mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Normed.Affine.AsymptoticCone

Goal state before:
pp: V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : NormedSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
v : V
hv : v ≠ 0
p : P
⊢ Tendsto (fun x ↦ ‖x.1‖ * ‖x.2‖) (atTop ×ˢ 𝓝 v) atTop

Tactic:
exact Tendsto.atTop_mul_pos (norm_pos_iff.mpr hv)
    (tendsto_norm_atTop_atTop.comp tendsto_id.fst)
    tendsto_snd.norm

Goal state after:
Module: Mathlib.Topology.Algebra.AsymptoticCone

Goal state before:
pp: k : Type u_1
V : Type u_2
inst✝⁹ : Field k
inst✝⁸ : LinearOrder k
inst✝⁷ : IsStrictOrderedRing k
inst✝⁶ : TopologicalSpace k
inst✝⁵ : OrderTopology k
inst✝⁴ : AddCommGroup V
inst✝³ : Module k V
inst✝² : TopologicalSpace V
inst✝¹ : IsTopologicalAddGroup V
inst✝ : ContinuousSMul k V
s✝ s : Set V
hs : Convex k s
hs' : IsClosed[inst✝²] s
p : V
hp : p ∈ s
v : V
hv : v ∈ asymptoticCone k s
u : V
hu : u ∈ asymptoticCone k s
a b : k
ha : 0 ≤ a
hb : 0 ≤ b
hab : a + b = 1
c : k
hc : 0 ≤ c
⊢ id c • (a • v + b • u) +ᵥ p ∈ s

Tactic:
simp_rw [id, smul_add, smul_smul]

Goal state after:
k : Type u_1
V : Type u_2
inst✝⁹ : Field k
inst✝⁸ : LinearOrder k
inst✝⁷ : IsStrictOrderedRing k
inst✝⁶ : TopologicalSpace k
inst✝⁵ : OrderTopology k
inst✝⁴ : AddCommGroup V
inst✝³ : Module k V
inst✝² : TopologicalSpace V
inst✝¹ : IsTopologicalAddGroup V
inst✝ : ContinuousSMul k V
s✝ s : Set V
hs : Convex k s
hs' : IsClosed[inst✝²] s
p : V
hp : p ∈ s
v : V
hv : v ∈ asymptoticCone k s
u : V
hu : u ∈ asymptoticCone k s
a b : k
ha : 0 ≤ a
hb : 0 ≤ b
hab : a + b = 1
c : k
hc : 0 ≤ c
⊢ ((c * a) • v + (c * b) • u) +ᵥ p ∈ s
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Beta

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

Tactic:
rcases eq_or_ne s 0 with rfl | hsne

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 (GammaSeq 0) atTop (𝓝 (Gamma 0))

case succ.inr
m : ℕ
IH : ∀ (s : ℂ), ⌊1 - s.re⌋₊ = m → Tendsto s.GammaSeq atTop (𝓝 (Gamma s))
s : ℂ
hs : ⌊1 - s.re⌋₊ = m + 1
hsne : s ≠ 0
⊢ Tendsto s.GammaSeq atTop (𝓝 (Gamma s))
Module: Mathlib.Analysis.Normed.Affine.MazurUlam

Goal state before:
pp: E : Type u_1
PE : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : MetricSpace PE
inst✝ : NormedAddTorsor E PE
x y : PE
⊢ ∀ (e : PE ≃ᵢ PE), e x = x → e y = y → e (midpoint ℝ x y) = midpoint ℝ x y

Tactic:
set z := midpoint ℝ x y

Goal state after:
E : Type u_1
PE : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : MetricSpace PE
inst✝ : NormedAddTorsor E PE
x y : PE
z : PE := midpoint ℝ x y
⊢ ∀ (e : PE ≃ᵢ PE), e x = x → e y = y → e z = z
Module: Mathlib.Analysis.Normed.Affine.Simplex

Goal state before:
pp: R : Type u_1
V : Type u_2
P : Type u_3
inst✝⁴ : Ring R
inst✝³ : SeminormedAddCommGroup V
inst✝² : PseudoMetricSpace P
inst✝¹ : Module R V
inst✝ : NormedAddTorsor V P
m n : ℕ
s : Simplex R P m
e : Fin (m + 1) ≃ Fin (n + 1)
⊢ (s.reindex e).Regular ↔ s.Regular

Tactic:
by
  refine ⟨fun h σ ↦ ?_, fun h σ ↦ ?_⟩
  · rcases h ((e.symm.trans σ).trans e) with ⟨x, hx⟩
    refine ⟨x, ?_⟩
    ext i
    simpa using congrFun hx (e i)
  · rcases h ((e.trans σ).trans e.symm) with ⟨x, hx⟩
    refine ⟨x, ?_⟩
    ext i
    simpa using congrFun hx (e.symm i)

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

Goal state before:
pp: k : Type u_1
V : Type u_2
P : Type u_3
inst✝⁴ : CommRing k
inst✝³ : NoZeroDivisors k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
t : Triangle k P
r : Fin 3 → k
h✝ : Nontrivial k
w : ↑Set.univ → Fin 3 → k := fun i ↦ Finset.affineCombinationLineMapWeights (↑i + 1) (↑i + 2) (r ↑i)
hw : ∀ (i : ↑Set.univ), ∑ j, w i j = 1
w' : Fin (2 + 1) → k
hw' : ∑ j, w' j = 1
h : ∀ (i : ↑Set.univ), ∃ r, ∀ (j : Fin (2 + 1)), r * {↑i}ᶜ.indicator (w i) j = {↑i}ᶜ.indicator w' j
hp' :
  ∀ (i : Fin 3),
    (Finset.affineCombination k Finset.univ t.points) w' ∈
      line[k, t.points i, (AffineMap.lineMap (t.points (i + 1)) (t.points (i + 2))) (r i)]
hp'w :
  ∀ (i : ↑Set.univ),
    (Finset.affineCombination k Finset.univ t.points) w' ∈
      line[k, t.points ↑i, (Finset.affineCombination k Finset.univ t.points) (w i)]
h' :
  ∀ (i : Fin 3), ∃ c, ∀ (j : Fin 3), j ≠ i → c * Finset.affineCombinationLineMapWeights (i + 1) (i + 2) (r i) j = w' j
c : Fin 3 → k := fun i ↦ ⋯.choose
hc : ∀ (i j : Fin 3), j ≠ i → c i * Finset.affineCombinationLineMapWeights (i + 1) (i + 2) (r i) j = w' j
hc1 : ∀ (i : Fin 3), c i * (1 - r i) = w' (i + 1)
i : Fin 3
⊢ c i * r i = w' (i + 2)

Tactic:
← hc i (i + 2) (by simp)

Goal state after:
k : Type u_1
V : Type u_2
P : Type u_3
inst✝⁴ : CommRing k
inst✝³ : NoZeroDivisors k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
t : Triangle k P
r : Fin 3 → k
h✝ : Nontrivial k
w : ↑Set.univ → Fin 3 → k := fun i ↦ Finset.affineCombinationLineMapWeights (↑i + 1) (↑i + 2) (r ↑i)
hw : ∀ (i : ↑Set.univ), ∑ j, w i j = 1
w' : Fin (2 + 1) → k
hw' : ∑ j, w' j = 1
h : ∀ (i : ↑Set.univ), ∃ r, ∀ (j : Fin (2 + 1)), r * {↑i}ᶜ.indicator (w i) j = {↑i}ᶜ.indicator w' j
hp' :
  ∀ (i : Fin 3),
    (Finset.affineCombination k Finset.univ t.points) w' ∈
      line[k, t.points i, (AffineMap.lineMap (t.points (i + 1)) (t.points (i + 2))) (r i)]
hp'w :
  ∀ (i : ↑Set.univ),
    (Finset.affineCombination k Finset.univ t.points) w' ∈
      line[k, t.points ↑i, (Finset.affineCombination k Finset.univ t.points) (w i)]
h' :
  ∀ (i : Fin 3), ∃ c, ∀ (j : Fin 3), j ≠ i → c * Finset.affineCombinationLineMapWeights (i + 1) (i + 2) (r i) j = w' j
c : Fin 3 → k := fun i ↦ ⋯.choose
hc : ∀ (i j : Fin 3), j ≠ i → c i * Finset.affineCombinationLineMapWeights (i + 1) (i + 2) (r i) j = w' j
hc1 : ∀ (i : Fin 3), c i * (1 - r i) = w' (i + 1)
i : Fin 3
⊢ c i * r i = c i * Finset.affineCombinationLineMapWeights (i + 1) (i + 2) (r i) (i + 2)
Module: Mathlib.Analysis.Quaternion

Goal state before:
pp: a : ℝ
⊢ √(a ^ 2) = ‖a‖

Tactic:
Real.sqrt_sq_eq_abs,

Goal state after:
a : ℝ
⊢ |a| = ‖a‖
Module: Mathlib.Analysis.Normed.Algebra.GelfandMazur

Goal state before:
pp: F : Type u_1
inst✝⁴ : NormedRing F
inst✝³ : NormOneClass F
inst✝² : NormMulClass F
inst✝¹ : NormedAlgebra ℂ F
inst✝ : Nontrivial F
x : F
z : ℂ
hz : (algebraMap ℂ F) z = x
⊢ (ofId ℂ F) z = x

Tactic:
← ofId_apply

Goal state after:
F : Type u_1
inst✝⁴ : NormedRing F
inst✝³ : NormOneClass F
inst✝² : NormMulClass F
inst✝¹ : NormedAlgebra ℂ F
inst✝ : Nontrivial F
x : F
z : ℂ
hz : (ofId ℂ F) z = x
⊢ (ofId ℂ F) z = x
Module: Mathlib.Analysis.Normed.Algebra.QuaternionExponential

Goal state before:
pp: q : ℍ
⊢ √(exp q.re ^ 2) = ‖exp q.re‖

Tactic:
Real.sqrt_sq_eq_abs,

Goal state after:
q : ℍ
⊢ |exp q.re| = ‖exp q.re‖
Module: Mathlib.Analysis.Normed.Unbundled.SmoothingSeminorm

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
μ : RingSeminorm R
hμ1 : μ 1 ≤ 1
x : R
hx : μ x ≠ 0
L : ℝ := ⨅ n, μ (x ^ ↑n) ^ (1 / ↑↑n)
hL0 : 0 ≤ L
ε : ℝ
⊢ ε > 0 → ∃ N, ∀ n ≥ N, dist (smoothingSeminormSeq μ x n) (smoothingFun μ x) < ε

Tactic:
hε

Goal state after:
R : Type u_1
inst✝ : CommRing R
μ : RingSeminorm R
hμ1 : μ 1 ≤ 1
x : R
hx : μ x ≠ 0
L : ℝ := ⨅ n, μ (x ^ ↑n) ^ (1 / ↑↑n)
hL0 : 0 ≤ L
ε : ℝ
hε : ε > 0
⊢ ∃ N, ∀ n ≥ N, dist (smoothingSeminormSeq μ x n) (smoothingFun μ x) < ε
Module: Mathlib.Analysis.Normed.Unbundled.InvariantExtension

Goal state before:
pp: K : Type u_1
inst✝² : NormedField K
L : Type u_2
inst✝¹ : Field L
inst✝ : Algebra K L
h_fin : FiniteDimensional K L
hu : IsUltrametricDist K
σ : L ≃ₐ[K] L
x : L
⊢ ∀ {n : ℕ}, 1 ≤ n → (algNormOfAlgEquiv σ) (x ^ n) = (algNormOfAlgEquiv σ) x ^ n

Tactic:
n

Goal state after:
K : Type u_1
inst✝² : NormedField K
L : Type u_2
inst✝¹ : Field L
inst✝ : Algebra K L
h_fin : FiniteDimensional K L
hu : IsUltrametricDist K
σ : L ≃ₐ[K] L
x : L
n : ℕ
⊢ 1 ≤ n → (algNormOfAlgEquiv σ) (x ^ n) = (algNormOfAlgEquiv σ) x ^ n
Module: Mathlib.Analysis.Normed.Unbundled.InvariantExtension

Goal state before:
pp: K : Type u_1
inst✝² : NormedField K
L : Type u_2
inst✝¹ : Field L
inst✝ : Algebra K L
h_fin : FiniteDimensional K L
hu : IsUltrametricDist K
x : L
⊢ ∀ {n : ℕ}, 1 ≤ n → (invariantExtension K L) (x ^ n) = (invariantExtension K L) x ^ n

Tactic:
n

Goal state after:
K : Type u_1
inst✝² : NormedField K
L : Type u_2
inst✝¹ : Field L
inst✝ : Algebra K L
h_fin : FiniteDimensional K L
hu : IsUltrametricDist K
x : L
n : ℕ
⊢ 1 ≤ n → (invariantExtension K L) (x ^ n) = (invariantExtension K L) x ^ n
Module: Mathlib.Analysis.Normed.Unbundled.SeminormFromConst

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
c : R
f : RingSeminorm R
hf1 : f 1 ≤ 1
hc : f c ≠ 0
hpm : IsPowMul ⇑f
x : R
m : ℕ
⊢ ∀ ⦃b : ℕ⦄, m ≤ b → seminormFromConst_seq c f x b ≤ seminormFromConst_seq c f x m

Tactic:
n

Goal state after:
R : Type u_1
inst✝ : CommRing R
c : R
f : RingSeminorm R
hf1 : f 1 ≤ 1
hc : f c ≠ 0
hpm : IsPowMul ⇑f
x : R
m n : ℕ
⊢ m ≤ n → seminormFromConst_seq c f x n ≤ seminormFromConst_seq c f x m
Module: Mathlib.Analysis.Normed.Field.Dense

Goal state before:
pp: case pos
K : Type u_1
L : Type u_2
inst✝⁶ : Field K
inst✝⁵ : NontriviallyNormedField L
inst✝⁴ : CompleteSpace L
inst✝³ : CharZero L
inst✝² : IsUltrametricDist L
inst✝¹ : Algebra K L
hi : DenseRange ⇑(algebraMap K L)
inst✝ : IsAlgClosed K
f : L[X]
fmon : f.Monic
firr : Irreducible f
fnatdeg0 : f.natDegree ≠ 0
F : Type u_2 := ⋯
this✝ : NormedField F := ⋯
this : NormedAlgebra L F := ⋯
a : f.SplittingField := ⋯
fa0 : (aeval a) f = 0
S : Finset F := ⋯
δ : ℝ := ⋯
a' : F
conj : IsConjRoot L a a'
ne : a ≠ a'
hS : S.Nonempty
⊢ a' ∈ S

Tactic:
simp only [minpoly.eq_of_irreducible_of_monic firr fa0 fmon, Finset.mem_filter,
        Set.mem_toFinset, S]

Goal state after:
case pos
K : Type u_1
L : Type u_2
inst✝⁶ : Field K
inst✝⁵ : NontriviallyNormedField L
inst✝⁴ : CompleteSpace L
inst✝³ : CharZero L
inst✝² : IsUltrametricDist L
inst✝¹ : Algebra K L
hi : DenseRange ⇑(algebraMap K L)
inst✝ : IsAlgClosed K
f : L[X]
fmon : f.Monic
firr : Irreducible f
fnatdeg0 : f.natDegree ≠ 0
F : Type u_2 := f.SplittingField
this✝ : NormedField F := spectralNorm.normedField L F
this : NormedAlgebra L F := spectralNorm.normedAlgebra L F
a : f.SplittingField := rootOfSplits ⋯ ⋯
fa0 : (aeval a) f = 0
S : Finset F := {x ∈ (f.rootSet F).toFinset | x ≠ a}
δ : ℝ := if hS : S.Nonempty then (Finset.image (fun x ↦ ‖a - x‖) S).min' ⋯ else 1
a' : F
conj : IsConjRoot L a a'
ne : a ≠ a'
hS : S.Nonempty
⊢ a' ∈ (minpoly L a).rootSet F ∧ a' ≠ a
Module: Mathlib.Analysis.Normed.Field.Dense

Goal state before:
pp: case pos
K : Type u_1
L : Type u_2
inst✝⁶ : Field K
inst✝⁵ : NontriviallyNormedField L
inst✝⁴ : CompleteSpace L
inst✝³ : CharZero L
inst✝² : IsUltrametricDist L
inst✝¹ : Algebra K L
hi : DenseRange ⇑(algebraMap K L)
inst✝ : IsAlgClosed K
f : L[X]
fmon : f.Monic
firr : Irreducible f
fnatdeg0 : f.natDegree ≠ 0
F : Type u_2 := f.SplittingField
this✝ : NormedField F := spectralNorm.normedField L F
this : NormedAlgebra L F := spectralNorm.normedAlgebra L F
a : f.SplittingField := rootOfSplits ⋯ ⋯
fa0 : (aeval a) f = 0
S : Finset F := {x ∈ (f.rootSet F).toFinset | x ≠ a}
δ : ℝ := if hS : S.Nonempty then (Finset.image (fun x ↦ ‖a - x‖) S).min' ⋯ else 1
a' : F
conj : IsConjRoot L a a'
ne : a ≠ a'
hS : S.Nonempty
⊢ (Finset.image (fun x ↦ ‖a - x‖) S).min' ⋯ ≤ ‖a - a'‖

pp: case neg
K : Type u_1
L : Type u_2
inst✝⁶ : Field K
inst✝⁵ : NontriviallyNormedField L
inst✝⁴ : CompleteSpace L
inst✝³ : CharZero L
inst✝² : IsUltrametricDist L
inst✝¹ : Algebra K L
hi : DenseRange ⇑(algebraMap K L)
inst✝ : IsAlgClosed K
f : L[X]
fmon : f.Monic
firr : Irreducible f
fnatdeg0 : f.natDegree ≠ 0
F : Type u_2 := f.SplittingField
this✝ : NormedField F := spectralNorm.normedField L F
this : NormedAlgebra L F := spectralNorm.normedAlgebra L F
a : f.SplittingField := rootOfSplits ⋯ ⋯
fa0 : (aeval a) f = 0
S : Finset F := {x ∈ (f.rootSet F).toFinset | x ≠ a}
δ : ℝ := if hS : S.Nonempty then (Finset.image (fun x ↦ ‖a - x‖) S).min' ⋯ else 1
a' : F
conj : IsConjRoot L a a'
ne : a ≠ a'
hS : ¬S.Nonempty
⊢ 1 ≤ ‖a - a'‖

Tactic:
· apply Finset.min'_le (S.image fun x => ‖a - x‖) (‖a - a'‖)
      apply Finset.mem_image_of_mem
      simp only [minpoly.eq_of_irreducible_of_monic firr fa0 fmon, Finset.mem_filter,
        Set.mem_toFinset, S]
      rw [← (isConjRoot_iff_mem_minpoly_rootSet ⟨f, fmon, fa0⟩)]
      exact ⟨conj, ne.symm⟩

Goal state after:
case neg
K : Type u_1
L : Type u_2
inst✝⁶ : Field K
inst✝⁵ : NontriviallyNormedField L
inst✝⁴ : CompleteSpace L
inst✝³ : CharZero L
inst✝² : IsUltrametricDist L
inst✝¹ : Algebra K L
hi : DenseRange ⇑(algebraMap K L)
inst✝ : IsAlgClosed K
f : L[X]
fmon : f.Monic
firr : Irreducible f
fnatdeg0 : f.natDegree ≠ 0
F : Type u_2 := f.SplittingField
this✝ : NormedField F := spectralNorm.normedField L F
this : NormedAlgebra L F := spectralNorm.normedAlgebra L F
a : f.SplittingField := rootOfSplits ⋯ ⋯
fa0 : (aeval a) f = 0
S : Finset F := {x ∈ (f.rootSet F).toFinset | x ≠ a}
δ : ℝ := if hS : S.Nonempty then (Finset.image (fun x ↦ ‖a - x‖) S).min' ⋯ else 1
a' : F
conj : IsConjRoot L a a'
ne : a ≠ a'
hS : ¬S.Nonempty
⊢ 1 ≤ ‖a - a'‖
Module: Mathlib.Analysis.Normed.Unbundled.SmoothingSeminorm

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
μ : RingSeminorm R
hμ1 : μ 1 ≤ 1
n : ℕ
hn : 1 ≤ n
| 1

Tactic:
← one_rpow (1 / n : ℝ)

Goal state after:
R : Type u_1
inst✝ : CommRing R
μ : RingSeminorm R
hμ1 : μ 1 ≤ 1
n : ℕ
hn : 1 ≤ n
| 1 ^ (1 / ↑n)
Module: Mathlib.Analysis.Normed.Unbundled.SmoothingSeminorm

Goal state before:
pp: case neg
R : Type u_1
inst✝ : CommRing R
μ : RingSeminorm R
s : ℕ → ℕ
hs_le : ∀ (n : ℕ), s n ≤ n
x : R
ψ : ℕ → ℕ
hψ_lim : Tendsto ((fun n ↦ ↑(s n) / ↑n) ∘ ψ) atTop (𝓝 0)
c : ℝ
hc_bd : ∀ (x_1 : ℝ) (x_2 : ℕ), (∀ (b : ℕ), x_2 ≤ b → μ x ^ (↑(s (ψ b)) * (1 / ↑(ψ b))) ≤ x_1) → c ≤ x_1
hμx : ¬μ x < 1
hμ_lim : ∀ (U : Set ℝ), 1 ∈ U → IsOpen U → ∃ N, ∀ (n : ℕ), N ≤ n → μ x ^ (↑(s (ψ n)) * (1 / ↑(ψ n))) ∈ U
⊢ c ≤ 1

Tactic:
apply le_of_forall_pos_le_add

Goal state after:
case neg
R : Type u_1
inst✝ : CommRing R
μ : RingSeminorm R
s : ℕ → ℕ
hs_le : ∀ (n : ℕ), s n ≤ n
x : R
ψ : ℕ → ℕ
hψ_lim : Tendsto ((fun n ↦ ↑(s n) / ↑n) ∘ ψ) atTop (𝓝 0)
c : ℝ
hc_bd : ∀ (x_1 : ℝ) (x_2 : ℕ), (∀ (b : ℕ), x_2 ≤ b → μ x ^ (↑(s (ψ b)) * (1 / ↑(ψ b))) ≤ x_1) → c ≤ x_1
hμx : ¬μ x < 1
hμ_lim : ∀ (U : Set ℝ), 1 ∈ U → IsOpen U → ∃ N, ∀ (n : ℕ), N ≤ n → μ x ^ (↑(s (ψ n)) * (1 / ↑(ψ n))) ∈ U
⊢ ∀ (ε : ℝ), 0 < ε → c ≤ 1 + ε
Module: Mathlib.Analysis.Normed.Unbundled.SmoothingSeminorm

Goal state before:
pp: case neg
R : Type u_1
inst✝ : CommRing R
μ : RingSeminorm R
s : ℕ → ℕ
hs_le : ∀ (n : ℕ), s n ≤ n
x : R
ψ : ℕ → ℕ
hψ_lim : Tendsto ((fun n ↦ ↑(s n) / ↑n) ∘ ψ) atTop (𝓝 0)
c : ℝ
hc_bd : ∀ (x_1 : ℝ) (x_2 : ℕ), (∀ (b : ℕ), x_2 ≤ b → μ x ^ (↑(s (ψ b)) * (1 / ↑(ψ b))) ≤ x_1) → c ≤ x_1
hμx : ¬μ x < 1
hμ_lim : ∀ (U : Set ℝ), 1 ∈ U → IsOpen U → ∃ N, ∀ (n : ℕ), N ≤ n → μ x ^ (↑(s (ψ n)) * (1 / ↑(ψ n))) ∈ U
ε : ℝ
⊢ 0 < ε → c ≤ 1 + ε

Tactic:
hε

Goal state after:
case neg
R : Type u_1
inst✝ : CommRing R
μ : RingSeminorm R
s : ℕ → ℕ
hs_le : ∀ (n : ℕ), s n ≤ n
x : R
ψ : ℕ → ℕ
hψ_lim : Tendsto ((fun n ↦ ↑(s n) / ↑n) ∘ ψ) atTop (𝓝 0)
c : ℝ
hc_bd : ∀ (x_1 : ℝ) (x_2 : ℕ), (∀ (b : ℕ), x_2 ≤ b → μ x ^ (↑(s (ψ b)) * (1 / ↑(ψ b))) ≤ x_1) → c ≤ x_1
hμx : ¬μ x < 1
hμ_lim : ∀ (U : Set ℝ), 1 ∈ U → IsOpen U → ∃ N, ∀ (n : ℕ), N ≤ n → μ x ^ (↑(s (ψ n)) * (1 / ↑(ψ n))) ∈ U
ε : ℝ
hε : 0 < ε
⊢ c ≤ 1 + ε
Module: Mathlib.Analysis.Normed.Unbundled.SmoothingSeminorm

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
μ : RingSeminorm R
hμ1 : μ 1 ≤ 1
s : ℕ → ℕ
hs_le : ∀ (n : ℕ), s n ≤ n
x : R
a : ℝ
a_in : a ∈ Set.Icc 0 1
ψ : ℕ → ℕ
hψ_mono : StrictMono ψ
hψ_lim : Tendsto ((fun n ↦ ↑(s n) / ↑n) ∘ ψ) atTop (𝓝 0)
ha : a = 0
b : ℝ
hb : ∃ a, ∀ (b_1 : ℕ), a ≤ b_1 → μ x ^ (↑(s (ψ b_1)) * (1 / ↑(ψ b_1))) ≤ b
⊢ ∃ a, ∀ (b_1 : ℕ), a ≤ b_1 → μ (x ^ s (ψ b_1)) ^ (1 / ↑(ψ b_1)) ≤ b

Tactic:
obtain ⟨m, hm⟩ := hb

Goal state after:
R : Type u_1
inst✝ : CommRing R
μ : RingSeminorm R
hμ1 : μ 1 ≤ 1
s : ℕ → ℕ
hs_le : ∀ (n : ℕ), s n ≤ n
x : R
a : ℝ
a_in : a ∈ Set.Icc 0 1
ψ : ℕ → ℕ
hψ_mono : StrictMono ψ
hψ_lim : Tendsto ((fun n ↦ ↑(s n) / ↑n) ∘ ψ) atTop (𝓝 0)
ha : a = 0
b : ℝ
m : ℕ
hm : ∀ (b_1 : ℕ), m ≤ b_1 → μ x ^ (↑(s (ψ b_1)) * (1 / ↑(ψ b_1))) ≤ b
⊢ ∃ a, ∀ (b_1 : ℕ), a ≤ b_1 → μ (x ^ s (ψ b_1)) ^ (1 / ↑(ψ b_1)) ≤ b
Module: Mathlib.Analysis.Normed.Unbundled.SmoothingSeminorm

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
μ : RingSeminorm R
hμ1 : μ 1 ≤ 1
hna : IsNonarchimedean ⇑μ
x y : R
hn : ∀ (n : ℕ), ∃ m < n + 1, μ ((x + y) ^ n) ^ (1 / ↑n) ≤ (μ (x ^ m) * μ (y ^ (n - m))) ^ (1 / ↑n)
mu : ℕ → ℕ := fun n ↦ _root_.mu μ hn n
nu : ℕ → ℕ := fun n ↦ n - mu n
hnu : nu = fun n ↦ n - mu n
hmu_le : ∀ (n : ℕ), mu n ≤ n
hmu_bdd : ∀ (n : ℕ), ↑(mu n) / ↑n ∈ Set.Icc 0 1
hs : Bornology.IsBounded (Set.Icc 0 1)
a : ℝ
a_in : a ∈ Set.Icc 0 1
ψ : ℕ → ℕ
hψ_mono : StrictMono ψ
hψ_lim : Tendsto ((fun n ↦ ↑(mu n) / ↑n) ∘ ψ) atTop (𝓝 a)
b : ℝ := 1 - a
hb : b = 1 - a
hb_lim : Tendsto ((fun n ↦ ↑(nu n) / ↑n) ∘ ψ) atTop (𝓝 b)
b_in : b ∈ Set.Icc 0 1
hnu_le : ∀ (n : ℕ), nu n ≤ n
hx : limsup (fun n ↦ μ (x ^ mu (ψ n)) ^ (1 / ↑(ψ n))) atTop ≤ smoothingFun μ x ^ a
hy : limsup (fun n ↦ μ (y ^ nu (ψ n)) ^ (1 / ↑(ψ n))) atTop ≤ smoothingFun μ y ^ b
hxy :
  limsup (fun n ↦ μ (x ^ mu (ψ n)) ^ (1 / ↑(ψ n)) * μ (y ^ nu (ψ n)) ^ (1 / ↑(ψ n))) atTop ≤
    smoothingFun μ x ^ a * smoothingFun μ y ^ b
ε : ℝ
⊢ ε > 0 → smoothingFun μ (x + y) - ε ≤ max (smoothingFun μ x) (smoothingFun μ y)

Tactic:
hε

Goal state after:
R : Type u_1
inst✝ : CommRing R
μ : RingSeminorm R
hμ1 : μ 1 ≤ 1
hna : IsNonarchimedean ⇑μ
x y : R
hn : ∀ (n : ℕ), ∃ m < n + 1, μ ((x + y) ^ n) ^ (1 / ↑n) ≤ (μ (x ^ m) * μ (y ^ (n - m))) ^ (1 / ↑n)
mu : ℕ → ℕ := fun n ↦ _root_.mu μ hn n
nu : ℕ → ℕ := fun n ↦ n - mu n
hnu : nu = fun n ↦ n - mu n
hmu_le : ∀ (n : ℕ), mu n ≤ n
hmu_bdd : ∀ (n : ℕ), ↑(mu n) / ↑n ∈ Set.Icc 0 1
hs : Bornology.IsBounded (Set.Icc 0 1)
a : ℝ
a_in : a ∈ Set.Icc 0 1
ψ : ℕ → ℕ
hψ_mono : StrictMono ψ
hψ_lim : Tendsto ((fun n ↦ ↑(mu n) / ↑n) ∘ ψ) atTop (𝓝 a)
b : ℝ := 1 - a
hb : b = 1 - a
hb_lim : Tendsto ((fun n ↦ ↑(nu n) / ↑n) ∘ ψ) atTop (𝓝 b)
b_in : b ∈ Set.Icc 0 1
hnu_le : ∀ (n : ℕ), nu n ≤ n
hx : limsup (fun n ↦ μ (x ^ mu (ψ n)) ^ (1 / ↑(ψ n))) atTop ≤ smoothingFun μ x ^ a
hy : limsup (fun n ↦ μ (y ^ nu (ψ n)) ^ (1 / ↑(ψ n))) atTop ≤ smoothingFun μ y ^ b
hxy :
  limsup (fun n ↦ μ (x ^ mu (ψ n)) ^ (1 / ↑(ψ n)) * μ (y ^ nu (ψ n)) ^ (1 / ↑(ψ n))) atTop ≤
    smoothingFun μ x ^ a * smoothingFun μ y ^ b
ε : ℝ
hε : ε > 0
⊢ smoothingFun μ (x + y) - ε ≤ max (smoothingFun μ x) (smoothingFun μ y)
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
h_fin : FiniteDimensional K L
hn : Normal K L
hu : IsUltrametricDist K
⊢ spectralNorm K L = ⇑(invariantExtension K L)

Tactic:
ext x
  have hna := hu.isNonarchimedean_norm
  set f := Classical.choose (exists_nonarchimedean_pow_mul_seminorm_of_finiteDimensional h_fin hna)
    with hf
  have hf_pow : IsPowMul f := (Classical.choose_spec
    (exists_nonarchimedean_pow_mul_seminorm_of_finiteDimensional h_fin hna)).1
  have hf_ext : ∀ (x : K), f (algebraMap K L x) = ‖x‖ := (Classical.choose_spec
    (exists_nonarchimedean_pow_mul_seminorm_of_finiteDimensional h_fin hna)).2.1
  have hf_na : IsNonarchimedean f := (Classical.choose_spec
    (exists_nonarchimedean_pow_mul_seminorm_of_finiteDimensional h_fin hna)).2.2
  rw [spectralNorm_eq_iSup_of_finiteDimensional_normal K L hf_pow hf_na hf_ext]
  simp only [invariantExtension_apply, algNormOfAlgEquiv_apply, hf]

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

Goal state before:
pp: K : Type u_2
inst✝³ : NormedField K
L : Type u_3
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : IsUltrametricDist K
h_alg : Algebra.IsAlgebraic K L
x : L
⊢ ∀ {n : ℕ}, 1 ≤ n → spectralNorm K L (x ^ n) = spectralNorm K L x ^ n

Tactic:
n

Goal state after:
K : Type u_2
inst✝³ : NormedField K
L : Type u_3
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : IsUltrametricDist K
h_alg : Algebra.IsAlgebraic K L
x : L
n : ℕ
⊢ 1 ≤ n → spectralNorm K L (x ^ n) = spectralNorm K L x ^ n
Module: Mathlib.Analysis.Normed.Group.Tannery

Goal state before:
pp: case inr.inr
α : Type u_1
β : Type u_2
G : Type u_3
𝓕 : Filter α
inst✝¹ : NormedAddCommGroup G
inst✝ : CompleteSpace G
f : α → β → G
g : β → G
bound : β → ℝ
h_sum : Summable bound
hab : ∀ (k : β), Tendsto (fun x ↦ f x k) 𝓕 (𝓝 (g k))
h_bound : ∀ᶠ (n : α) in 𝓕, ∀ (k : β), ‖f n k‖ ≤ bound k
h✝¹ : Nonempty β
h✝ : 𝓕.NeBot
h_g_le : ∀ (k : β), ‖g k‖ ≤ bound k
⊢ Tendsto (fun x ↦ ∑' (k : β), f x k) 𝓕 (𝓝 (∑' (k : β), g k))

Tactic:
have h_sumg : Summable (‖g ·‖) :=
    h_sum.of_norm_bounded (fun k ↦ (norm_norm (g k)).symm ▸ h_g_le k)

Goal state after:
case inr.inr
α : Type u_1
β : Type u_2
G : Type u_3
𝓕 : Filter α
inst✝¹ : NormedAddCommGroup G
inst✝ : CompleteSpace G
f : α → β → G
g : β → G
bound : β → ℝ
h_sum : Summable bound
hab : ∀ (k : β), Tendsto (fun x ↦ f x k) 𝓕 (𝓝 (g k))
h_bound : ∀ᶠ (n : α) in 𝓕, ∀ (k : β), ‖f n k‖ ≤ bound k
h✝¹ : Nonempty β
h✝ : 𝓕.NeBot
h_g_le : ∀ (k : β), ‖g k‖ ≤ bound k
h_sumg : Summable fun x ↦ ‖g x‖
⊢ Tendsto (fun x ↦ ∑' (k : β), f x k) 𝓕 (𝓝 (∑' (k : β), g k))
Module: Mathlib.Analysis.Normed.Unbundled.SmoothingSeminorm

Goal state before:
pp: case h
R : Type u_1
inst✝ : CommRing R
μ : RingSeminorm R
hμ1 : μ 1 ≤ 1
x : R
hx : ∀ (n : ℕ), 1 ≤ n → μ (x ^ n) = μ x ^ n
n : ℕ
hn : 1 ≤ n
⊢ μ (x ^ n) ^ (1 / ↑n) = μ x

Tactic:
have hn0 : (n : ℝ) ≠ 0 := cast_ne_zero.mpr (one_le_iff_ne_zero.mp hn)

Goal state after:
case h
R : Type u_1
inst✝ : CommRing R
μ : RingSeminorm R
hμ1 : μ 1 ≤ 1
x : R
hx : ∀ (n : ℕ), 1 ≤ n → μ (x ^ n) = μ x ^ n
n : ℕ
hn : 1 ≤ n
hn0 : ↑n ≠ 0
⊢ μ (x ^ n) ^ (1 / ↑n) = μ x
Module: Mathlib.Analysis.Normed.Group.Tannery

Goal state before:
pp: case inr.inr
α : Type u_1
β : Type u_2
G : Type u_3
𝓕 : Filter α
inst✝¹ : NormedAddCommGroup G
inst✝ : CompleteSpace G
f : α → β → G
g : β → G
bound : β → ℝ
h_sum : Summable bound
hab : ∀ (k : β), Tendsto (fun x ↦ f x k) 𝓕 (𝓝 (g k))
h_bound : ∀ᶠ (n : α) in 𝓕, ∀ (k : β), ‖f n k‖ ≤ bound k
h✝¹ : Nonempty β
h✝ : 𝓕.NeBot
h_g_le : ∀ (k : β), ‖g k‖ ≤ bound k
h_sumg : Summable fun x ↦ ‖g x‖
h_suma : ∀ᶠ (n : α) in 𝓕, Summable fun x ↦ ‖f n x‖
ε : ℝ
⊢ ε > 0 → ∀ᶠ (x : α) in 𝓕, dist (∑' (k : β), f x k) (∑' (k : β), g k) < ε

Tactic:
hε

Goal state after:
case inr.inr
α : Type u_1
β : Type u_2
G : Type u_3
𝓕 : Filter α
inst✝¹ : NormedAddCommGroup G
inst✝ : CompleteSpace G
f : α → β → G
g : β → G
bound : β → ℝ
h_sum : Summable bound
hab : ∀ (k : β), Tendsto (fun x ↦ f x k) 𝓕 (𝓝 (g k))
h_bound : ∀ᶠ (n : α) in 𝓕, ∀ (k : β), ‖f n k‖ ≤ bound k
h✝¹ : Nonempty β
h✝ : 𝓕.NeBot
h_g_le : ∀ (k : β), ‖g k‖ ≤ bound k
h_sumg : Summable fun x ↦ ‖g x‖
h_suma : ∀ᶠ (n : α) in 𝓕, Summable fun x ↦ ‖f n x‖
ε : ℝ
hε : ε > 0
⊢ ∀ᶠ (x : α) in 𝓕, dist (∑' (k : β), f x k) (∑' (k : β), g k) < ε