mathlib-initiative/mathlib-tactics huggingface dataset:
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)]
i : Fin 3
⊢ i ∈ Set.univ

Tactic:
by simp

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Beta

Goal state before:
pp: s : ℂ
⊢ (Gamma s)⁻¹ = s * (Gamma (s + 1))⁻¹

Tactic:
rcases ne_or_eq s 0 with (h | rfl)

Goal state after:
case inl
s : ℂ
h : s ≠ 0
⊢ (Gamma s)⁻¹ = s * (Gamma (s + 1))⁻¹

case inr
⊢ (Gamma 0)⁻¹ = 0 * (Gamma (0 + 1))⁻¹
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
i : Fin 3
⊢ i + 1 ≠ i

Tactic:
by simp

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
⊢ i + 2 ≠ i

Tactic:
by simp

Goal state after:
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
⊢ ↑t + 1 / 2 = ↑(t + 1 / 2)

Tactic:
by simp

Goal state after:
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
⊢ 2 * ↑t = ↑(2 * t)

Tactic:
by simp

Goal state after:
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
n : ℕ
⊢ ((↑(n + 1).factorial)⁻¹ • x ^ (n + 1)).snd = ((↑n.factorial)⁻¹ • x.fst ^ n) • x.snd

Tactic:
snd_smul,

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
n : ℕ
⊢ (↑(n + 1).factorial)⁻¹ • (x ^ (n + 1)).snd = ((↑n.factorial)⁻¹ • x.fst ^ n) • x.snd
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)
hc2 : ∀ (i : Fin 3), c i * r i = w' (i + 2)
hcr : (∏ i, c i) * ∏ i, r i = (∏ i, c i) * ∏ i, (1 - r i)
i : Fin 3
hi : c i = 0
hw'i1 : w' (i + 1) = 0
hw'i2 : w' (i + 2) = 0
hw'i0 : w' i = 1
hi1 : c (i + 1) * r (i + 1) = 1
hi1' : c (i + 1) * (1 - r (i + 1)) = 0
hci1 : c (i + 1) = 1
hri1 : r (i + 1) = 1
hi2 : c (i + 2) * (1 - r (i + 2)) = 1
hi2' : c (i + 2) * r (i + 2) = 0
hci2 : c (i + 2) = 1
hri2 : r (i + 2) = 0
⊢ i + 2 ∈ Finset.univ

Tactic:
by simp

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)
hc2 : ∀ (i : Fin 3), c i * r i = w' (i + 2)
hcr : (∏ i, c i) * ∏ i, r i = (∏ i, c i) * ∏ i, (1 - r i)
i : Fin 3
hi : c i = 0
hw'i1 : w' (i + 1) = 0
hw'i2 : w' (i + 2) = 0
hw'i0 : w' i = 1
hi1 : c (i + 1) * r (i + 1) = 1
hi1' : c (i + 1) * (1 - r (i + 1)) = 0
hci1 : c (i + 1) = 1
hri1 : r (i + 1) = 1
hi2 : c (i + 2) * (1 - r (i + 2)) = 1
hi2' : c (i + 2) * r (i + 2) = 0
hci2 : c (i + 2) = 1
hri2 : r (i + 2) = 0
⊢ i + 1 ∈ Finset.univ

Tactic:
by simp

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

Goal state before:
pp: 𝕜 : Type u_1
F : Type u_2
inst✝⁴ : NormedField 𝕜
inst✝³ : ProperSpace 𝕜
inst✝² : SeminormedRing F
inst✝¹ : NormedAlgebra 𝕜 F
inst✝ : NormOneClass F
x : F
⊢ Tendsto (⇑(algebraMap 𝕜 F)) (cobounded 𝕜) (cobounded F)

Tactic:
by simp

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Series

Goal state before:
pp: case e'_6
z : ℂ
this :
  HasSum
    (fun x ↦ ((z * I) ^ (2 * x.1 + ↑x.2) / ↑(2 * x.1 + ↑x.2)! + (-z * I) ^ (2 * x.1 + ↑x.2) / ↑(2 * x.1 + ↑x.2)!) / 2)
    ((NormedSpace.exp (z * I) + NormedSpace.exp (-z * I)) / 2)
k : ℕ
⊢ (z ^ 2) ^ k * (I ^ 2) ^ k / ↑(2 * k)! =
    2 * ((z ^ 2) ^ k * (I ^ 2) ^ k / ↑(2 * k)!) / 2 +
      ((z ^ 2) ^ k * (I ^ 2) ^ k * (z * I) / ↑(2 * k + 1)! + (z ^ 2) ^ k * (I ^ 2) ^ k * (-z * I) / ↑(2 * k + 1)!) / 2

Tactic:
neg_mul,

Goal state after:
case e'_6
z : ℂ
this :
  HasSum
    (fun x ↦ ((z * I) ^ (2 * x.1 + ↑x.2) / ↑(2 * x.1 + ↑x.2)! + (-z * I) ^ (2 * x.1 + ↑x.2) / ↑(2 * x.1 + ↑x.2)!) / 2)
    ((NormedSpace.exp (z * I) + NormedSpace.exp (-z * I)) / 2)
k : ℕ
⊢ (z ^ 2) ^ k * (I ^ 2) ^ k / ↑(2 * k)! =
    2 * ((z ^ 2) ^ k * (I ^ 2) ^ k / ↑(2 * k)!) / 2 +
      ((z ^ 2) ^ k * (I ^ 2) ^ k * (z * I) / ↑(2 * k + 1)! + (z ^ 2) ^ k * (I ^ 2) ^ k * -(z * I) / ↑(2 * k + 1)!) / 2
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Series

Goal state before:
pp: case e'_6
z : ℂ
this :
  HasSum
    (fun x ↦
      ((-z * I) ^ (2 * x.1 + ↑x.2) / ↑(2 * x.1 + ↑x.2)! - (z * I) ^ (2 * x.1 + ↑x.2) / ↑(2 * x.1 + ↑x.2)!) * I / 2)
    ((NormedSpace.exp (-z * I) - NormedSpace.exp (z * I)) * I / 2)
k : ℕ
⊢ (z ^ 2) ^ k * (I ^ 2) ^ k * (z * I) / ↑(2 * k + 1)! / I =
    ((z ^ 2) ^ k * (I ^ 2) ^ k * (-z * I) / ↑(2 * k + 1)! - (z ^ 2) ^ k * (I ^ 2) ^ k * (z * I) / ↑(2 * k + 1)!) * I / 2

Tactic:
neg_mul,

Goal state after:
case e'_6
z : ℂ
this :
  HasSum
    (fun x ↦
      ((-z * I) ^ (2 * x.1 + ↑x.2) / ↑(2 * x.1 + ↑x.2)! - (z * I) ^ (2 * x.1 + ↑x.2) / ↑(2 * x.1 + ↑x.2)!) * I / 2)
    ((NormedSpace.exp (-z * I) - NormedSpace.exp (z * I)) * I / 2)
k : ℕ
⊢ (z ^ 2) ^ k * (I ^ 2) ^ k * (z * I) / ↑(2 * k + 1)! / I =
    ((z ^ 2) ^ k * (I ^ 2) ^ k * -(z * I) / ↑(2 * k + 1)! - (z ^ 2) ^ k * (I ^ 2) ^ k * (z * I) / ↑(2 * k + 1)!) * I / 2
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Series

Goal state before:
pp: case e'_6
z : ℂ
this :
  HasSum
    (fun x ↦
      ((-z * I) ^ (2 * x.1 + ↑x.2) / ↑(2 * x.1 + ↑x.2)! - (z * I) ^ (2 * x.1 + ↑x.2) / ↑(2 * x.1 + ↑x.2)!) * I / 2)
    ((NormedSpace.exp (-z * I) - NormedSpace.exp (z * I)) * I / 2)
k : ℕ
⊢ (z ^ 2) ^ k * (I ^ 2) ^ k * (z * I) / ↑(2 * k + 1)! / I =
    -(2 * ((z ^ 2) ^ k * (I ^ 2) ^ k * (z * I) / ↑(2 * k + 1)!)) * I / 2

Tactic:
neg_mul,

Goal state after:
case e'_6
z : ℂ
this :
  HasSum
    (fun x ↦
      ((-z * I) ^ (2 * x.1 + ↑x.2) / ↑(2 * x.1 + ↑x.2)! - (z * I) ^ (2 * x.1 + ↑x.2) / ↑(2 * x.1 + ↑x.2)!) * I / 2)
    ((NormedSpace.exp (-z * I) - NormedSpace.exp (z * I)) * I / 2)
k : ℕ
⊢ (z ^ 2) ^ k * (I ^ 2) ^ k * (z * I) / ↑(2 * k + 1)! / I =
    -(2 * ((z ^ 2) ^ k * (I ^ 2) ^ k * (z * I) / ↑(2 * k + 1)!) * I) / 2
Module: Mathlib.Analysis.Normed.Algebra.QuaternionExponential

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

Tactic:
expSeries_even_of_imaginary hq

Goal state after:
q : ℍ
hq : q.re = 0
c s : ℝ
hc : HasSum (fun a ↦ ↑((-1) ^ a * ‖q‖ ^ (2 * a) / ↑(2 * a)!)) ↑c
hs : HasSum (fun z ↦ ((-1) ^ z * ‖q‖ ^ (2 * z + 1) / ↑(2 * z + 1)! / ‖q‖) • q) ((s / ‖q‖) • q)
n : ℕ
⊢ ↑((-1) ^ n * ‖q‖ ^ (2 * n) / ↑(2 * n)!) = ↑((-1) ^ n * ‖q‖ ^ (2 * n) / ↑(2 * n)!)
Module: Mathlib.Analysis.Normed.Algebra.QuaternionExponential

Goal state before:
pp: q : ℍ
⊢ exp q.re ^ 2 * (Real.cos ‖q.im‖ ^ 2 + Real.sin ‖q.im‖ ^ 2) = exp q.re ^ 2

Tactic:
by rw [Real.cos_sq_add_sin_sq, mul_one]

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

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
f : R → ℝ
f_neg : ∀ (x : R), f (-x) = f x
x y : R
⊢ f (-x * y) / f y = f (x * y) / f y

Tactic:
neg_mul,

Goal state after:
R : Type u_1
inst✝ : CommRing R
f : R → ℝ
f_neg : ∀ (x : R), f (-x) = f x
x y : R
⊢ f (-(x * y)) / f y = f (x * y) / f y
Module: Mathlib.Analysis.Normed.Unbundled.SeminormFromBounded

Goal state before:
pp: case neg
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
f_add : ∀ (a b : R), f (a + b) ≤ f a + f b
x y z : R
hz : ¬f z = 0
⊢ f ((x + y) * z) ≤ f (x * z) + f (y * z)

Tactic:
add_mul

Goal state after:
case neg
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
f_add : ∀ (a b : R), f (a + b) ≤ f a + f b
x y z : R
hz : ¬f z = 0
⊢ f (x * z + y * z) ≤ f (x * z) + f (y * z)
Module: Mathlib.Analysis.Normed.Unbundled.FiniteExtension

Goal state before:
pp: K : Type u_1
L : Type u_2
inst✝⁴ : NormedField K
inst✝³ : Ring L
inst✝² : Algebra K L
ι : Type u_3
inst✝¹ : Fintype ι
inst✝ : Nonempty ι
B : Basis ι K L
i : ι
hBi : B i = 1
x : K
⊢ ‖x‖ ≤ ‖if i = i then x else 0‖

Tactic:
by simp

Goal state after:
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
⊢ Tendsto (smoothingSeminormSeq μ x) atTop (𝓝 (smoothingFun μ x))

Tactic:
have h0 (n : ℕ) (hn : 1 ≤ n) : μ (x ^ n) ^ (1 / (n : ℝ)) = 0 := by
    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)
    rw [hμn, zero_rpow (one_div_cast_ne_zero (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
h0 : ∀ (n : ℕ), 1 ≤ n → μ (x ^ n) ^ (1 / ↑n) = 0
⊢ Tendsto (smoothingSeminormSeq μ x) atTop (𝓝 (smoothingFun μ x))
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
⊢ (Classical.choose ⋯) (σ 0) = 0

Tactic:
by simp

Goal state after:
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
hm : 1 ≤ m
⊢ Tendsto (fun n ↦ seminormFromConst_seq c f (x ^ m) (m * n)) atTop (𝓝 (seminormFromConst' c f (x ^ m)))

Tactic:
apply (tendsto_seminormFromConst_seq_atTop hf1 hc hpm (x ^ m)).comp
      (tendsto_atTop_atTop_of_monotone (fun _ _ hnk ↦ mul_le_mul_right hnk m) _)

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
hm : 1 ≤ m
⊢ ∀ (b : ℕ), ∃ a, b ≤ m * a
Module: Mathlib.Analysis.Normed.Unbundled.SpectralNorm

Goal state before:
pp: case refine_1.inr.inl
R : Type u_1
inst✝ : NormedDivisionRing R
P : R[X]
hP : P.Monic
h : iSup (spectralValueTerms P) ≤ 1
n : ℕ
hPn : P.natDegree = n
⊢ ‖P.coeff n‖ ≤ 1

Tactic:
rw [← hPn, hP.coeff_natDegree, norm_one]

Goal state after:
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)
⊢ CompleteSpace E

Tactic:
apply Metric.complete_of_cauchySeq_tendsto
  intro u hu
  obtain ⟨f, hf₁, hf₂⟩ := Metric.exists_subseq_summable_dist_of_cauchySeq u hu
  simp only [dist_eq_norm] at hf₂
  let v n := u (f (n + 1)) - u (f n)
  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
  obtain ⟨a, ha⟩ := h v hf₂
  refine ⟨a + u (f 0), ?_⟩
  refine tendsto_nhds_of_cauchySeq_of_subseq hu hf₁.tendsto_atTop ?_
  rw [hv_sum] at ha
  have h₁ : Tendsto (fun n => u (f n) - u (f 0) + u (f 0)) atTop (𝓝 (a + u (f 0))) :=
    Tendsto.add_const _ ha
  simpa only [sub_add_cancel] using! h₁

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

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
μ : RingSeminorm R
hμ1 : μ 1 ≤ 1
⊢ smoothingFun μ 1 ≤ 1

Tactic:
apply le_of_tendsto (tendsto_smoothingFun_of_map_one_le_one μ hμ1 (1 : R))

Goal state after:
R : Type u_1
inst✝ : CommRing R
μ : RingSeminorm R
hμ1 : μ 1 ≤ 1
⊢ ∀ᶠ (c : ℕ) in atTop, smoothingSeminormSeq μ 1 c ≤ 1
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
⊢ 0 < 1 / 2

Tactic:
by simp

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

Goal state before:
pp: case inr
K : Type u_2
inst✝³ : NormedField K
L : Type u_3
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : DecidableEq L
f : AlgebraNorm K L
hf_pm : IsPowMul ⇑f
hf_na : IsNonarchimedean ⇑f
hf1 : f 1 = 1
p : K[X]
s : Multiset L
hp : (mapAlg K L) p = (Multiset.map (fun a ↦ X - C a) s).prod
h_le : 0 ≤ ⨆ x, if x ∈ s then f x else 0
m : ℕ
hm : m < p.natDegree
h : 0 < ↑p.natDegree - ↑m
hps : s.card = p.natDegree
hc : ‖p.coeff m‖ = f (((mapAlg K L) p).coeff m)
hn1 : (-1) ^ (s.card - m) = -1
⊢ f (-1 * s.esymm (s.card - m)) = f (s.esymm (s.card - m))

Tactic:
neg_mul,

Goal state after:
case inr
K : Type u_2
inst✝³ : NormedField K
L : Type u_3
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : DecidableEq L
f : AlgebraNorm K L
hf_pm : IsPowMul ⇑f
hf_na : IsNonarchimedean ⇑f
hf1 : f 1 = 1
p : K[X]
s : Multiset L
hp : (mapAlg K L) p = (Multiset.map (fun a ↦ X - C a) s).prod
h_le : 0 ≤ ⨆ x, if x ∈ s then f x else 0
m : ℕ
hm : m < p.natDegree
h : 0 < ↑p.natDegree - ↑m
hps : s.card = p.natDegree
hc : ‖p.coeff m‖ = f (((mapAlg K L) p).coeff m)
hn1 : (-1) ^ (s.card - m) = -1
⊢ f (-(1 * s.esymm (s.card - m))) = f (s.esymm (s.card - m))
Module: Mathlib.Analysis.Normed.Unbundled.SpectralNorm

Goal state before:
pp: case pos
K : Type u_2
inst✝³ : NormedField K
L : Type u_3
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : DecidableEq L
f : AlgebraNorm K L
hf_pm : IsPowMul ⇑f
hf_na : IsNonarchimedean ⇑f
hf1 : f 1 = 1
p : K[X]
s : Multiset L
hp : (mapAlg K L) p = (Multiset.map (fun a ↦ X - C a) s).prod
h_le : 0 ≤ ⨆ x, if x ∈ s then f x else 0
m : ℕ
hm : m < p.natDegree
h✝ : 0 < ↑p.natDegree - ↑m
hps : s.card = p.natDegree
hc : ‖p.coeff m‖ = f (((mapAlg K L) p).coeff m)
h : f ((-1) ^ (s.card - m) * s.esymm (s.card - m)) = f (s.esymm (s.card - m))
⊢ f (Multiset.map prod (powersetCard (s.card - m) s)).sum ≤ (⨆ x, if x ∈ s then f x else 0) ^ (p.natDegree - m)

Tactic:
obtain ⟨t, ht_card, hts, ht_ge⟩ : ∃ t : Multiset L, card t = card s - m ∧
          (∀ x : L, x ∈ t → x ∈ s) ∧ f (map prod (powersetCard (card s - m) s)).sum ≤ f t.prod :=
        hf_na.multiset_powerset_image_add s m

Goal state after:
case pos
K : Type u_2
inst✝³ : NormedField K
L : Type u_3
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : DecidableEq L
f : AlgebraNorm K L
hf_pm : IsPowMul ⇑f
hf_na : IsNonarchimedean ⇑f
hf1 : f 1 = 1
p : K[X]
s : Multiset L
hp : (mapAlg K L) p = (Multiset.map (fun a ↦ X - C a) s).prod
h_le : 0 ≤ ⨆ x, if x ∈ s then f x else 0
m : ℕ
hm : m < p.natDegree
h✝ : 0 < ↑p.natDegree - ↑m
hps : s.card = p.natDegree
hc : ‖p.coeff m‖ = f (((mapAlg K L) p).coeff m)
h : f ((-1) ^ (s.card - m) * s.esymm (s.card - m)) = f (s.esymm (s.card - m))
t : Multiset L
ht_card : t.card = s.card - m
hts : ∀ x ∈ t, x ∈ s
ht_ge : f (Multiset.map prod (powersetCard (s.card - m) s)).sum ≤ f t.prod
⊢ f (Multiset.map prod (powersetCard (s.card - m) s)).sum ≤ (⨆ x, if x ∈ s then f x else 0) ^ (p.natDegree - m)
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
this✝ : CauchySeq s
g : G
hg : Tendsto s atTop (𝓝 g)
n : ℕ
hnorm₀ : ‖u 0‖ ≤ C * b 0 + C * ‖h‖
this : ∑ k ∈ range (n + 1), C * b k ≤ ε * ‖h‖
⊢ C * ‖h‖ + ∑ k ∈ range (n + 1), C * b k ≤ (C + ε) * ‖h‖

Tactic:
add_mul

Goal state after:
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
this✝ : CauchySeq s
g : G
hg : Tendsto s atTop (𝓝 g)
n : ℕ
hnorm₀ : ‖u 0‖ ≤ C * b 0 + C * ‖h‖
this : ∑ k ∈ range (n + 1), C * b k ≤ ε * ‖h‖
⊢ C * ‖h‖ + ∑ k ∈ range (n + 1), C * b k ≤ C * ‖h‖ + ε * ‖h‖
Module: Mathlib.Analysis.Normed.Group.HomCompletion

Goal state before:
pp: G : Type u_1
inst✝² : SeminormedAddCommGroup G
H : Type u_2
inst✝¹ : SeminormedAddCommGroup H
K : Type u_3
inst✝ : SeminormedAddCommGroup K
⊢ ∀ (v : G), ‖↑v‖ ≤ 1 * ‖v‖

Tactic:
by simp

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

Goal state before:
pp: M : SemiNormedGrp
r : M.carrier
⊢ (ConcreteCategory.hom (𝟙 M)) r = r

Tactic:
by simp

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

Goal state before:
pp: M N O : SemiNormedGrp
f : M ⟶ N
g : N ⟶ O
r : M.carrier
⊢ (ConcreteCategory.hom (f ≫ g)) r = (ConcreteCategory.hom g) ((ConcreteCategory.hom f) r)

Tactic:
by simp

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

Goal state before:
pp: M : SemiNormedGrp₁
r : M.carrier
⊢ ↑(Hom.hom (𝟙 M)) r = r

Tactic:
by simp

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

Goal state before:
pp: M N O : SemiNormedGrp₁
f : M ⟶ N
g : N ⟶ O
r : M.carrier
⊢ ↑(Hom.hom (f ≫ g)) r = ↑(Hom.hom g) (↑(Hom.hom f) r)

Tactic:
by simp

Goal state after: