mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.MellinTransform

Goal state before:
pp: case hbc
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
a b : ℝ
f : ℝ → E
s : ℂ
hfc : LocallyIntegrableOn f (Ioi 0) volume
hf_top : f =O[atTop] fun x ↦ x ^ (-a)
hs_top : s.re < a
hf_bot : f =O[𝓝[>] 0] fun x ↦ x ^ (-b)
hs_bot : b < s.re
F : ℂ → ℝ → E := ⋯
F' : ℂ → ℝ → E := ⋯
v : ℝ
hv0 : 0 < v
hv1 : v < s.re - b
hv2 : v < a - s.re
bound : ℝ → ℝ := ⋯
h1 : ∀ᶠ (z : ℂ) in 𝓝 s, AEStronglyMeasurable (F z) (volume.restrict (Ioi 0))
h2 : IntegrableOn (F s) (Ioi 0) volume
h3 : AEStronglyMeasurable (F' s) (volume.restrict (Ioi 0))
t : ℝ
ht : t ∈ Ioi 0
z : ℂ
hz : z ∈ Metric.ball s v
⊢ ‖↑t ^ (z - 1)‖ ≤ t ^ (s.re + v - 1) + t ^ (s.re - v - 1)

Tactic:
norm_cpow_eq_rpow_re_of_pos ht

Goal state after:
case hbc
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
a b : ℝ
f : ℝ → E
s : ℂ
hfc : LocallyIntegrableOn f (Ioi 0) volume
hf_top : f =O[atTop] fun x ↦ x ^ (-a)
hs_top : s.re < a
hf_bot : f =O[𝓝[>] 0] fun x ↦ x ^ (-b)
hs_bot : b < s.re
F : ℂ → ℝ → E := fun z t ↦ ↑t ^ (z - 1) • f t
F' : ℂ → ℝ → E := fun z t ↦ (↑t ^ (z - 1) * ↑(log t)) • f t
v : ℝ
hv0 : 0 < v
hv1 : v < s.re - b
hv2 : v < a - s.re
bound : ℝ → ℝ := fun t ↦ (t ^ (s.re + v - 1) + t ^ (s.re - v - 1)) * |log t| * ‖f t‖
h1 : ∀ᶠ (z : ℂ) in 𝓝 s, AEStronglyMeasurable (F z) (volume.restrict (Ioi 0))
h2 : IntegrableOn (F s) (Ioi 0) volume
h3 : AEStronglyMeasurable (F' s) (volume.restrict (Ioi 0))
t : ℝ
ht : t ∈ Ioi 0
z : ℂ
hz : z ∈ Metric.ball s v
⊢ t ^ (z - 1).re ≤ t ^ (s.re + v - 1) + t ^ (s.re - v - 1)
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Beta

Goal state before:
pp: case hg
u : ℂ
hu : 0 < u.re
v : ℂ
x : ℝ
hx : x ∈ Icc 0 (1 / 2)
⊢ ContinuousAt (fun x ↦ (1 - ↑x) ^ (v - 1)) x

Tactic:
apply ContinuousAt.cpow (by fun_prop) (by fun_prop)

Goal state after:
case hg
u : ℂ
hu : 0 < u.re
v : ℂ
x : ℝ
hx : x ∈ Icc 0 (1 / 2)
⊢ 1 - ↑x ∈ slitPlane
Module: Mathlib.Analysis.Meromorphic.LogDeriv

Goal state before:
pp: case a
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
𝕜' : Type u_2
inst✝¹ : NontriviallyNormedField 𝕜'
inst✝ : NormedAlgebra 𝕜 𝕜'
U : Set 𝕜
ι : Type u_3
F : ι → 𝕜 → 𝕜'
d : ι → ℤ
hd : (support d).Finite
h : ∀ (i : ι), MeromorphicOn (F i) U
h' : ∀ (i : ι), ∀ x ∈ U, meromorphicOrderAt (F i) x ≠ ⊤
hA : ∀ᶠ (y : 𝕜) in codiscreteWithin U, ∀ i ∈ hd.toFinset, AnalyticAt 𝕜 (F i) y
hN : ∀ᶠ (y : 𝕜) in codiscreteWithin U, ∀ i ∈ hd.toFinset, F i y ≠ 0
y : 𝕜
h₁y : ∀ i ∈ hd.toFinset, AnalyticAt 𝕜 (F i) y
h₂y : ∀ i ∈ hd.toFinset, F i y ≠ 0
h₀ : ∏ᶠ (i : ι), F i ^ d i = ∏ i ∈ hd.toFinset, F i ^ d i
hsub : (support fun i ↦ d i • logDeriv (F i) y) ⊆ ↑hd.toFinset
i : ι
hi : i ∈ hd.toFinset
⊢ logDeriv (F i ^ d i) y = ↑(d i) * logDeriv (F i) y

Tactic:
Pi.pow_def

Goal state after:
case a
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
𝕜' : Type u_2
inst✝¹ : NontriviallyNormedField 𝕜'
inst✝ : NormedAlgebra 𝕜 𝕜'
U : Set 𝕜
ι : Type u_3
F : ι → 𝕜 → 𝕜'
d : ι → ℤ
hd : (support d).Finite
h : ∀ (i : ι), MeromorphicOn (F i) U
h' : ∀ (i : ι), ∀ x ∈ U, meromorphicOrderAt (F i) x ≠ ⊤
hA : ∀ᶠ (y : 𝕜) in codiscreteWithin U, ∀ i ∈ hd.toFinset, AnalyticAt 𝕜 (F i) y
hN : ∀ᶠ (y : 𝕜) in codiscreteWithin U, ∀ i ∈ hd.toFinset, F i y ≠ 0
y : 𝕜
h₁y : ∀ i ∈ hd.toFinset, AnalyticAt 𝕜 (F i) y
h₂y : ∀ i ∈ hd.toFinset, F i y ≠ 0
h₀ : ∏ᶠ (i : ι), F i ^ d i = ∏ i ∈ hd.toFinset, F i ^ d i
hsub : (support fun i ↦ d i • logDeriv (F i) y) ⊆ ↑hd.toFinset
i : ι
hi : i ∈ hd.toFinset
⊢ logDeriv (fun i_1 ↦ F i i_1 ^ d i) y = ↑(d i) * logDeriv (F i) y
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Beta

Goal state before:
pp: s : ℂ
n : ℕ
hn : n ≠ 0
| ↑n * ↑n ^ s * ↑n ! / (((↑n + 1 + s) * ∏ k ∈ Finset.range n, (s + ↑(k + 1))) * s)

Tactic:
mul_comm _ (Finset.prod _ _)

Goal state after:
s : ℂ
n : ℕ
hn : n ≠ 0
| ↑n * ↑n ^ s * ↑n ! / ((∏ k ∈ Finset.range n, (s + ↑(k + 1))) * (↑n + 1 + s) * s)
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Beta

Goal state before:
pp: case succ.inl
m : ℕ
IH : ∀ (s : ℂ), ⌊1 - s.re⌋₊ = m → Tendsto s.GammaSeq atTop (𝓝 (Gamma s))
hs : ⌊1 - re 0⌋₊ = m + 1
⊢ Tendsto (GammaSeq 0) atTop (𝓝 (Gamma 0))

Tactic:
unfold GammaSeq

Goal state after:
case succ.inl
m : ℕ
IH : ∀ (s : ℂ), ⌊1 - s.re⌋₊ = m → Tendsto s.GammaSeq atTop (𝓝 (Gamma s))
hs : ⌊1 - re 0⌋₊ = m + 1
⊢ Tendsto (fun n ↦ ↑n ^ 0 * ↑n ! / ∏ j ∈ Finset.range (n + 1), (0 + ↑j)) atTop (𝓝 (Gamma 0))
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Beta

Goal state before:
pp: case pos
z : ℂ
pi_ne : ↑π ≠ 0
hs : ∃ k, -z * ↑π = ↑k * ↑π
⊢ Gamma z * Gamma (1 - z) = 0

Tactic:
obtain ⟨k, hk⟩ := hs

Goal state after:
case pos
z : ℂ
pi_ne : ↑π ≠ 0
k : ℤ
hk : -z * ↑π = ↑k * ↑π
⊢ Gamma z * Gamma (1 - z) = 0
Module: Mathlib.Analysis.Quaternion

Goal state before:
pp: a : ℍ
⊢ ⟪a, a⟫ = ‖a‖ * ‖a‖

Tactic:
real_inner_self_eq_norm_mul_norm

Goal state after:
a : ℍ
⊢ ‖a‖ * ‖a‖ = ‖a‖ * ‖a‖
Module: Mathlib.Analysis.Normed.Algebra.QuaternionExponential

Goal state before:
pp: q : ℍ
⊢ ‖exp q‖ = ‖exp q.re‖

Tactic:
norm_eq_sqrt_real_inner (exp q),

Goal state after:
q : ℍ
⊢ √(inner ℝ (exp q) (exp q)) = ‖exp q.re‖
Module: Mathlib.Analysis.Normed.Algebra.GelfandMazur

Goal state before:
pp: case refine_1
F : Type u_1
inst✝² : NormedRing F
inst✝¹ : NormedAlgebra ℝ F
inst✝ : NormOneClass F
x : F
c : ℝ
hc₀ : 0 < c
hbd : ∀ (r : ℝ), c ≤ ‖x - (algebraMap ℝ F) r‖
s : Set ℝ
hs : s ∈ cobounded ℝ
⊢ Tendsto (fun x_1 ↦ ‖x_1.1 • x - (algebraMap ℝ F) x_1.2‖) (𝓟 sᶜ ×ˢ cobounded ℝ) atTop

pp: case refine_2
F : Type u_1
inst✝² : NormedRing F
inst✝¹ : NormedAlgebra ℝ F
inst✝ : NormOneClass F
x : F
c : ℝ
hc₀ : 0 < c
hbd : ∀ (r : ℝ), c ≤ ‖x - (algebraMap ℝ F) r‖
⊢ Tendsto (fun x_1 ↦ ‖x_1.1 • x - (algebraMap ℝ F) x_1.2‖) (cobounded ℝ ×ˢ ⊤) atTop

Tactic:
· rw [← isCobounded_def, ← isBounded_compl_iff] at hs
    obtain ⟨M, hM_pos, hM⟩ : ∃ M > 0, ∀ y ∈ sᶜ, ‖y‖ ≤ M := hs.exists_pos_norm_le
    suffices Tendsto (‖algebraMap ℝ F ·.2‖ - M * ‖x‖) (𝓟 sᶜ ×ˢ cobounded ℝ) atTop by
      refine tendsto_atTop_mono' _ ?_ this
      filter_upwards [prod_mem_prod (mem_principal_self sᶜ) univ_mem] with w hw
      rw [norm_sub_rev]
      refine le_trans ?_ (norm_sub_norm_le ..)
      specialize hM _ (Set.mem_prod.mp hw).1
      simp only [norm_algebraMap', norm_smul]
      gcongr
    simp only [norm_algebraMap', sub_eq_add_neg]
    exact tendsto_atTop_add_const_right _ _ <| tendsto_norm_atTop_iff_cobounded.mpr tendsto_snd

Goal state after:
case refine_2
F : Type u_1
inst✝² : NormedRing F
inst✝¹ : NormedAlgebra ℝ F
inst✝ : NormOneClass F
x : F
c : ℝ
hc₀ : 0 < c
hbd : ∀ (r : ℝ), c ≤ ‖x - (algebraMap ℝ F) r‖
⊢ Tendsto (fun x_1 ↦ ‖x_1.1 • x - (algebraMap ℝ F) x_1.2‖) (cobounded ℝ ×ˢ ⊤) atTop
Module: Mathlib.Analysis.Normed.Field.Ultra

Goal state before:
pp: case hmn
R : Type u_1
inst✝ : NormedDivisionRing R
x : R
m : ℕ
h : ∀ (x : R) (n : ℕ), ‖n • x‖ ≤ ‖x‖
hm : max 1 (‖x‖ ^ m) = ‖x‖ ^ m
hx : 1 < ‖x‖ ^ m
i : ℕ
hi : i ∈ Finset.range (m + 1)
⊢ i ≤ m

Tactic:
simpa [Nat.lt_succ_iff] using hi

Goal state after:
Module: Mathlib.Analysis.Normed.Unbundled.RingSeminorm

Goal state before:
pp: case pos
R : Type u_1
inst✝¹ : NonUnitalRing R
inst✝ : DecidableEq R
x y : R
h : x * y = 0
⊢ AddGroupSeminorm.toFun 1 (x * y) ≤ AddGroupSeminorm.toFun 1 x * AddGroupSeminorm.toFun 1 y

Tactic:
refine (if_pos h).trans_le (mul_nonneg ?_ ?_)

Goal state after:
case pos.refine_1
R : Type u_1
inst✝¹ : NonUnitalRing R
inst✝ : DecidableEq R
x y : R
h : x * y = 0
⊢ 0 ≤ AddGroupSeminorm.toFun 1 x

case pos.refine_2
R : Type u_1
inst✝¹ : NonUnitalRing R
inst✝ : DecidableEq R
x y : R
h : x * y = 0
⊢ 0 ≤ AddGroupSeminorm.toFun 1 y
Module: Mathlib.Analysis.Normed.Unbundled.RingSeminorm

Goal state before:
pp: case inr
R : Type u_1
inst✝ : Ring R
p : RingSeminorm R
hp : p 1 ≤ 1
s : ℕ → ℕ
hs_le : ∀ (n : ℕ), s n ≤ n
x : R
ψ : ℕ → ℕ
h_le : ∀ (m : ℕ), p (x ^ s (ψ m)) ^ (1 / ↑(ψ m)) ≤ p x ^ (↑(s (ψ m)) / ↑(ψ m))
hfx : 1 < p x
⊢ ∃ b, ∀ (x_1 : ℕ), p (x ^ s (ψ x_1)) ^ (1 / ↑(ψ x_1)) ≤ b

Tactic:
| inr hfx =>

Goal state after:
Module: Mathlib.Analysis.Normed.Unbundled.SeminormFromBounded

Goal state before:
pp: case a
R : Type u_1
inst✝ : CommRing R
f : R → ℝ
f_nonneg : 0 ≤ f
x : R
hx : ∀ (y : R), f (x * y) ≤ f x * f y
h_one : f 1 ≤ 1
h_bdd : BddAbove (Set.range fun y ↦ f (x * y) / f y)
⊢ f x ≤ ⨆ y, f (x * y) / f y

Tactic:
convert! le_ciSup h_bdd (1 : R)

Goal state after:
R : Type u_1
inst✝ : CommRing R
f : R → ℝ
f_nonneg : 0 ≤ f
x : R
hx : ∀ (y : R), f (x * y) ≤ f x * f y
h_one : f 1 ≤ 1
h_bdd : BddAbove (Set.range fun y ↦ f (x * y) / f y)
⊢ f x = f (x * 1) / f 1
Module: Mathlib.Analysis.Normed.Unbundled.IsPowMulFaithful

Goal state before:
pp: F : Type u_1
α : Type u_2
inst✝² : Ring α
inst✝¹ : FunLike F α ℝ
inst✝ : RingSeminormClass F α ℝ
f g : F
hfpm : IsPowMul ⇑f
hgpm : IsPowMul ⇑g
r : ℝ
hr0 : 0 < r
hr : ∀ (a : α), f a ≤ r * g a
s : ℝ
hs0 : 0 < s
hs : ∀ (a : α), g a ≤ s * f a
hle : RingHom.IsBoundedWrt (⇑f) (⇑g) (RingHom.id α)
hge : RingHom.IsBoundedWrt (⇑g) (⇑f) (RingHom.id α)
⊢ f = g

Tactic:
← Function.Injective.eq_iff DFunLike.coe_injective

Goal state after:
F : Type u_1
α : Type u_2
inst✝² : Ring α
inst✝¹ : FunLike F α ℝ
inst✝ : RingSeminormClass F α ℝ
f g : F
hfpm : IsPowMul ⇑f
hgpm : IsPowMul ⇑g
r : ℝ
hr0 : 0 < r
hr : ∀ (a : α), f a ≤ r * g a
s : ℝ
hs0 : 0 < s
hs : ∀ (a : α), g a ≤ s * f a
hle : RingHom.IsBoundedWrt (⇑f) (⇑g) (RingHom.id α)
hge : RingHom.IsBoundedWrt (⇑g) (⇑f) (RingHom.id α)
⊢ ⇑f = ⇑g
Module: Mathlib.Analysis.Normed.Field.Krasner

Goal state before:
pp: case refine_1
K : Type u_1
L : Type u_2
inst✝⁵ : NormedField L
inst✝⁴ : NontriviallyNormedField K
inst✝³ : CompleteSpace K
inst✝² : IsUltrametricDist K
inst✝¹ : NormedAlgebra K L
inst✝ : Algebra.IsAlgebraic K L
x y : L
xsep : IsSeparable K x
sp : (Polynomial.map (algebraMap K L) (minpoly K x)).Splits
yint : IsIntegral K y
kr : ∀ (x' : L), IsConjRoot K x x' → x ≠ x' → ‖x - y‖ < ‖x - x'‖
C : Type u_1 := AlgebraicClosure K
this✝² : NontriviallyNormedField C := spectralNorm.nontriviallyNormedField K C
this✝¹ : NormedAlgebra K C := spectralNorm.normedAlgebra K C
iL : L →ₐ[K] C := IsAlgClosed.lift
algInst✝ : Algebra L C := iL.toAlgebra
this✝ : NormedAlgebra L C := spectralNorm.normedAlgebra' K L C
this : IsKrasner K C := of_completeSpace_of_normal K C
norm_iL : ∀ (x : L), ‖iL x‖ = ‖x‖
⊢ (Polynomial.map (algebraMap K C) (minpoly K (iL x))).Splits

pp: case refine_2
K : Type u_1
L : Type u_2
inst✝⁵ : NormedField L
inst✝⁴ : NontriviallyNormedField K
inst✝³ : CompleteSpace K
inst✝² : IsUltrametricDist K
inst✝¹ : NormedAlgebra K L
inst✝ : Algebra.IsAlgebraic K L
x y : L
xsep : IsSeparable K x
sp : (Polynomial.map (algebraMap K L) (minpoly K x)).Splits
yint : IsIntegral K y
kr : ∀ (x' : L), IsConjRoot K x x' → x ≠ x' → ‖x - y‖ < ‖x - x'‖
C : Type u_1 := AlgebraicClosure K
this✝² : NontriviallyNormedField C := spectralNorm.nontriviallyNormedField K C
this✝¹ : NormedAlgebra K C := spectralNorm.normedAlgebra K C
iL : L →ₐ[K] C := IsAlgClosed.lift
algInst✝ : Algebra L C := iL.toAlgebra
this✝ : NormedAlgebra L C := spectralNorm.normedAlgebra' K L C
this : IsKrasner K C := of_completeSpace_of_normal K C
norm_iL : ∀ (x : L), ‖iL x‖ = ‖x‖
⊢ ∀ (x' : C), IsConjRoot K (iL x) x' → iL x ≠ x' → ‖iL x - iL y‖ < ‖iL x - x'‖

Tactic:
· exact Polynomial.Splits.of_dvd (sp.of_isScalarTower C)
        (Polynomial.map_ne_zero (minpoly.ne_zero xsep.isIntegral))
        (Polynomial.map_dvd _ (by simpa using! minpoly.dvd_map_of_isScalarTower' K K C x))

Goal state after:
case refine_2
K : Type u_1
L : Type u_2
inst✝⁵ : NormedField L
inst✝⁴ : NontriviallyNormedField K
inst✝³ : CompleteSpace K
inst✝² : IsUltrametricDist K
inst✝¹ : NormedAlgebra K L
inst✝ : Algebra.IsAlgebraic K L
x y : L
xsep : IsSeparable K x
sp : (Polynomial.map (algebraMap K L) (minpoly K x)).Splits
yint : IsIntegral K y
kr : ∀ (x' : L), IsConjRoot K x x' → x ≠ x' → ‖x - y‖ < ‖x - x'‖
C : Type u_1 := AlgebraicClosure K
this✝² : NontriviallyNormedField C := spectralNorm.nontriviallyNormedField K C
this✝¹ : NormedAlgebra K C := spectralNorm.normedAlgebra K C
iL : L →ₐ[K] C := IsAlgClosed.lift
algInst✝ : Algebra L C := iL.toAlgebra
this✝ : NormedAlgebra L C := spectralNorm.normedAlgebra' K L C
this : IsKrasner K C := of_completeSpace_of_normal K C
norm_iL : ∀ (x : L), ‖iL x‖ = ‖x‖
⊢ ∀ (x' : C), IsConjRoot K (iL x) x' → iL x ≠ x' → ‖iL x - iL y‖ < ‖iL x - x'‖
Module: Mathlib.Analysis.Normed.Group.Completeness

Goal state before:
pp: E : Type u_1
inst✝ : NormedAddCommGroup E
h : ∀ (u : ℕ → E), (Summable fun x ↦ ‖u x‖) → ∃ a, Tendsto (fun n ↦ ∑ i ∈ range n, u i) atTop (𝓝 a)
u : ℕ → E
hu : CauchySeq u
f : ℕ → ℕ
hf₁ : StrictMono f
hf₂ : Summable fun i ↦ ‖u (f (i + 1)) - u (f i)‖
v : ℕ → E := fun n ↦ u (f (n + 1)) - u (f n)
⊢ ∃ a, Tendsto u atTop (𝓝 a)

Tactic:
have hv_sum : (fun n => (∑ i ∈ range n, v i)) = fun n => u (f n) - u (f 0) := by
    ext n
    exact sum_range_sub (u ∘ f) n

Goal state after:
E : Type u_1
inst✝ : NormedAddCommGroup E
h : ∀ (u : ℕ → E), (Summable fun x ↦ ‖u x‖) → ∃ a, Tendsto (fun n ↦ ∑ i ∈ range n, u i) atTop (𝓝 a)
u : ℕ → E
hu : CauchySeq u
f : ℕ → ℕ
hf₁ : StrictMono f
hf₂ : Summable fun i ↦ ‖u (f (i + 1)) - u (f i)‖
v : ℕ → E := fun n ↦ u (f (n + 1)) - u (f n)
hv_sum : (fun n ↦ ∑ i ∈ range n, v i) = fun n ↦ u (f n) - u (f 0)
⊢ ∃ a, Tendsto u atTop (𝓝 a)
Module: Mathlib.Analysis.Normed.Unbundled.SpectralNorm

Goal state before:
pp: K : Type u_2
inst✝² : NormedField K
L : Type u_3
inst✝¹ : Field L
inst✝ : Algebra K L
f : AlgebraNorm K L
hf_pm : IsPowMul ⇑f
hf_na : IsNonarchimedean ⇑f
p : K[X]
hp : p.Monic
x : L
hx : (aeval x) p = 0
hx0 : ¬f x = 0
h_ge : ¬f x ≤ spectralValue p
n : ℕ
hn : n < p.natDegree
hexp : (‖p.coeff n‖ ^ (1 / (↑p.natDegree - ↑n))) ^ (p.natDegree - n) = ‖p.coeff n‖
⊢ ‖p.coeff n‖ < f x ^ (p.natDegree - n)

Tactic:
have h_base : ‖p.coeff n‖ ^ (1 / (p.natDegree - n : ℝ)) < f x := by
        rw [spectralValue, iSup, not_le, Set.Finite.csSup_lt_iff (spectralValueTerms_finite_range p)
          (Set.range_nonempty (spectralValueTerms p))] at h_ge
        have h_rg : ‖p.coeff n‖ ^ (1 / (p.natDegree - n : ℝ)) ∈
          Set.range (spectralValueTerms p) := by use n; simp only [spectralValueTerms, if_pos hn]
        exact h_ge (‖p.coeff n‖₊ ^ (1 / (p.natDegree - n : ℝ))) h_rg

Goal state after:
K : Type u_2
inst✝² : NormedField K
L : Type u_3
inst✝¹ : Field L
inst✝ : Algebra K L
f : AlgebraNorm K L
hf_pm : IsPowMul ⇑f
hf_na : IsNonarchimedean ⇑f
p : K[X]
hp : p.Monic
x : L
hx : (aeval x) p = 0
hx0 : ¬f x = 0
h_ge : ¬f x ≤ spectralValue p
n : ℕ
hn : n < p.natDegree
hexp : (‖p.coeff n‖ ^ (1 / (↑p.natDegree - ↑n))) ^ (p.natDegree - n) = ‖p.coeff n‖
h_base : ‖p.coeff n‖ ^ (1 / (↑p.natDegree - ↑n)) < f x
⊢ ‖p.coeff n‖ < f x ^ (p.natDegree - n)
Module: Mathlib.Analysis.Normed.Unbundled.SpectralNorm

Goal state before:
pp: case pos
K : Type u_2
inst✝² : NormedField K
L : Type u_3
inst✝¹ : Field L
inst✝ : Algebra K L
f : AlgebraNorm K L
hf_pm : IsPowMul ⇑f
hf_na : IsNonarchimedean ⇑f
p : K[X]
hp : p.Monic
x : L
hx : (aeval x) p = 0
hx0 : ¬f x = 0
h_ge : ¬f x ≤ spectralValue p
hn_lt : ∀ n < p.natDegree, ‖p.coeff n‖ < f x ^ (p.natDegree - n)
h_deg : 0 < p.natDegree
n : ℕ
hn : n < p.natDegree
hn0 : n = 0
⊢ ‖p.coeff 0‖ * f 1 < f x ^ (p.natDegree - 0)

Tactic:
← hn0

Goal state after:
case pos
K : Type u_2
inst✝² : NormedField K
L : Type u_3
inst✝¹ : Field L
inst✝ : Algebra K L
f : AlgebraNorm K L
hf_pm : IsPowMul ⇑f
hf_na : IsNonarchimedean ⇑f
p : K[X]
hp : p.Monic
x : L
hx : (aeval x) p = 0
hx0 : ¬f x = 0
h_ge : ¬f x ≤ spectralValue p
hn_lt : ∀ n < p.natDegree, ‖p.coeff n‖ < f x ^ (p.natDegree - n)
h_deg : 0 < p.natDegree
n : ℕ
hn : n < p.natDegree
hn0 : n = 0
⊢ ‖p.coeff n‖ * f 1 < f x ^ (p.natDegree - n)
Module: Mathlib.Analysis.Normed.Unbundled.SpectralNorm

Goal state before:
pp: K : Type u_2
inst✝² : NormedField K
L : Type u_3
inst✝¹ : Field L
inst✝ : Algebra K L
f : AlgebraNorm K L
hf_pm : IsPowMul ⇑f
hf_na : IsNonarchimedean ⇑f
p : K[X]
hp : p.Monic
x : L
hx : (aeval x) p = 0
hx0 : ¬f x = 0
h_ge : ¬f x ≤ spectralValue p
hn_lt : ∀ n < p.natDegree, ‖p.coeff n‖ < f x ^ (p.natDegree - n)
h_deg : 0 < p.natDegree
hn' : ∀ n < p.natDegree, f (p.coeff n • x ^ n) < f (x ^ p.natDegree)
g : ℕ → L := fun i ↦ p.coeff i • x ^ i
m : ℕ
hm : (Finset.range p.natDegree).Nonempty → m ∈ Finset.range p.natDegree
h : f ((Finset.range p.natDegree).sum g) ≤ f (g m)
⊢ ∃ m, ∃ (_ : 0 < p.natDegree → m < p.natDegree), f ((Finset.range p.natDegree).sum g) ≤ f (g m)

Tactic:
rw [Finset.nonempty_range_iff, ← zero_lt_iff, Finset.mem_range] at hm

Goal state after:
K : Type u_2
inst✝² : NormedField K
L : Type u_3
inst✝¹ : Field L
inst✝ : Algebra K L
f : AlgebraNorm K L
hf_pm : IsPowMul ⇑f
hf_na : IsNonarchimedean ⇑f
p : K[X]
hp : p.Monic
x : L
hx : (aeval x) p = 0
hx0 : ¬f x = 0
h_ge : ¬f x ≤ spectralValue p
hn_lt : ∀ n < p.natDegree, ‖p.coeff n‖ < f x ^ (p.natDegree - n)
h_deg : 0 < p.natDegree
hn' : ∀ n < p.natDegree, f (p.coeff n • x ^ n) < f (x ^ p.natDegree)
g : ℕ → L := fun i ↦ p.coeff i • x ^ i
m : ℕ
hm : 0 < p.natDegree → m < p.natDegree
h : f ((Finset.range p.natDegree).sum g) ≤ f (g m)
⊢ ∃ m, ∃ (_ : 0 < p.natDegree → m < p.natDegree), f ((Finset.range p.natDegree).sum g) ≤ f (g m)
Module: Mathlib.Analysis.Normed.Unbundled.SpectralNorm

Goal state before:
pp: K : Type u_2
inst✝² : NormedField K
L : Type u_3
inst✝¹ : Field L
inst✝ : Algebra K L
f : AlgebraNorm K L
hf_pm : IsPowMul ⇑f
hf_na : IsNonarchimedean ⇑f
p : K[X]
hp : p.Monic
x : L
hx : (aeval x) p = 0
hx0 : ¬f x = 0
h_ge : ¬f x ≤ spectralValue p
hn_lt : ∀ n < p.natDegree, ‖p.coeff n‖ < f x ^ (p.natDegree - n)
h_deg : 0 < p.natDegree
h_lt : f (∑ i ∈ Finset.range p.natDegree, p.coeff i • x ^ i) < f (x ^ p.natDegree)
⊢ f (p.coeff p.natDegree • x ^ p.natDegree + ∑ x_1 ∈ Finset.range p.natDegree, p.coeff x_1 • x ^ x_1) =
    f (x ^ p.natDegree)

Tactic:
hp.coeff_natDegree,

Goal state after:
K : Type u_2
inst✝² : NormedField K
L : Type u_3
inst✝¹ : Field L
inst✝ : Algebra K L
f : AlgebraNorm K L
hf_pm : IsPowMul ⇑f
hf_na : IsNonarchimedean ⇑f
p : K[X]
hp : p.Monic
x : L
hx : (aeval x) p = 0
hx0 : ¬f x = 0
h_ge : ¬f x ≤ spectralValue p
hn_lt : ∀ n < p.natDegree, ‖p.coeff n‖ < f x ^ (p.natDegree - n)
h_deg : 0 < p.natDegree
h_lt : f (∑ i ∈ Finset.range p.natDegree, p.coeff i • x ^ i) < f (x ^ p.natDegree)
⊢ f (1 • x ^ p.natDegree + ∑ x_1 ∈ Finset.range p.natDegree, p.coeff x_1 • x ^ x_1) = f (x ^ p.natDegree)
Module: Mathlib.Analysis.Normed.Group.ControlledClosure

Goal state before:
pp: G : Type u_1
inst✝² : NormedAddCommGroup G
inst✝¹ : CompleteSpace G
H : Type u_2
inst✝ : NormedAddCommGroup H
f : NormedAddGroupHom G H
K : AddSubgroup H
C ε : ℝ
hC : 0 < C
hε : 0 < ε
hyp : f.SurjectiveOnWith K C
h : H
h_in : h ∈ K.topologicalClosure
hyp_h : ¬h = 0
b : ℕ → ℝ := fun i ↦ (1 / 2) ^ i * (ε * ‖h‖ / 2) / C
b_pos : ∀ (i : ℕ), 0 < b i
v : ℕ → H
lim_v : Tendsto (fun n ↦ ∑ k ∈ range (n + 1), v k) atTop (𝓝 h)
v_in : ∀ (n : ℕ), v n ∈ K
hv₀ : ‖-v 0 + h‖ < b 0
hv : ∀ n > 0, ‖v n‖ < b n
u : ℕ → G
hu : ∀ (n : ℕ), f (u n) = v n
hnorm_u : ∀ (n : ℕ), ‖u n‖ ≤ C * ‖v n‖
s : ℕ → G := fun n ↦ ∑ k ∈ range (n + 1), u k
n : ℕ
hn : n ≥ 1
⊢ C * b n = (1 / 2) ^ n * (ε * ‖h‖ / 2)

Tactic:
by simp [b, mul_div_cancel₀ _ hC.ne.symm]

Goal state after:
Module: Mathlib.Analysis.Normed.Group.SemiNormedGrp.Kernels

Goal state before:
pp: X Y : SemiNormedGrp₁
f : X ⟶ Y
x : X.carrier
⊢ ↑(Hom.hom (f ≫ mkHom (↑(Hom.hom f)).range.normedMk ⋯)) x = ↑(Hom.hom (0 ≫ mkHom (↑(Hom.hom f)).range.normedMk ⋯)) x

Tactic:
Limits.zero_comp,

Goal state after:
X Y : SemiNormedGrp₁
f : X ⟶ Y
x : X.carrier
⊢ ↑(Hom.hom (f ≫ mkHom (↑(Hom.hom f)).range.normedMk ⋯)) x = ↑(Hom.hom 0) x
Module: Mathlib.Analysis.Normed.Group.SemiNormedGrp.Kernels

Goal state before:
pp: X Y : SemiNormedGrp
f : X ⟶ Y
⊢ explicitCokernelπ f ≫ (explicitCokernelIso f).hom = cokernel.π f

Tactic:
simp [explicitCokernelπ, explicitCokernelIso, IsColimit.coconePointUniqueUpToIso]

Goal state after:
Module: Mathlib.Analysis.Normed.Unbundled.SpectralNorm

Goal state before:
pp: K : Type u
inst✝⁴ : NontriviallyNormedField K
L : Type v
inst✝³ : Field L
inst✝² : Algebra K L
inst✝¹ : Algebra.IsAlgebraic K L
hu : IsUltrametricDist K
inst✝ : CompleteSpace K
f : AbsoluteValue L ℝ
hf_ext : ∀ (x : K), f ((algebraMap K L) x) = ‖x‖
x : L
g : AlgebraNorm K L :=
  let __src := MulRingNorm.mulRingNormEquivAbsoluteValue.invFun f;
  { toAddGroupSeminorm := __src.toAddGroupSeminorm, mul_le' := ⋯, eq_zero_of_map_eq_zero' := ⋯, smul' := ⋯ }
hg_pow : IsPowMul ⇑g
⊢ f x = spectralNorm K L x

Tactic:
have hgx : f x = g x := rfl

Goal state after:
K : Type u
inst✝⁴ : NontriviallyNormedField K
L : Type v
inst✝³ : Field L
inst✝² : Algebra K L
inst✝¹ : Algebra.IsAlgebraic K L
hu : IsUltrametricDist K
inst✝ : CompleteSpace K
f : AbsoluteValue L ℝ
hf_ext : ∀ (x : K), f ((algebraMap K L) x) = ‖x‖
x : L
g : AlgebraNorm K L :=
  let __src := MulRingNorm.mulRingNormEquivAbsoluteValue.invFun f;
  { toAddGroupSeminorm := __src.toAddGroupSeminorm, mul_le' := ⋯, eq_zero_of_map_eq_zero' := ⋯, smul' := ⋯ }
hg_pow : IsPowMul ⇑g
hgx : f x = g x
⊢ f x = spectralNorm K L x
Module: Mathlib.Analysis.Normed.Module.ContinuousInverse

Goal state before:
pp: R : Type u_7
E : Type u_8
F : Type u_10
inst✝⁶ : Ring R
inst✝⁵ : TopologicalSpace E
inst✝⁴ : AddCommGroup E
inst✝³ : Module R E
inst✝² : TopologicalSpace F
inst✝¹ : AddCommGroup F
inst✝ : Module R F
f : E →L[R] F
hf : f.HasLeftInverse
⊢ (↑f).range.ClosedComplemented

Tactic:
use (f.comp hf.leftInverse).codRestrict f.range (by intro y; simp)

Goal state after:
case h
R : Type u_7
E : Type u_8
F : Type u_10
inst✝⁶ : Ring R
inst✝⁵ : TopologicalSpace E
inst✝⁴ : AddCommGroup E
inst✝³ : Module R E
inst✝² : TopologicalSpace F
inst✝¹ : AddCommGroup F
inst✝ : Module R F
f : E →L[R] F
hf : f.HasLeftInverse
⊢ ∀ (x : ↥(↑f).range), ((f ∘SL hf.leftInverse).codRestrict (↑f).range ⋯) ↑x = x
Module: Mathlib.Analysis.SpecialFunctions.Log.Summable

Goal state before:
pp: ι : Type u_1
R : Type u_2
inst✝³ : NormedCommRing R
inst✝² : NormOneClass R
f : ι → R
inst✝¹ : CompleteSpace R
inst✝ : NormMulClass R
hf : ∀ (i : ι), 1 + f i ≠ 0
hu : Summable fun x ↦ ‖f x‖
⊢ Real.exp (∑' (i : ι), Real.log ‖1 + f i‖) ≠ 0

Tactic:
apply Real.exp_ne_zero

Goal state after:
Module: Mathlib.Analysis.Normed.Module.Ball.RadialEquiv

Goal state before:
pp: case h
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
r : ℝ
hr : r ≠ 0
V : Set ↑(sphere 0 r)
hV : IsOpen[instTopologicalSpaceSubtype] V
y : { x // x ∈ sphere 0 r }
hyV : y ∈ V
U : Set ℝ
hU : IsOpen U
hU₀ : 0 ∉ U
hr₀ : 0 < r
x : { x // x ∈ Ioi 0 }
hxU : ↑x ∈ U
this :
  Subtype.val ∘ ⇑(homeomorphSphereProd E r hr₀).symm '' V ×ˢ Ioi 0 ↓∩ U ∈
    𝓝[{0}ᶜ] ↑((homeomorphSphereProd E r hr₀).symm (y, x))
⊢ IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] {0}ᶜ

pp: case ha
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
r : ℝ
hr : r ≠ 0
V : Set ↑(sphere 0 r)
hV : IsOpen[instTopologicalSpaceSubtype] V
y : { x // x ∈ sphere 0 r }
hyV : y ∈ V
U : Set ℝ
hU : IsOpen U
hU₀ : 0 ∉ U
hr₀ : 0 < r
x : { x // x ∈ Ioi 0 }
hxU : ↑x ∈ U
this :
  Subtype.val ∘ ⇑(homeomorphSphereProd E r hr₀).symm '' V ×ˢ Ioi 0 ↓∩ U ∈
    𝓝[{0}ᶜ] ↑((homeomorphSphereProd E r hr₀).symm (y, x))
⊢ ↑((homeomorphSphereProd E r hr₀).symm (y, x)) ∈ {0}ᶜ

Tactic:
· exact isOpen_compl_singleton

Goal state after:
case ha
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
r : ℝ
hr : r ≠ 0
V : Set ↑(sphere 0 r)
hV : IsOpen[instTopologicalSpaceSubtype] V
y : { x // x ∈ sphere 0 r }
hyV : y ∈ V
U : Set ℝ
hU : IsOpen U
hU₀ : 0 ∉ U
hr₀ : 0 < r
x : { x // x ∈ Ioi 0 }
hxU : ↑x ∈ U
this :
  Subtype.val ∘ ⇑(homeomorphSphereProd E r hr₀).symm '' V ×ˢ Ioi 0 ↓∩ U ∈
    𝓝[{0}ᶜ] ↑((homeomorphSphereProd E r hr₀).symm (y, x))
⊢ ↑((homeomorphSphereProd E r hr₀).symm (y, x)) ∈ {0}ᶜ
Module: Mathlib.Analysis.Normed.Module.MStructure

Goal state before:
pp: X : Type u_1
inst✝³ : NormedAddCommGroup X
M : Type u_2
inst✝² : Ring M
inst✝¹ : Module M X
inst✝ : FaithfulSMul M X
P : { P // IsLprojection X P }
⊢ ↑⊤ = ↑(P ⊔ Pᶜ)

Tactic:
rw [coe_top, coe_sup, coe_compl, add_sub_cancel, ← coe_compl, mul_compl_self,
            sub_zero]

Goal state after: