mathlib-initiative/mathlib-tactics huggingface dataset:
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 : Set V
c : k
v p : V
hs₁ : StarConvex k p s
hs₂ : IsClosed[inst✝²] s
hc : 0 ≤ c
hv : v ∈ asymptoticCone k s
⊢ c • v +ᵥ p ∈ s

Tactic:
refine isClosed_iff_frequently.mp hs₂ _ <|
    tendsto_snd (f := atTop (α := k)) |>.const_smul _ |>.vadd_const _ |>.frequently ?_
  rw [mem_asymptoticCone_iff, asymptoticNhds_eq_smul_vadd v p, vadd_pure, frequently_map,
    ← map₂_smul, ← map_prod_eq_map₂, frequently_map] at hv
  apply hv.mp
  filter_upwards [tendsto_fst.eventually (eventually_ge_atTop c)]
    with ⟨t, u⟩ (ht : c ≤ t) (h : t • u +ᵥ p ∈ s)
  change c • u +ᵥ p ∈ s
  apply hs₁.segment_subset h
  simp_rw [mem_segment_iff_sameRay, ← vsub_eq_sub, vadd_vsub, vadd_vsub_vadd_cancel_right,
    ← sub_smul]
  exact (SameRay.sameRay_nonneg_smul_left _ hc).nonneg_smul_right (sub_nonneg.mpr ht)

Goal state after:
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
z : PE := midpoint ℝ x y
s : Set (PE ≃ᵢ PE) := {e | e x = x ∧ e y = y}
this : Nonempty ↑s
h_bdd : BddAbove (range fun e ↦ dist (↑e z) z)
R : PE ≃ᵢ PE := (pointReflection ℝ z).toIsometryEquiv
f : PE ≃ᵢ PE → PE ≃ᵢ PE := fun e ↦ ((e.trans R).trans e.symm).trans R
hf_dist : ∀ (e : PE ≃ᵢ PE), dist ((f e) z) z = 2 * dist (e z) z
hf_maps_to : MapsTo f s s
c : ℝ := ⨆ e, dist (↑e z) z
e : PE ≃ᵢ PE
he : e ∈ s
⊢ dist (↑⟨e, he⟩ z) z ≤ c / 2

Tactic:
simp only [le_div_iff₀' (zero_lt_two' ℝ), ← hf_dist]

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
s : Set (PE ≃ᵢ PE) := {e | e x = x ∧ e y = y}
this : Nonempty ↑s
h_bdd : BddAbove (range fun e ↦ dist (↑e z) z)
R : PE ≃ᵢ PE := (pointReflection ℝ z).toIsometryEquiv
f : PE ≃ᵢ PE → PE ≃ᵢ PE := fun e ↦ ((e.trans R).trans e.symm).trans R
hf_dist : ∀ (e : PE ≃ᵢ PE), dist ((f e) z) z = 2 * dist (e z) z
hf_maps_to : MapsTo f s s
c : ℝ := ⨆ e, dist (↑e z) z
e : PE ≃ᵢ PE
he : e ∈ s
⊢ dist ((f e) z) z ≤ c
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Beta

Goal state before:
pp: case pos.negSucc
z : ℂ
pi_ne : ↑π ≠ 0
a✝ : ℕ
hk : z = -↑(Int.negSucc a✝)
⊢ Gamma (1 + ↑a✝) * 0 = 0

Tactic:
mul_zero

Goal state after:
case pos.negSucc
z : ℂ
pi_ne : ↑π ≠ 0
a✝ : ℕ
hk : z = -↑(Int.negSucc a✝)
⊢ 0 = 0
Module: Mathlib.Analysis.Normed.Affine.Simplex

Goal state before:
pp: case e'_3.e'_4
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
n : ℕ
s : Simplex R P n
hr : s.Regular
i j : Fin (n + 1)
hij : i ≠ j
hn : n ≠ 0
hi : ¬i = 1
x : P ≃ᵢ P
hx : s.points ∘ ⇑(Equiv.trans (Equiv.swap 0 i) (Equiv.swap 1 j)) = ⇑x ∘ s.points
⊢ (Equiv.swap 1 j) ((Equiv.swap 0 i) 1) = j

Tactic:
rw [Equiv.swap_apply_of_ne_of_ne (by simp [hn]) (Ne.symm hi)]

Goal state after:
case e'_3.e'_4
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
n : ℕ
s : Simplex R P n
hr : s.Regular
i j : Fin (n + 1)
hij : i ≠ j
hn : n ≠ 0
hi : ¬i = 1
x : P ≃ᵢ P
hx : s.points ∘ ⇑(Equiv.trans (Equiv.swap 0 i) (Equiv.swap 1 j)) = ⇑x ∘ s.points
⊢ (Equiv.swap 1 j) 1 = j
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Beta

Goal state before:
pp: s : ℂ
h1 : AnalyticOnNhd ℂ (fun z ↦ (Gamma z)⁻¹ * (Gamma (z + 1 / 2))⁻¹) univ
h2 : AnalyticOnNhd ℂ (fun z ↦ (Gamma (2 * z))⁻¹ * 2 ^ (2 * z - 1) / ↑√π) univ
h3 : Tendsto ofReal (𝓝[≠] 1) (𝓝[≠] 1)
t : ℝ
ht : 0 < t
⊢ (↑(Real.Gamma t) * ↑(Real.Gamma (t + 1 / 2)))⁻¹ = (Gamma (2 * ↑t))⁻¹ * 2 ^ (2 * ↑t - 1) / ↑√π

Tactic:
←
    ofReal_mul,

Goal state after:
s : ℂ
h1 : AnalyticOnNhd ℂ (fun z ↦ (Gamma z)⁻¹ * (Gamma (z + 1 / 2))⁻¹) univ
h2 : AnalyticOnNhd ℂ (fun z ↦ (Gamma (2 * z))⁻¹ * 2 ^ (2 * z - 1) / ↑√π) univ
h3 : Tendsto ofReal (𝓝[≠] 1) (𝓝[≠] 1)
t : ℝ
ht : 0 < t
⊢ (↑(Real.Gamma t * Real.Gamma (t + 1 / 2)))⁻¹ = (Gamma (2 * ↑t))⁻¹ * 2 ^ (2 * ↑t - 1) / ↑√π
Module: Mathlib.Analysis.Normed.Algebra.TrivSqZeroExt

Goal state before:
pp: 𝕜 : Type u_1
R : Type u_3
M : Type u_4
inst✝¹⁶ : Field 𝕜
inst✝¹⁵ : CharZero 𝕜
inst✝¹⁴ : Ring R
inst✝¹³ : AddCommGroup M
inst✝¹² : Algebra 𝕜 R
inst✝¹¹ : Module 𝕜 M
inst✝¹⁰ : Module R M
inst✝⁹ : Module Rᵐᵒᵖ M
inst✝⁸ : SMulCommClass R Rᵐᵒᵖ M
inst✝⁷ : IsScalarTower 𝕜 R M
inst✝⁶ : IsScalarTower 𝕜 Rᵐᵒᵖ M
inst✝⁵ : TopologicalSpace R
inst✝⁴ : TopologicalSpace M
inst✝³ : IsTopologicalRing R
inst✝² : IsTopologicalAddGroup M
inst✝¹ : ContinuousSMul R M
inst✝ : ContinuousSMul Rᵐᵒᵖ M
x : tsze R M
hx : MulOpposite.op x.fst • x.snd = x.fst • x.snd
e : R
h : HasSum (fun n ↦ (expSeries 𝕜 R n) fun x_1 ↦ x.fst) e
⊢ HasSum (fun n ↦ ((expSeries 𝕜 R n) fun x_1 ↦ x.fst) • x.snd)
    (e • x.snd - ∑ i ∈ Finset.range 1, ((expSeries 𝕜 (tsze R M) i) fun x_1 ↦ x).snd)

Tactic:
expSeries_apply_eq

Goal state after:
𝕜 : Type u_1
R : Type u_3
M : Type u_4
inst✝¹⁶ : Field 𝕜
inst✝¹⁵ : CharZero 𝕜
inst✝¹⁴ : Ring R
inst✝¹³ : AddCommGroup M
inst✝¹² : Algebra 𝕜 R
inst✝¹¹ : Module 𝕜 M
inst✝¹⁰ : Module R M
inst✝⁹ : Module Rᵐᵒᵖ M
inst✝⁸ : SMulCommClass R Rᵐᵒᵖ M
inst✝⁷ : IsScalarTower 𝕜 R M
inst✝⁶ : IsScalarTower 𝕜 Rᵐᵒᵖ M
inst✝⁵ : TopologicalSpace R
inst✝⁴ : TopologicalSpace M
inst✝³ : IsTopologicalRing R
inst✝² : IsTopologicalAddGroup M
inst✝¹ : ContinuousSMul R M
inst✝ : ContinuousSMul Rᵐᵒᵖ M
x : tsze R M
hx : MulOpposite.op x.fst • x.snd = x.fst • x.snd
e : R
h : HasSum (fun n ↦ (↑n.factorial)⁻¹ • x.fst ^ n) e
⊢ HasSum (fun n ↦ ((↑n.factorial)⁻¹ • x.fst ^ n) • x.snd)
    (e • x.snd - ∑ x_1 ∈ Finset.range 1, ((↑x_1.factorial)⁻¹ • x ^ x_1).snd)
Module: Mathlib.Analysis.Normed.Algebra.TrivSqZeroExt

Goal state before:
pp: R : Type u_3
M : Type u_4
inst✝¹⁴ : Ring R
inst✝¹³ : AddCommGroup M
inst✝¹² : Module R M
inst✝¹¹ : Module Rᵐᵒᵖ M
inst✝¹⁰ : SMulCommClass R Rᵐᵒᵖ M
inst✝⁹ : TopologicalSpace R
inst✝⁸ : TopologicalSpace M
inst✝⁷ : IsTopologicalRing R
inst✝⁶ : IsTopologicalAddGroup M
inst✝⁵ : ContinuousSMul R M
inst✝⁴ : ContinuousSMul Rᵐᵒᵖ M
inst✝³ : Algebra ℚ R
inst✝² : Module ℚ M
inst✝¹ : T2Space R
inst✝ : T2Space M
x : tsze R M
hx : MulOpposite.op x.fst • x.snd = x.fst • x.snd
⊢ exp x = inl (exp x.fst) + inr (exp x.fst • x.snd)

Tactic:
simp_rw [exp_eq_expSeries_sum ℚ, FormalMultilinearSeries.sum]

Goal state after:
R : Type u_3
M : Type u_4
inst✝¹⁴ : Ring R
inst✝¹³ : AddCommGroup M
inst✝¹² : Module R M
inst✝¹¹ : Module Rᵐᵒᵖ M
inst✝¹⁰ : SMulCommClass R Rᵐᵒᵖ M
inst✝⁹ : TopologicalSpace R
inst✝⁸ : TopologicalSpace M
inst✝⁷ : IsTopologicalRing R
inst✝⁶ : IsTopologicalAddGroup M
inst✝⁵ : ContinuousSMul R M
inst✝⁴ : ContinuousSMul Rᵐᵒᵖ M
inst✝³ : Algebra ℚ R
inst✝² : Module ℚ M
inst✝¹ : T2Space R
inst✝ : T2Space M
x : tsze R M
hx : MulOpposite.op x.fst • x.snd = x.fst • x.snd
⊢ (∑' (n : ℕ), (expSeries ℚ (tsze R M) n) fun x_1 ↦ x) =
    inl (∑' (n : ℕ), (expSeries ℚ R n) fun x_1 ↦ x.fst) + inr ((∑' (n : ℕ), (expSeries ℚ R n) fun x_1 ↦ x.fst) • x.snd)
Module: Mathlib.Analysis.Normed.Algebra.GelfandMazur

Goal state before:
pp: F : Type u_1
inst✝³ : NormedRing F
inst✝² : NormOneClass F
inst✝¹ : NormMulClass F
inst✝ : NormedAlgebra ℂ F
x : F
z : ℂ
hz : IsMinOn (fun x_1 ↦ ‖x - (algebraMap ℂ F) x_1‖) univ z
H : ∀ (z' : ℂ), ‖x - (algebraMap ℂ F) z'‖ ≠ 0
c : ℂ
M : ℝ := ‖x - (algebraMap ℂ F) z‖
hMdef : M = ‖x - (algebraMap ℂ F) z‖
hM₀ : 0 < M
y w : ℂ
hy : ‖x - (algebraMap ℂ F) y‖ = M
n : ℕ
hn : n > 0
⊢ (C (w - y) ^ n).natDegree < n

Tactic:
compute_degree!

Goal state after:
Module: Mathlib.Analysis.Normed.Algebra.QuaternionExponential

Goal state before:
pp: q : ℍ
hq : q.re = 0
n : ℕ
⊢ ((expSeries ℝ ℍ (2 * n)) fun x ↦ q) = ↑((-1) ^ n * ‖q‖ ^ (2 * n) / ↑(2 * n)!)

Tactic:
expSeries_apply_eq

Goal state after:
q : ℍ
hq : q.re = 0
n : ℕ
⊢ (↑(2 * n)!)⁻¹ • q ^ (2 * n) = ↑((-1) ^ n * ‖q‖ ^ (2 * n) / ↑(2 * n)!)
Module: Mathlib.Analysis.Normed.Algebra.QuaternionExponential

Goal state before:
pp: q : ℍ
hq : q.re = 0
n : ℕ
⊢ ((expSeries ℝ ℍ (2 * n + 1)) fun x ↦ q) = ((-1) ^ n * ‖q‖ ^ (2 * n + 1) / ↑(2 * n + 1)! / ‖q‖) • q

Tactic:
expSeries_apply_eq

Goal state after:
q : ℍ
hq : q.re = 0
n : ℕ
⊢ (↑(2 * n + 1)!)⁻¹ • q ^ (2 * n + 1) = ((-1) ^ n * ‖q‖ ^ (2 * n + 1) / ↑(2 * n + 1)! / ‖q‖) • q
Module: Mathlib.Analysis.Normed.Algebra.GelfandMazur

Goal state before:
pp: F : Type u_1
inst✝³ : NormedRing F
inst✝² : NormedAlgebra ℝ F
inst✝¹ : NormOneClass F
inst✝ : NormMulClass F
x : F
z : ℝ × ℝ
h : IsMinOn (fun x_1 ↦ ‖φ x x_1‖) Set.univ z
w✝ : ℝ × ℝ
M : ℝ := ‖φ x z‖
H : M ≠ 0
hM : M = ‖φ x z‖
hM₀ : 0 < M
w u : ℝ × ℝ
hw : ‖φ x w‖ = M
n : ℕ
hn : n > 0
q : ℝ × ℝ → ℝ[X] := fun y ↦ X ^ 2 - C y.1 * X + C y.2
hq : ∀ (y : ℝ × ℝ), (q y).IsMonicOfDegree 2
p : ℝ[X]
hp : p.IsMonicOfDegree (2 * (n - 1))
hrel : ‖(aeval x) (q u)‖ * ‖(aeval x) p‖ = ‖(aeval x) (q w ^ n - (q w - q u) ^ n)‖
⊢ ‖φ x u‖ * ‖(aeval x) p‖ ≤ M ^ n + ‖φ x u - φ x w‖ ^ n

Tactic:
map_sub,

Goal state after:
F : Type u_1
inst✝³ : NormedRing F
inst✝² : NormedAlgebra ℝ F
inst✝¹ : NormOneClass F
inst✝ : NormMulClass F
x : F
z : ℝ × ℝ
h : IsMinOn (fun x_1 ↦ ‖φ x x_1‖) Set.univ z
w✝ : ℝ × ℝ
M : ℝ := ‖φ x z‖
H : M ≠ 0
hM : M = ‖φ x z‖
hM₀ : 0 < M
w u : ℝ × ℝ
hw : ‖φ x w‖ = M
n : ℕ
hn : n > 0
q : ℝ × ℝ → ℝ[X] := fun y ↦ X ^ 2 - C y.1 * X + C y.2
hq : ∀ (y : ℝ × ℝ), (q y).IsMonicOfDegree 2
p : ℝ[X]
hp : p.IsMonicOfDegree (2 * (n - 1))
hrel : ‖(aeval x) (q u)‖ * ‖(aeval x) p‖ = ‖(aeval x) (q w ^ n) - (aeval x) ((q w - q u) ^ n)‖
⊢ ‖φ x u‖ * ‖(aeval x) p‖ ≤ M ^ n + ‖φ x u - φ x w‖ ^ n
Module: Mathlib.Analysis.Normed.Field.Ultra

Goal state before:
pp: case inr
R : Type u_1
inst✝ : NormedDivisionRing R
h : ∀ (n : ℕ), ‖↑n‖ ≤ 1
x✝ : R
m : ℕ
x : R
n : ℕ
hx : 0 < ‖x‖
⊢ ‖↑n‖ * ‖x‖ ≤ ‖x‖

Tactic:
· simpa only [mul_le_iff_le_one_left hx] using h _

Goal state after:
Module: Mathlib.Analysis.Normed.Field.Ultra

Goal state before:
pp: case h.inr
R : Type u_1
inst✝ : NormedDivisionRing R
x : R
m : ℕ
h : ∀ (x : R) (n : ℕ), ‖n • x‖ ≤ ‖x‖
hm✝ : max 1 (‖x‖ ^ m) = 1
hx : ‖x‖ ^ m ≤ 1
i : ℕ
hi : i ∈ Finset.range (m + 1)
hm : m ≠ 0
⊢ ‖x‖ ^ i ≤ 1

Tactic:
rw [pow_le_one_iff_of_nonneg (norm_nonneg _) hm] at hx
      exact pow_le_one₀ (norm_nonneg _) hx

Goal state after:
Module: Mathlib.Topology.Algebra.Order.LiminfLimsup

Goal state before:
pp: ι : Type u_1
α : Type u_2
inst✝⁴ : AddCommGroup α
inst✝³ : ConditionallyCompleteLinearOrder α
inst✝² : DenselyOrdered α
inst✝¹ : AddLeftMono α
f : Filter ι
inst✝ : f.NeBot
u v : ι → α
h₁ : IsBoundedUnder (fun x1 x2 ↦ x1 ≥ x2) f u
h₂ : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) f u
h₃ : IsCoboundedUnder (fun x1 x2 ↦ x1 ≤ x2) f v
h₄ : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) f v
h : IsCoboundedUnder (fun x1 x2 ↦ x1 ≤ x2) f (u + v)
h' : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) f (u + v)
a : α
a_u : a > limsup u f
b : α
b_v : b > limsup v f
⊢ ∀ y > a + b, ∀ᶠ (a : ι) in f, (u + v) a < y

Tactic:
intro c

Goal state after:
ι : Type u_1
α : Type u_2
inst✝⁴ : AddCommGroup α
inst✝³ : ConditionallyCompleteLinearOrder α
inst✝² : DenselyOrdered α
inst✝¹ : AddLeftMono α
f : Filter ι
inst✝ : f.NeBot
u v : ι → α
h₁ : IsBoundedUnder (fun x1 x2 ↦ x1 ≥ x2) f u
h₂ : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) f u
h₃ : IsCoboundedUnder (fun x1 x2 ↦ x1 ≤ x2) f v
h₄ : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) f v
h : IsCoboundedUnder (fun x1 x2 ↦ x1 ≤ x2) f (u + v)
h' : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) f (u + v)
a : α
a_u : a > limsup u f
b : α
b_v : b > limsup v f
c : α
⊢ c > a + b → ∀ᶠ (a : ι) in f, (u + v) a < c
Module: Mathlib.Analysis.Normed.Unbundled.RingSeminorm

Goal state before:
pp: R : Type u_1
K : Type u_2
inst✝ : Field K
f : RingSeminorm K
hnt : f ≠ 0
x : K
hx : f.toFun x = 0
c : K
hc : f c ≠ 0
hn0 : ¬x = 0
⊢ f c = 0

Tactic:
rw [← mul_one c, ← mul_inv_cancel₀ hn0, ← mul_assoc, mul_comm c, mul_assoc]

Goal state after:
R : Type u_1
K : Type u_2
inst✝ : Field K
f : RingSeminorm K
hnt : f ≠ 0
x : K
hx : f.toFun x = 0
c : K
hc : f c ≠ 0
hn0 : ¬x = 0
⊢ f (x * (c * x⁻¹)) = 0
Module: Mathlib.Topology.Algebra.Order.LiminfLimsup

Goal state before:
pp: ι : Type u_1
α : Type u_2
inst✝⁴ : AddCommGroup α
inst✝³ : ConditionallyCompleteLinearOrder α
inst✝² : DenselyOrdered α
inst✝¹ : AddLeftMono α
f : Filter ι
inst✝ : f.NeBot
u v : ι → α
h₁ : IsBoundedUnder (fun x1 x2 ↦ x1 ≥ x2) f u
h₂ : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) f u
h₃ : IsBoundedUnder (fun x1 x2 ↦ x1 ≥ x2) f v
h₄ : IsCoboundedUnder (fun x1 x2 ↦ x1 ≥ x2) f v
h : IsCoboundedUnder (fun x1 x2 ↦ x2 ≤ x1) f (u + v)
h' : IsBoundedUnder (fun x1 x2 ↦ x2 ≤ x1) f (u + v)
a : α
a_u : a < liminf u f
b : α
b_v : b < liminf v f
⊢ ∀ y < a + b, ∀ᶠ (a : ι) in f, y < (u + v) a

Tactic:
intro c

Goal state after:
ι : Type u_1
α : Type u_2
inst✝⁴ : AddCommGroup α
inst✝³ : ConditionallyCompleteLinearOrder α
inst✝² : DenselyOrdered α
inst✝¹ : AddLeftMono α
f : Filter ι
inst✝ : f.NeBot
u v : ι → α
h₁ : IsBoundedUnder (fun x1 x2 ↦ x1 ≥ x2) f u
h₂ : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) f u
h₃ : IsBoundedUnder (fun x1 x2 ↦ x1 ≥ x2) f v
h₄ : IsCoboundedUnder (fun x1 x2 ↦ x1 ≥ x2) f v
h : IsCoboundedUnder (fun x1 x2 ↦ x2 ≤ x1) f (u + v)
h' : IsBoundedUnder (fun x1 x2 ↦ x2 ≤ x1) f (u + v)
a : α
a_u : a < liminf u f
b : α
b_v : b < liminf v f
c : α
⊢ c < a + b → ∀ᶠ (a : ι) in f, c < (u + v) a
Module: Mathlib.Analysis.Normed.Unbundled.SeminormFromBounded

Goal state before:
pp: case pos
R : Type u_1
inst✝ : CommRing R
f : R → ℝ
c : ℝ
f_nonneg : 0 ≤ f
x : R
f_mul : f x ≤ c * f x * 0
h1 : f 1 = 0
⊢ f x ≤ f 1 * seminormFromBounded' f x

Tactic:
mul_zero

Goal state after:
case pos
R : Type u_1
inst✝ : CommRing R
f : R → ℝ
c : ℝ
f_nonneg : 0 ≤ f
x : R
f_mul : f x ≤ 0
h1 : f 1 = 0
⊢ f x ≤ f 1 * seminormFromBounded' f x
Module: Mathlib.Analysis.Normed.Unbundled.SeminormFromBounded

Goal state before:
pp: case refine_1
R : Type u_1
inst✝ : CommRing R
f : R → ℝ
c : ℝ
f_nonneg : 0 ≤ f
f_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y
x : R
h : seminormFromBounded' f x = 0
hf : f x ≤ f 1 * 0
⊢ f x = 0

Tactic:
mul_zero

Goal state after:
case refine_1
R : Type u_1
inst✝ : CommRing R
f : R → ℝ
c : ℝ
f_nonneg : 0 ≤ f
f_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y
x : R
h : seminormFromBounded' f x = 0
hf : f x ≤ 0
⊢ f x = 0
Module: Mathlib.Analysis.Normed.Unbundled.SeminormFromBounded

Goal state before:
pp: case refine_2
R : Type u_1
inst✝ : CommRing R
f : R → ℝ
c : ℝ
f_nonneg : 0 ≤ f
f_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y
x : R
h : f x = 0
hf : seminormFromBounded' f x ≤ c * 0
⊢ seminormFromBounded' f x = 0

Tactic:
mul_zero

Goal state after:
case refine_2
R : Type u_1
inst✝ : CommRing R
f : R → ℝ
c : ℝ
f_nonneg : 0 ≤ f
f_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y
x : R
h : f x = 0
hf : seminormFromBounded' f x ≤ 0
⊢ seminormFromBounded' f x = 0
Module: Mathlib.Analysis.Normed.Field.Approximation

Goal state before:
pp: K : Type u_1
L : Type u_2
inst✝² : Field K
inst✝¹ : NormedField L
inst✝ : Algebra K L
hd : DenseRange ⇑(algebraMap K L)
f : L[X]
hf : f.Monic
ε : ℝ
hε : ε > 0
h : ¬f.natDegree = 0
c : ℕ → K
hc : ∀ (i : ℕ), dist (f.coeff i) ((algebraMap K L) (c i)) < ε
⊢ (C 1 * X ^ f.natDegree + ∑ i < f.natDegree, C (c i) * X ^ i).natDegree = (C 1 * X ^ f.natDegree).natDegree

Tactic:
by
        apply Polynomial.natDegree_add_eq_left_of_natDegree_lt
        simp only [map_one, one_mul, natDegree_pow, natDegree_X, mul_one]
        rw [← Nat.le_sub_one_iff_lt (Nat.pos_of_ne_zero h)]
        apply Polynomial.natDegree_sum_le_of_forall_le
        refine fun i hi ↦ (Polynomial.natDegree_C_mul_X_pow_le _ _).trans ?_
        simpa [Nat.le_sub_one_iff_lt (Nat.pos_of_ne_zero h)] using hi

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

Goal state before:
pp: case pos
R : Type u_1
inst✝ : CommRing R
f : R → ℝ
c : ℝ
f_nonneg : 0 ≤ f
f_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y
x✝ : R
hx✝ : ∀ (y : R), f (x✝ * y) = f x✝ * f y
x : R
hx : f x = 0
⊢ f x✝ * 0 ≤ f x✝

Tactic:
mul_zero

Goal state after:
case pos
R : Type u_1
inst✝ : CommRing R
f : R → ℝ
c : ℝ
f_nonneg : 0 ≤ f
f_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y
x✝ : R
hx✝ : ∀ (y : R), f (x✝ * y) = f x✝ * f y
x : R
hx : f x = 0
⊢ 0 ≤ f x✝
Module: Mathlib.Analysis.Normed.Unbundled.SmoothingSeminorm

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
μ : RingSeminorm R
x : R
hx : μ x = 0
n : ℕ
hn : 1 ≤ n
⊢ μ (x ^ n) ^ (1 / ↑n) = 0

Tactic:
have hμn : μ (x ^ n) = 0 := by
      apply le_antisymm _ (apply_nonneg μ _)
      rw [← zero_pow (pos_iff_ne_zero.mp hn), ← hx]
      exact map_pow_le_pow _ x (one_le_iff_ne_zero.mp hn)

Goal state after:
R : Type u_1
inst✝ : CommRing R
μ : RingSeminorm R
x : R
hx : μ x = 0
n : ℕ
hn : 1 ≤ n
hμn : μ (x ^ n) = 0
⊢ μ (x ^ n) ^ (1 / ↑n) = 0
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
hx : ∀ (y : R), f (x * y) = f x * f y
y : R
⊢ Tendsto (seminormFromConst_seq c f (x * y)) atTop (𝓝 (seminormFromConst' c f x * seminormFromConst' c f y))

Tactic:
rw [seminormFromConst_apply_of_isMul hf1 hc hpm hx]

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
hx : ∀ (y : R), f (x * y) = f x * f y
y : R
⊢ Tendsto (seminormFromConst_seq c f (x * y)) atTop (𝓝 (f x * seminormFromConst' c f y))
Module: Mathlib.Analysis.Normed.Unbundled.SpectralNorm

Goal state before:
pp: case pos
R : Type u_1
inst✝¹ : SeminormedRing R
inst✝ : Nontrivial R
r : R
this : (⨆ n, if n = 0 then ‖r‖ else 0) = ‖r‖
n : ℕ
hn : n = 0
⊢ ‖X.coeff 0 - (C r).coeff 0‖ ^ 1⁻¹ = ‖r‖

Tactic:
coeff_X_zero,

Goal state after:
case pos
R : Type u_1
inst✝¹ : SeminormedRing R
inst✝ : Nontrivial R
r : R
this : (⨆ n, if n = 0 then ‖r‖ else 0) = ‖r‖
n : ℕ
hn : n = 0
⊢ ‖0 - (C r).coeff 0‖ ^ 1⁻¹ = ‖r‖
Module: Mathlib.Analysis.Normed.Unbundled.SpectralNorm

Goal state before:
pp: case neg
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
⊢ f x ≤ spectralValue p

Tactic:
by_contra h_ge

Goal state after:
case neg
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
⊢ False
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‖
h_base : ‖p.coeff n‖ ^ (1 / (↑p.natDegree - ↑n)) < f x
⊢ ‖p.coeff n‖ < f x ^ (p.natDegree - n)

Tactic:
rw [← hexp, ← rpow_natCast, ← rpow_natCast]

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‖ ^ (1 / (↑p.natDegree - ↑n))) ^ ↑(p.natDegree - n) < f x ^ ↑(p.natDegree - n)
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
norm_sub_le : ∀ (a' : F), IsConjRoot L a a' → a ≠ a' → δ ≤ ‖a - a'‖
hS : S.Nonempty
a' : f.SplittingField
ha' : a' ∈ f.rootSet F ∧ a' ≠ a
⊢ a - a' ≠ 0

Tactic:
sub_ne_zero

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
norm_sub_le : ∀ (a' : F), IsConjRoot L a a' → a ≠ a' → δ ≤ ‖a - a'‖
hS : S.Nonempty
a' : f.SplittingField
ha' : a' ∈ f.rootSet F ∧ a' ≠ a
⊢ a ≠ a'
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
s : ℕ → ℕ
hs : ∀ (n : ℕ), s n ≤ n
x : R
ψ : ℕ → ℕ
hψ : ∀ (n : ℕ), 0 ≤ 1 / ↑(ψ n)
hx : 1 < μ x
n : ℕ
⊢ μ x ^ (↑(s (ψ n)) / ↑(ψ n)) ≤ μ x

Tactic:
conv_rhs => rw [← rpow_one (μ x)]

Goal state after:
case h
R : Type u_1
inst✝ : CommRing R
μ : RingSeminorm R
hμ1 : μ 1 ≤ 1
s : ℕ → ℕ
hs : ∀ (n : ℕ), s n ≤ n
x : R
ψ : ℕ → ℕ
hψ : ∀ (n : ℕ), 0 ≤ 1 / ↑(ψ n)
hx : 1 < μ x
n : ℕ
⊢ μ x ^ (↑(s (ψ n)) / ↑(ψ n)) ≤ μ x ^ 1
Module: Mathlib.Analysis.Normed.Unbundled.SmoothingSeminorm

Goal state before:
pp: 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)
⊢ ∀ b ∈ lowerBounds {a | ∀ᶠ (n : ℝ) in map (fun n ↦ μ x ^ (↑(s (ψ n)) * (1 / ↑(ψ n)))) atTop, n ≤ a}, b ≤ 1

Tactic:
intro c

Goal state after:
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
⊢ c ∈ lowerBounds {a | ∀ᶠ (n : ℝ) in map (fun n ↦ μ x ^ (↑(s (ψ n)) * (1 / ↑(ψ n)))) atTop, n ≤ a} → c ≤ 1
Module: Mathlib.Analysis.Normed.Group.HomCompletion

Goal state before:
pp: G : Type u_1
inst✝¹ : SeminormedAddCommGroup G
H : Type u_2
inst✝ : SeminormedAddCommGroup H
f : NormedAddGroupHom G H
C : ℝ
h : f.SurjectiveOnWith f.range C
hatg : Completion G
hatg_in : f.completion hatg = 0
ε : ℝ
ε_pos : 0 < ε
C' : ℝ
C'_pos : C' > 0
hC' : f.SurjectiveOnWith f.range C'
δ : ℝ
δ_pos : 0 < δ
hδ : (1 + C' * ‖f‖) * δ < ε
g : G
hg : ‖hatg - ↑g‖ < δ
g' : G
hgg' : f g' = f g
hfg : ‖g'‖ ≤ C' * ‖f g‖
⊢ f (g - g') = 0

Tactic:
map_sub,

Goal state after:
G : Type u_1
inst✝¹ : SeminormedAddCommGroup G
H : Type u_2
inst✝ : SeminormedAddCommGroup H
f : NormedAddGroupHom G H
C : ℝ
h : f.SurjectiveOnWith f.range C
hatg : Completion G
hatg_in : f.completion hatg = 0
ε : ℝ
ε_pos : 0 < ε
C' : ℝ
C'_pos : C' > 0
hC' : f.SurjectiveOnWith f.range C'
δ : ℝ
δ_pos : 0 < δ
hδ : (1 + C' * ‖f‖) * δ < ε
g : G
hg : ‖hatg - ↑g‖ < δ
g' : G
hgg' : f g' = f g
hfg : ‖g'‖ ≤ C' * ‖f g‖
⊢ f g - f g' = 0
Module: Mathlib.Analysis.Normed.Group.SemiNormedGrp.Kernels

Goal state before:
pp: V W : SemiNormedGrp
f g : V ⟶ W
c : Fork f g
⊢ (NormedAddGroupHom.compHom (Hom.hom f - Hom.hom g)) (Hom.hom c.ι) = 0

Tactic:
map_sub,

Goal state after:
V W : SemiNormedGrp
f g : V ⟶ W
c : Fork f g
⊢ (NormedAddGroupHom.compHom (Hom.hom f) - NormedAddGroupHom.compHom (Hom.hom g)) (Hom.hom c.ι) = 0