → ℝ)
e : E ≃ₗ[ℝ] ι → ℝ
g : (ι → ℝ) →ₗ[ℝ] ι → ℝ
hg : g = ↑e ∘ₗ f ∘ₗ ↑e.symm
⊢ LinearMap.det (↑e ∘ₗ f ∘ₗ ↑e.symm) = LinearMap.det f

Tactic:
exact LinearMap.det_conj f e

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Lebesgue.Basic

Goal state before:
pp: case e'_2
α : Type u_1
inst✝¹ : MeasurableSpace α
μ : Measure α
f g : α → ℝ
s : Set α
inst✝ : SFinite μ
hf : AEMeasurable f (μ.restrict s)
hg : AEMeasurable g (μ.restrict s)
hs : MeasurableSet s
h₁ : (fun y ↦ ofReal ((g - f) y)) =ᵐ[μ.restrict s] fun y ↦ ofReal ((AEMeasurable.mk g hg - AEMeasurable.mk f hf) y)
h₂ :
  ((μ.restrict s).prod volume) (regionBetween f g s) =
    ((μ.restrict s).prod volume) (regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s)
⊢ (μ.prod volume) (regionBetween f g s) = ((μ.restrict s).prod volume) (regionBetween f g s)

pp: case e'_3
α : Type u_1
inst✝¹ : MeasurableSpace α
μ : Measure α
f g : α → ℝ
s : Set α
inst✝ : SFinite μ
hf : AEMeasurable f (μ.restrict s)
hg : AEMeasurable g (μ.restrict s)
hs : MeasurableSet s
h₁ : (fun y ↦ ofReal ((g - f) y)) =ᵐ[μ.restrict s] fun y ↦ ofReal ((AEMeasurable.mk g hg - AEMeasurable.mk f hf) y)
h₂ :
  ((μ.restrict s).prod volume) (regionBetween f g s) =
    ((μ.restrict s).prod volume) (regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s)
⊢ (μ.prod volume) (regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s) =
    ((μ.restrict s).prod volume) (regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s)

Tactic:
· rw [Measure.restrict_prod_eq_prod_univ]
    exact (Measure.restrict_eq_self _ (regionBetween_subset f g s)).symm

Goal state after:
case e'_3
α : Type u_1
inst✝¹ : MeasurableSpace α
μ : Measure α
f g : α → ℝ
s : Set α
inst✝ : SFinite μ
hf : AEMeasurable f (μ.restrict s)
hg : AEMeasurable g (μ.restrict s)
hs : MeasurableSet s
h₁ : (fun y ↦ ofReal ((g - f) y)) =ᵐ[μ.restrict s] fun y ↦ ofReal ((AEMeasurable.mk g hg - AEMeasurable.mk f hf) y)
h₂ :
  ((μ.restrict s).prod volume) (regionBetween f g s) =
    ((μ.restrict s).prod volume) (regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s)
⊢ (μ.prod volume) (regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s) =
    ((μ.restrict s).prod volume) (regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s)
Module: Mathlib.Analysis.BoxIntegral.Box.SubboxInduction

Goal state before:
pp: ι : Type u_1
p : Box ι → Prop
I : Box ι
H_nhds :
  ∀ z ∈ Box.Icc I,
    ∃ U ∈ 𝓝[Box.Icc I] z,
      ∀ J ≤ I,
        ∀ (m : ℕ),
          z ∈ Box.Icc J → Box.Icc J ⊆ U → (∀ (i : ι), J.upper i - J.lower i = (I.upper i - I.lower i) / 2 ^ m) → p J
J : ℕ → Box ι
hJmono : Antitone J
hJle : ∀ (m : ℕ), J m ≤ I
hJp : ∀ (m : ℕ), ¬p (J m)
hJsub : ∀ (m : ℕ) (i : ι), (J m).upper i - (J m).lower i = (I.upper i - I.lower i) / 2 ^ m
h0 : J 0 = I
z : ι → ℝ := ⨆ m, (J m).lower
hzJ : ∀ (m : ℕ), z ∈ Box.Icc (J m)
hJl_mem : ∀ (m : ℕ), (J m).lower ∈ Box.Icc I
hJu_mem : ∀ (m : ℕ), (J m).upper ∈ Box.Icc I
hJlz : Tendsto (fun m ↦ (J m).lower) atTop (𝓝 z)
this : Tendsto (fun m ↦ (J m).upper - (J m).lower) atTop (𝓝 0)
⊢ Tendsto (fun m ↦ (J m).upper) atTop (𝓝 z)

Tactic:
by simpa using hJlz.add this

Goal state after:
Module: Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction

Goal state before:
pp: ι : Type u_1
inst✝ : Fintype ι
I : Box ι
π₁ : TaggedPrepartition I
π₂ : Prepartition I
hU : π₂.iUnion = ↑I \ π₁.iUnion
r : (ι → ℝ) → ↑(Ioi 0)
⊢ (π₁.unionComplToSubordinate π₂ hU r).distortion = max π₁.distortion π₂.distortion

Tactic:
by
  simp [unionComplToSubordinate]

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.PiL2

Goal state before:
pp: ι : Type u_1
ι' : Type u_2
𝕜 : Type u_3
inst✝⁴ : RCLike 𝕜
E : Type u_4
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : Fintype ι
inst✝ : Fintype ι'
b : OrthonormalBasis ι 𝕜 E
e : ι ≃ ι'
x : E
i' : ι'
⊢ ⟪(b.reindex e) i', x⟫ = ⟪b (e.symm i'), x⟫

Tactic:
OrthonormalBasis.coe_reindex,

Goal state after:
ι : Type u_1
ι' : Type u_2
𝕜 : Type u_3
inst✝⁴ : RCLike 𝕜
E : Type u_4
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : Fintype ι
inst✝ : Fintype ι'
b : OrthonormalBasis ι 𝕜 E
e : ι ≃ ι'
x : E
i' : ι'
⊢ ⟪(⇑b ∘ ⇑e.symm) i', x⟫ = ⟪b (e.symm i'), x⟫
Module: Mathlib.Analysis.BoxIntegral.Partition.Filter

Goal state before:
pp: case refine_1.mpr
ι : Type u_1
inst✝ : Fintype ι
I : Box ι
c : ℝ≥0
l : IntegrationParams
π : TaggedPrepartition I
r : (ι → ℝ) → ↑(Set.Ioi 0)
hπ : l.MemBaseSet I c r π
p : Box ι → Prop
hD : l.bDistortion = true
π₁ : Prepartition I
hπ₁U : π₁.iUnion = ↑I \ π.iUnion
hc : π₁.distortion ≤ c
π₂ : TaggedPrepartition I := ⋯
this : Disjoint π₁.iUnion π₂.iUnion
h : (π.filter p).iUnion ⊆ π.iUnion
x : ι → ℝ
⊢ x ∈ ↑I \ (π.filter p).iUnion → x ∈ ↑I \ π.iUnion ∪ π.iUnion \ (π.filter p).iUnion

Tactic:
rintro ⟨hxI, hxp⟩
      by_cases hxπ : x ∈ π.iUnion
      exacts [Or.inr ⟨hxπ, hxp⟩, Or.inl ⟨hxI, hxπ⟩]

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.PiL2

Goal state before:
pp: case refine_1
𝕜 : Type u_3
inst✝³ : RCLike 𝕜
E : Type u_4
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
v : Set E
inst✝ : FiniteDimensional 𝕜 E
hv : Orthonormal 𝕜 Subtype.val
u₀ : Set E
hu₀s : u₀ ⊇ v
hu₀ : Orthonormal 𝕜 Subtype.val
hu₀_max : (span 𝕜 u₀)ᗮ = ⊥
hu₀_finite : u₀.Finite
u : Finset E := hu₀_finite.toFinset
fu : ↥u ≃ ↑u₀ := hu₀_finite.subtypeEquivToFinset.symm
hu : Orthonormal 𝕜 Subtype.val
⊢ (span 𝕜 (range Subtype.val))ᗮ = ⊥

pp: case refine_2
𝕜 : Type u_3
inst✝³ : RCLike 𝕜
E : Type u_4
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
v : Set E
inst✝ : FiniteDimensional 𝕜 E
hv : Orthonormal 𝕜 Subtype.val
u₀ : Set E
hu₀s : u₀ ⊇ v
hu₀ : Orthonormal 𝕜 Subtype.val
hu₀_max : (span 𝕜 u₀)ᗮ = ⊥
hu₀_finite : u₀.Finite
u : Finset E := hu₀_finite.toFinset
fu : ↥u ≃ ↑u₀ := hu₀_finite.subtypeEquivToFinset.symm
hu : Orthonormal 𝕜 Subtype.val
⊢ v ⊆ ↑u

pp: case refine_3
𝕜 : Type u_3
inst✝³ : RCLike 𝕜
E : Type u_4
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
v : Set E
inst✝ : FiniteDimensional 𝕜 E
hv : Orthonormal 𝕜 Subtype.val
u₀ : Set E
hu₀s : u₀ ⊇ v
hu₀ : Orthonormal 𝕜 Subtype.val
hu₀_max : (span 𝕜 u₀)ᗮ = ⊥
hu₀_finite : u₀.Finite
u : Finset E := hu₀_finite.toFinset
fu : ↥u ≃ ↑u₀ := hu₀_finite.subtypeEquivToFinset.symm
hu : Orthonormal 𝕜 Subtype.val
⊢ ⇑(OrthonormalBasis.mkOfOrthogonalEqBot hu ?refine_1) = Subtype.val

Tactic:
· simpa [u] using! hu₀_max

Goal state after:
case refine_2
𝕜 : Type u_3
inst✝³ : RCLike 𝕜
E : Type u_4
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
v : Set E
inst✝ : FiniteDimensional 𝕜 E
hv : Orthonormal 𝕜 Subtype.val
u₀ : Set E
hu₀s : u₀ ⊇ v
hu₀ : Orthonormal 𝕜 Subtype.val
hu₀_max : (span 𝕜 u₀)ᗮ = ⊥
hu₀_finite : u₀.Finite
u : Finset E := hu₀_finite.toFinset
fu : ↥u ≃ ↑u₀ := hu₀_finite.subtypeEquivToFinset.symm
hu : Orthonormal 𝕜 Subtype.val
⊢ v ⊆ ↑u

case refine_3
𝕜 : Type u_3
inst✝³ : RCLike 𝕜
E : Type u_4
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
v : Set E
inst✝ : FiniteDimensional 𝕜 E
hv : Orthonormal 𝕜 Subtype.val
u₀ : Set E
hu₀s : u₀ ⊇ v
hu₀ : Orthonormal 𝕜 Subtype.val
hu₀_max : (span 𝕜 u₀)ᗮ = ⊥
hu₀_finite : u₀.Finite
u : Finset E := hu₀_finite.toFinset
fu : ↥u ≃ ↑u₀ := hu₀_finite.subtypeEquivToFinset.symm
hu : Orthonormal 𝕜 Subtype.val
⊢ ⇑(OrthonormalBasis.mkOfOrthogonalEqBot hu ⋯) = Subtype.val
Module: Mathlib.Analysis.BoxIntegral.UnitPartition

Goal state before:
pp: case neg
ι : Type u_1
n : ℕ
inst✝¹ : NeZero n
inst✝ : Fintype ι
B I : Box ι
hI : ¬∃ ν ∈ admissibleIndex n B, I = box n ν
⊢ (if hI : ∃ ν ∈ admissibleIndex n B, I = box n ν then tag n hI.choose else ⋯.choose) ∈ Box.Icc B

Tactic:
simp_rw [dif_neg hI]
      exact Box.coe_subset_Icc B.exists_mem.choose_spec

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁵ : RCLike 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace 𝕜 E
ι : Type u_3
inst✝² : LinearOrder ι
inst✝¹ : LocallyFiniteOrderBot ι
inst✝ : WellFoundedLT ι
f : ι → E
hf : Pairwise fun x1 x2 ↦ ⟪f x1, f x2⟫_𝕜 = 0
i : ι
⊢ f i - ∑ i_1 < i, (𝕜 ∙ gramSchmidt 𝕜 f i_1).starProjection (f i) = f i - 0

Tactic:
congr
    apply Finset.sum_eq_zero
    intro j hj
    rw [Submodule.starProjection_apply, Submodule.coe_eq_zero]
    suffices span 𝕜 (f '' Set.Iic j) ⟂ 𝕜 ∙ f i by
      apply orthogonalProjectionOnto_apply_of_mem_orthogonal
      rw [mem_orthogonal_singleton_iff_inner_left, ← mem_orthogonal_singleton_iff_inner_right]
      exact this (gramSchmidt_mem_span 𝕜 f (le_refl j))
    rw [isOrtho_span]
    rintro u ⟨k, hk, rfl⟩ v (rfl : v = f i)
    apply hf
    exact (lt_of_le_of_lt hk (Finset.mem_Iio.mp hj)).ne

Goal state after:
Module: Mathlib.LinearAlgebra.Orientation

Goal state before:
pp: R : Type u_1
inst✝⁸ : CommRing R
inst✝⁷ : PartialOrder R
inst✝⁶ : IsStrictOrderedRing R
M : Type u_2
N : Type u_3
inst✝⁵ : AddCommGroup M
inst✝⁴ : AddCommGroup N
inst✝³ : Module R M
inst✝² : Module R N
ι : Type u_4
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
e : Basis ι R M
f : M ≃ₗ[R] N
⊢ (e.map f).orientation = (Orientation.map ι f) e.orientation

Tactic:
simp_rw [Basis.orientation, Orientation.map_apply, Basis.det_map']

Goal state after:
Module: Mathlib.Analysis.BoxIntegral.UnitPartition

Goal state before:
pp: ι : Type u_1
s : Set (ι → ℝ)
inst✝ : Fintype ι
⊢ (fun x ↦
      ↑(Nat.card ↑(s ∩ (↑⌊x⌋₊)⁻¹ • ↑(span ℤ (Set.range ⇑(Pi.basisFun ℝ ι))))) / ↑⌊x⌋₊ ^ card ι *
        (↑⌊x⌋₊ / x) ^ card ι) =ᶠ[atTop]
    fun x ↦ ↑(Nat.card ↑(s ∩ (↑⌊x⌋₊)⁻¹ • ↑(span ℤ (Set.range ⇑(Pi.basisFun ℝ ι))))) / x ^ card ι

Tactic:
by
  filter_upwards [eventually_ge_atTop 1] with x hx
  have : 0 < ⌊x⌋₊ := Nat.floor_pos.mpr hx
  rw [div_pow, mul_div, div_mul_cancel₀ _ (by positivity)]

Goal state after:
Module: Mathlib.Analysis.PSeries

Goal state before:
pp: case hf₁
b : ℝ
hb : 1 < b
⊢ Summable fun n ↦ ↑n ^ (-b)

Tactic:
rwa [summable_nat_rpow, neg_lt_neg_iff]

Goal state after:
Module: Mathlib.Analysis.PSeries

Goal state before:
pp: case hf₂
b : ℝ
hb : 1 < b
⊢ Summable fun n ↦ ↑n ^ (-b)

Tactic:
rwa [summable_nat_rpow, neg_lt_neg_iff]

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

Goal state before:
pp: case h
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
L : Submodule ℤ E
inst✝² : DiscreteTopology ↥L
ι : Type u_3
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
b : Basis ι ℤ ↥L
n : ℕ
r : ℝ
s : ℕ → Finset (ι → ℤ) := fun n ↦ Fintype.piFinset fun i ↦ Icc (-↑n) ↑n
d : ℕ := Fintype.card ι
hr : r < -↑d
hr' : r < 0
hd : 1 ≤ d
hs0 : s 0 = {0}
hs : ∀ {a b : ℕ}, a ≤ b → s a ⊆ s b
this : ∀ (k : ℕ), #(s (k + 1) \ s k) ≤ 2 * d * (2 * k + 3) ^ (d - 1)
ε : ℝ := normBound b
hε : 0 < ε
k : ℕ
hk : k ∈ range n
v : ι → ℤ
hv : v ∈ s (k + 1) \ s k
j : ι
hj : v j ∉ Icc (-↑k) ↑k
H : -↑k ≤ v j ∧ v j ≤ ↑k
⊢ False

Tactic:
← Finset.mem_Icc

Goal state after:
case h
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
L : Submodule ℤ E
inst✝² : DiscreteTopology ↥L
ι : Type u_3
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
b : Basis ι ℤ ↥L
n : ℕ
r : ℝ
s : ℕ → Finset (ι → ℤ) := fun n ↦ Fintype.piFinset fun i ↦ Icc (-↑n) ↑n
d : ℕ := Fintype.card ι
hr : r < -↑d
hr' : r < 0
hd : 1 ≤ d
hs0 : s 0 = {0}
hs : ∀ {a b : ℕ}, a ≤ b → s a ⊆ s b
this : ∀ (k : ℕ), #(s (k + 1) \ s k) ≤ 2 * d * (2 * k + 3) ^ (d - 1)
ε : ℝ := normBound b
hε : 0 < ε
k : ℕ
hk : k ∈ range n
v : ι → ℤ
hv : v ∈ s (k + 1) \ s k
j : ι
hj : v j ∉ Icc (-↑k) ↑k
H : v j ∈ Icc (-↑k) ↑k
⊢ False
Module: Mathlib.Algebra.Module.ZLattice.Summable

Goal state before:
pp: E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : FiniteDimensional ℝ E
L : Submodule ℤ E
inst✝ : DiscreteTopology ↥L
h✝ : Nontrivial ↥L
I : Type u_1 := Free.ChooseBasisIndex ℤ ↥L
this : Fintype I
b : Basis I ℤ ↥L := Free.chooseBasis ℤ ↥L
d : ℕ := Fintype.card I
hd : d ≠ 0
ε : ℝ := normBound b
⊢ ∃ A > 0, ∃ B > 0, ∀ r < -↑d, ∀ (s : Finset ↥L), ∑ z ∈ s, ‖z‖ ^ r ≤ A * B ^ r * ∑' (k : ℕ), ↑k ^ (↑d - 1 + r)

Tactic:
refine ⟨2 * d * 3 ^ (d - 1), by positivity, ε, normBound_pos b, fun r hr u ↦ ?_⟩

Goal state after:
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : FiniteDimensional ℝ E
L : Submodule ℤ E
inst✝ : DiscreteTopology ↥L
h✝ : Nontrivial ↥L
I : Type u_1 := Free.ChooseBasisIndex ℤ ↥L
this : Fintype I
b : Basis I ℤ ↥L := Free.chooseBasis ℤ ↥L
d : ℕ := Fintype.card I
hd : d ≠ 0
ε : ℝ := normBound b
r : ℝ
hr : r < -↑d
u : Finset ↥L
⊢ ∑ z ∈ u, ‖z‖ ^ r ≤ 2 * ↑d * 3 ^ (d - 1) * ε ^ r * ∑' (k : ℕ), ↑k ^ (↑d - 1 + r)
Module: Mathlib.Data.Nat.Factorial.DoubleFactorial

Goal state before:
pp: n : ℕ
⊢ (2 * (n + 1) + 1)‼ = ∏ i ∈ Finset.range (n + 1), (2 * (i + 1) + 1)

Tactic:
by
    rw [Finset.prod_range_succ, ← doubleFactorial_eq_prod_odd _, mul_comm (2 * n + 1)‼,
      (by ring : 2 * (n + 1) + 1 = 2 * n + 1 + 2)]
    rfl

Goal state after:
Module: Mathlib.Data.Nat.Choose.Multinomial

Goal state before:
pp: α : Type u_1
s : Finset α
a : α
inst✝ : DecidableEq α
ha : a ∉ s
f : α → ℕ
⊢ multinomial (cons a s ha) f = (f a + ∑ i ∈ s, f i).choose (f a) * multinomial s f

Tactic:
multinomial_cons

Goal state after:
α : Type u_1
s : Finset α
a : α
inst✝ : DecidableEq α
ha : a ∉ s
f : α → ℕ
⊢ (f a + ∑ i ∈ s, f i).choose (f a) * multinomial s f = (f a + ∑ i ∈ s, f i).choose (f a) * multinomial s f
Module: Mathlib.Data.Nat.Choose.Multinomial

Goal state before:
pp: α : Type u_1
a : α
f : α → ℕ
⊢ multinomial (cons a ∅ ⋯) f = 1

Tactic:
multinomial_cons

Goal state after:
α : Type u_1
a : α
f : α → ℕ
⊢ (f a + ∑ i ∈ ∅, f i).choose (f a) * multinomial ∅ f = 1
Module: Mathlib.Data.Nat.Choose.Multinomial

Goal state before:
pp: a b : ℕ
⊢ multinomial univ ![a, b] = (a + b)! / (a ! * b !)

Tactic:
rw [multinomial, Fin.sum_univ_two, Fin.prod_univ_two]

Goal state after:
a b : ℕ
⊢ (![a, b] 0 + ![a, b] 1)! / ((![a, b] 0)! * (![a, b] 1)!) = (a + b)! / (a ! * b !)
Module: Mathlib.Data.Nat.Choose.Multinomial

Goal state before:
pp: case neg
α : Type u_1
a : α
f : α →₀ ℕ
h : a ∉ f.support
⊢ Nat.multinomial f.support ⇑f =
    (f.sum fun x ↦ id).choose (f a) * Nat.multinomial (f.update a 0).support ⇑(f.update a 0)

Tactic:
notMem_support_iff

Goal state after:
case neg
α : Type u_1
a : α
f : α →₀ ℕ
h : f a = 0
⊢ Nat.multinomial f.support ⇑f =
    (f.sum fun x ↦ id).choose (f a) * Nat.multinomial (f.update a 0).support ⇑(f.update a 0)
Module: Mathlib.Algebra.MvPolynomial.Division

Goal state before:
pp: case refine_1
σ : Type u_1
R : Type u_3
inst✝ : CommRing R
i : σ
p r : MvPolynomial σ R
h✝ : X i ∣ p - r
h : r = p.modMonomial (Finsupp.single i 1)
n : σ →₀ ℕ
hn : n i ≠ 0
⊢ n ∉ (p.modMonomial (Finsupp.single i 1)).support

Tactic:
notMem_support_iff

Goal state after:
case refine_1
σ : Type u_1
R : Type u_3
inst✝ : CommRing R
i : σ
p r : MvPolynomial σ R
h✝ : X i ∣ p - r
h : r = p.modMonomial (Finsupp.single i 1)
n : σ →₀ ℕ
hn : n i ≠ 0
⊢ coeff n (p.modMonomial (Finsupp.single i 1)) = 0
Module: Mathlib.Algebra.MvPolynomial.Division

Goal state before:
pp: σ : Type u_1
R : Type u_3
inst✝¹ : CommRing R
i : σ
p q : MvPolynomial σ R
inst✝ : IsCancelMulZero R
hi : X i ∣ p
hq : p.divMonomial (Finsupp.single i 1) ∣ q
this : p = X i * p.divMonomial (Finsupp.single i 1)
⊢ p ∣ X i * q

Tactic:
by
        rw [this]
        exact mul_dvd_mul_left (X i) hq

Goal state after:
Module: Mathlib.Data.Nat.Choose.Multinomial

Goal state before:
pp: case cons.e_a.e_a
α : Type u_1
R : Type u_2
inst✝¹ : DecidableEq α
inst✝ : Semiring R
f✝ : α → R
a : α
s : Finset α
has : a ∉ s
ih :
  ∀ (hc : (↑s).Pairwise (Commute on f✝)) (n : ℕ),
    (∑ i ∈ s, f✝ i) ^ n = ∑ k ∈ s.piAntidiag n, ↑(multinomial s k) * s.noncommProd (fun i ↦ f✝ i ^ k i) ⋯
hc : (↑(cons a s has)).Pairwise (Commute on f✝)
n : ℕ
p : ℕ × ℕ
hp : p ∈ antidiagonal n
f : α → ℕ
hf : s.sum f = p.2 ∧ ∀ (i : α), f i ≠ 0 → i ∈ s
t : α
ht : t ∈ s
⊢ (f t + if t = a then p.1 else 0) = f t

Tactic:
if_neg,

Goal state after:
case cons.e_a.e_a
α : Type u_1
R : Type u_2
inst✝¹ : DecidableEq α
inst✝ : Semiring R
f✝ : α → R
a : α
s : Finset α
has : a ∉ s
ih :
  ∀ (hc : (↑s).Pairwise (Commute on f✝)) (n : ℕ),
    (∑ i ∈ s, f✝ i) ^ n = ∑ k ∈ s.piAntidiag n, ↑(multinomial s k) * s.noncommProd (fun i ↦ f✝ i ^ k i) ⋯
hc : (↑(cons a s has)).Pairwise (Commute on f✝)
n : ℕ
p : ℕ × ℕ
hp : p ∈ antidiagonal n
f : α → ℕ
hf : s.sum f = p.2 ∧ ∀ (i : α), f i ≠ 0 → i ∈ s
t : α
ht : t ∈ s
⊢ f t + 0 = f t

case cons.e_a.e_a.hnc
α : Type u_1
R : Type u_2
inst✝¹ : DecidableEq α
inst✝ : Semiring R
f✝ : α → R
a : α
s : Finset α
has : a ∉ s
ih :
  ∀ (hc : (↑s).Pairwise (Commute on f✝)) (n : ℕ),
    (∑ i ∈ s, f✝ i) ^ n = ∑ k ∈ s.piAntidiag n, ↑(multinomial s k) * s.noncommProd (fun i ↦ f✝ i ^ k i) ⋯
hc : (↑(cons a s has)).Pairwise (Commute on f✝)
n : ℕ
p : ℕ × ℕ
hp : p ∈ antidiagonal n
f : α → ℕ
hf : s.sum f = p.2 ∧ ∀ (i : α), f i ≠ 0 → i ∈ s
t : α
ht : t ∈ s
⊢ ¬t = a
Module: Mathlib.Data.Nat.Choose.Multinomial

Goal state before:
pp: case cons.e_a.e_a
α : Type u_1
R : Type u_2
inst✝¹ : DecidableEq α
inst✝ : Semiring R
f✝ : α → R
a : α
s : Finset α
has : a ∉ s
ih :
  ∀ (hc : (↑s).Pairwise (Commute on f✝)) (n : ℕ),
    (∑ i ∈ s, f✝ i) ^ n = ∑ k ∈ s.piAntidiag n, ↑(multinomial s k) * s.noncommProd (fun i ↦ f✝ i ^ k i) ⋯
hc : (↑(cons a s has)).Pairwise (Commute on f✝)
n : ℕ
p : ℕ × ℕ
hp : p ∈ antidiagonal n
f : α → ℕ
hf : s.sum f = p.2 ∧ ∀ (i : α), f i ≠ 0 → i ∈ s
t : α
ht : t ∈ s
⊢ f✝ t ^ (f t + if t = a then p.1 else 0) = f✝ t ^ f t

Tactic:
if_neg,

Goal state after:
case cons.e_a.e_a
α : Type u_1
R : Type u_2
inst✝¹ : DecidableEq α
inst✝ : Semiring R
f✝ : α → R
a : α
s : Finset α
has : a ∉ s
ih :
  ∀ (hc : (↑s).Pairwise (Commute on f✝)) (n : ℕ),
    (∑ i ∈ s, f✝ i) ^ n = ∑ k ∈ s.piAntidiag n, ↑(multinomial s k) * s.noncommProd (fun i ↦ f✝ i ^ k i) ⋯
hc : (↑(cons a s has)).Pairwise (Commute on f✝)
n : ℕ
p : ℕ × ℕ
hp : p ∈ antidiagonal n
f : α → ℕ
hf : s.sum f = p.2 ∧ ∀ (i : α), f i ≠ 0 → i ∈ s
t : α
ht : t ∈ s
⊢ f✝ t ^ (f t + 0) = f✝ t ^ f t

case cons.e_a.e_a.hnc
α : Type u_1
R : Type u_2
inst✝¹ : DecidableEq α
inst✝ : Semiring R
f✝ : α → R
a : α
s : Finset α
has : a ∉ s
ih :
  ∀ (hc : (↑s).Pairwise (Commute on f✝)) (n : ℕ),
    (∑ i ∈ s, f✝ i) ^ n = ∑ k ∈ s.piAntidiag n, ↑(multinomial s k) * s.noncommProd (fun i ↦ f✝ i ^ k i) ⋯
hc : (↑(cons a s has)).Pairwise (Commute on f✝)
n : ℕ
p : ℕ × ℕ
hp : p ∈ antidiagonal n
f : α → ℕ
hf : s.sum f = p.2 ∧ ∀ (i : α), f i ≠ 0 → i ∈ s
t : α
ht : t ∈ s
⊢ ¬t = a