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 : ∃ᶠ (a : k × V) in atTop ×ˢ 𝓝 v, a.1 • a.2 +ᵥ p ∈ s
⊢ ∃ᶠ (x : k × V) in atTop ×ˢ 𝓝 v, c • x.2 +ᵥ p ∈ s

Tactic:
]

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 : Set V
c : k
v p : V
hs₁ : StarConvex k p s
hs₂ : IsClosed[inst✝²] s
hc : 0 ≤ c
hv : ∃ᶠ (a : k × V) in atTop ×ˢ 𝓝 v, a.1 • a.2 +ᵥ p ∈ s
⊢ ∃ᶠ (x : k × V) in atTop ×ˢ 𝓝 v, c • x.2 +ᵥ p ∈ s
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Beta

Goal state before:
pp: case e'_3
s : ℂ
hs : 0 < s.re
f : ℕ → ℝ → ℂ := fun n ↦ (Ioc 0 ↑n).indicator fun x ↦ ↑((1 - x / ↑n) ^ n) * ↑x ^ (s - 1)
f_ible : ∀ (n : ℕ), Integrable (f n) (volume.restrict (Ioi 0))
f_tends : ∀ x ∈ Ioi 0, Tendsto (fun n ↦ f n x) atTop (𝓝 (↑(rexp (-x)) * ↑x ^ (s - 1)))
n : ℕ
⊢ ∫ (x : ℝ) in Ioc 0 ↑n, ↑((1 - x / ↑n) ^ n) * ↑x ^ (s - 1) ∂volume =
    ∫ (x : ℝ) in Ioc 0 ↑n, ↑((1 - x / ↑n) ^ n) * ↑x ^ (s - 1) ∂volume

Tactic:
]

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

Goal state before:
pp: case refine_3
k : Type u_1
V : Type u_2
P : Type u_3
ι : Type u_4
inst✝³ : Ring k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
p : ι → P
hp : AffineIndependent k p
s : Set ι
fs : ↑s → Finset ι
hfs : ∀ (i : ↑s), ↑i ∈ fs i
w : ↑s → ι → k
hw : ∀ (i : ↑s), ∑ j ∈ fs i, w i j = 1
p' : P
hp' : ∀ (i : ↑s), p' ∈ line[k, p ↑i, (Finset.affineCombination k (fs i) p) (w i)]
hp'' : ∀ (i : ↑s), ∃ r, (Finset.affineCombination k (fs i) p) ((AffineMap.lineMap (Pi.single (↑i) 1) (w i)) r) = p'
i' : ι
hi' : i' ∈ s
ri' : k
hri' :
  (Finset.affineCombination k (fs ⟨i', hi'⟩) p) ((AffineMap.lineMap (Pi.single (↑⟨i', hi'⟩) 1) (w ⟨i', hi'⟩)) ri') = p'
w' : ι → k := (AffineMap.lineMap (Pi.single i' 1) (w ⟨i', hi'⟩)) ri'
i : ↑s
r : k
hr :
  (Finset.affineCombination k (fs i) p) ((AffineMap.lineMap (Pi.single (↑i) 1) (w i)) r) =
    (Finset.affineCombination k (fs ⟨i', hi'⟩) p) ((AffineMap.lineMap (Pi.single (↑⟨i', hi'⟩) 1) (w ⟨i', hi'⟩)) ri')
⊢ ∀ (j : ι), r * (↑(fs i) \ {↑i}).indicator (w i) j = (↑(fs ⟨i', hi'⟩) \ {↑i}).indicator w' j

Tactic:
]

Goal state after:
case refine_3
k : Type u_1
V : Type u_2
P : Type u_3
ι : Type u_4
inst✝³ : Ring k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
p : ι → P
hp : AffineIndependent k p
s : Set ι
fs : ↑s → Finset ι
hfs : ∀ (i : ↑s), ↑i ∈ fs i
w : ↑s → ι → k
hw : ∀ (i : ↑s), ∑ j ∈ fs i, w i j = 1
p' : P
hp' : ∀ (i : ↑s), p' ∈ line[k, p ↑i, (Finset.affineCombination k (fs i) p) (w i)]
hp'' : ∀ (i : ↑s), ∃ r, (Finset.affineCombination k (fs i) p) ((AffineMap.lineMap (Pi.single (↑i) 1) (w i)) r) = p'
i' : ι
hi' : i' ∈ s
ri' : k
hri' :
  (Finset.affineCombination k (fs ⟨i', hi'⟩) p) ((AffineMap.lineMap (Pi.single (↑⟨i', hi'⟩) 1) (w ⟨i', hi'⟩)) ri') = p'
w' : ι → k := (AffineMap.lineMap (Pi.single i' 1) (w ⟨i', hi'⟩)) ri'
i : ↑s
r : k
hr :
  (Finset.affineCombination k (fs i) p) ((AffineMap.lineMap (Pi.single (↑i) 1) (w i)) r) =
    (Finset.affineCombination k (fs ⟨i', hi'⟩) p) ((AffineMap.lineMap (Pi.single (↑⟨i', hi'⟩) 1) (w ⟨i', hi'⟩)) ri')
⊢ ∀ (j : ι), r * (↑(fs i) \ {↑i}).indicator (w i) j = (↑(fs ⟨i', hi'⟩) \ {↑i}).indicator w' j
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Beta

Goal state before:
pp: s : ℂ
hs : 0 < s.re
f : ℕ → ℝ → ℂ := fun n ↦ (Ioc 0 ↑n).indicator fun x ↦ ↑((1 - x / ↑n) ^ n) * ↑x ^ (s - 1)
f_ible : ∀ (n : ℕ), Integrable (f n) (volume.restrict (Ioi 0))
f_tends : ∀ x ∈ Ioi 0, Tendsto (fun n ↦ f n x) atTop (𝓝 (↑(rexp (-x)) * ↑x ^ (s - 1)))
n : ℕ
⊢ ∀ᵐ (x : ℝ), x ∈ Ioi 0 → ‖f n x‖ ≤ rexp (-x) * x ^ (s.re - 1)

Tactic:
]

Goal state after:
s : ℂ
hs : 0 < s.re
f : ℕ → ℝ → ℂ := fun n ↦ (Ioc 0 ↑n).indicator fun x ↦ ↑((1 - x / ↑n) ^ n) * ↑x ^ (s - 1)
f_ible : ∀ (n : ℕ), Integrable (f n) (volume.restrict (Ioi 0))
f_tends : ∀ x ∈ Ioi 0, Tendsto (fun n ↦ f n x) atTop (𝓝 (↑(rexp (-x)) * ↑x ^ (s - 1)))
n : ℕ
⊢ ∀ᵐ (x : ℝ), x ∈ Ioi 0 → ‖f n x‖ ≤ rexp (-x) * x ^ (s.re - 1)
Module: Mathlib.Topology.Algebra.AsymptoticCone

Goal state before:
pp: case inr
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
hs : Convex k s
this : ∀ {s : Set V}, Convex k s → IsClosed[inst✝²] s → Convex k (asymptoticCone k s)
hs' : ¬IsClosed[inst✝²] s
⊢ Convex k (asymptoticCone k (closure[inst✝²] s))

Tactic:
]

Goal state after:
case inr
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
hs : Convex k s
this : ∀ {s : Set V}, Convex k s → IsClosed[inst✝²] s → Convex k (asymptoticCone k s)
hs' : ¬IsClosed[inst✝²] s
⊢ Convex k (asymptoticCone k (closure[inst✝²] s))
Module: Mathlib.Topology.Algebra.AsymptoticCone

Goal state before:
pp: case inl
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
hs : Convex k ∅
hs' : IsClosed[inst✝²] ∅
⊢ Convex k ∅

Tactic:
]

Goal state after:
case inl
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
hs : Convex k ∅
hs' : IsClosed[inst✝²] ∅
⊢ Convex k ∅
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Beta

Goal state before:
pp: case inl
s : ℂ
hs : 0 < s.re
f : ℕ → ℝ → ℂ := fun n ↦ (Ioc 0 ↑n).indicator fun x ↦ ↑((1 - x / ↑n) ^ n) * ↑x ^ (s - 1)
f_ible : ∀ (n : ℕ), Integrable (f n) (volume.restrict (Ioi 0))
f_tends : ∀ x ∈ Ioi 0, Tendsto (fun n ↦ f n x) atTop (𝓝 (↑(rexp (-x)) * ↑x ^ (s - 1)))
n : ℕ
x : ℝ
hx : 0 < x
hxn : ↑n < x
⊢ 0 ≤ x ^ (s.re - 1)

Tactic:
]

Goal state after:
case inl
s : ℂ
hs : 0 < s.re
f : ℕ → ℝ → ℂ := fun n ↦ (Ioc 0 ↑n).indicator fun x ↦ ↑((1 - x / ↑n) ^ n) * ↑x ^ (s - 1)
f_ible : ∀ (n : ℕ), Integrable (f n) (volume.restrict (Ioi 0))
f_tends : ∀ x ∈ Ioi 0, Tendsto (fun n ↦ f n x) atTop (𝓝 (↑(rexp (-x)) * ↑x ^ (s - 1)))
n : ℕ
x : ℝ
hx : 0 < x
hxn : ↑n < x
⊢ 0 ≤ x ^ (s.re - 1)
Module: Mathlib.Topology.Algebra.AsymptoticCone

Goal state before:
pp: case inr
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
⊢ ∃ᶠ (p : V) in asymptoticNhds k V (a • v + b • u), p ∈ s

Tactic:
]

Goal state after:
case inr
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
⊢ ∃ᶠ (p : V) in asymptoticNhds k V (a • v + b • u), p ∈ s
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Beta

Goal state before:
pp: case inr
s : ℂ
hs : 0 < s.re
f : ℕ → ℝ → ℂ := fun n ↦ (Ioc 0 ↑n).indicator fun x ↦ ↑((1 - x / ↑n) ^ n) * ↑x ^ (s - 1)
f_ible : ∀ (n : ℕ), Integrable (f n) (volume.restrict (Ioi 0))
f_tends : ∀ x ∈ Ioi 0, Tendsto (fun n ↦ f n x) atTop (𝓝 (↑(rexp (-x)) * ↑x ^ (s - 1)))
n : ℕ
x : ℝ
hx : 0 < x
hxn : x ≤ ↑n
⊢ (1 - x / ↑n) ^ n * x ^ (s.re - 1) ≤ rexp (-x) * x ^ (s.re - 1)

Tactic:
]

Goal state after:
case inr
s : ℂ
hs : 0 < s.re
f : ℕ → ℝ → ℂ := fun n ↦ (Ioc 0 ↑n).indicator fun x ↦ ↑((1 - x / ↑n) ^ n) * ↑x ^ (s - 1)
f_ible : ∀ (n : ℕ), Integrable (f n) (volume.restrict (Ioi 0))
f_tends : ∀ x ∈ Ioi 0, Tendsto (fun n ↦ f n x) atTop (𝓝 (↑(rexp (-x)) * ↑x ^ (s - 1)))
n : ℕ
x : ℝ
hx : 0 < x
hxn : x ≤ ↑n
⊢ (1 - x / ↑n) ^ n * x ^ (s.re - 1) ≤ rexp (-x) * x ^ (s.re - 1)
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Beta

Goal state before:
pp: case zero
s : ℂ
hs : 0 < s.re
⊢ Tendsto s.GammaSeq atTop (𝓝 (Gamma s))

Tactic:
]

Goal state after:
case zero
s : ℂ
hs : 0 < s.re
⊢ Tendsto s.GammaSeq atTop (𝓝 (Gamma s))
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
h₁ : c • v +ᵥ p ∈ s
h₂ : c • u +ᵥ p ∈ s
⊢ (c * a) • v + (c * b) • u ∈ [c • v -[k] c • u]

Tactic:
]

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
h₁ : c • v +ᵥ p ∈ s
h₂ : c • u +ᵥ p ∈ s
⊢ (c * a) • v + (c * b) • u ∈ [c • v -[k] c • u]
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Beta

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

Tactic:
]

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

Goal state before:
pp: 𝕜 : Type u_1
V : Type u_3
W : Type u_4
inst✝⁴ : SeminormedAddCommGroup V
inst✝³ : SeminormedAddCommGroup W
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedSpace 𝕜 V
inst✝ : NormedSpace 𝕜 W
f : V →ᴬ[𝕜] W
h : f 0 = 0
⊢ max ‖f 0‖ ‖f.contLinear‖ = max ‖f 0‖ ‖f.contLinear‖

Tactic:
]

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

Goal state before:
pp: m : ℕ
IH : ∀ (s : ℂ), ⌊1 - s.re⌋₊ = m → Tendsto s.GammaSeq atTop (𝓝 (Gamma s))
s : ℂ
hs : ↑(m + 1) ≤ 1 - s.re ∧ 1 - s.re < ↑(m + 1) + 1
hsne : s ≠ 0
this : s.re ≤ -↑m
⊢ 0 ≤ 1 - (s + 1).re

Tactic:
simpa using (show s.re ≤ 0 by grind)

Goal state after:
Module: Mathlib.Analysis.Normed.Affine.ContinuousAffineMap

Goal state before:
pp: 𝕜 : Type u_1
V : Type u_3
W : Type u_4
inst✝⁴ : SeminormedAddCommGroup V
inst✝³ : SeminormedAddCommGroup W
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedSpace 𝕜 V
inst✝ : NormedSpace 𝕜 W
f : V →ᴬ[𝕜] W
h : f 0 = 0
⊢ max 0 ‖f.contLinear‖ = max 0 ‖f.contLinear‖

Tactic:
]

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

Goal state before:
pp: case succ.inr
m : ℕ
IH : ∀ (s : ℂ), ⌊1 - s.re⌋₊ = m → Tendsto s.GammaSeq atTop (𝓝 (Gamma s))
s : ℂ
hs : ↑(m + 1) ≤ 1 - s.re ∧ 1 - s.re < ↑(m + 1) + 1
hsne : s ≠ 0
this : s.re ≤ -↑m
⊢ ↑m ≤ 1 - (s + 1).re ∧ 1 - (s + 1).re < ↑m + 1

Tactic:
]

Goal state after:
case succ.inr
m : ℕ
IH : ∀ (s : ℂ), ⌊1 - s.re⌋₊ = m → Tendsto s.GammaSeq atTop (𝓝 (Gamma s))
s : ℂ
hs : ↑(m + 1) ≤ 1 - s.re ∧ 1 - s.re < ↑(m + 1) + 1
hsne : s ≠ 0
this : s.re ≤ -↑m
⊢ ↑m ≤ 1 - (s + 1).re ∧ 1 - (s + 1).re < ↑m + 1
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Beta

Goal state before:
pp: case succ.inr
m : ℕ
s : ℂ
IH : Tendsto (s + 1).GammaSeq atTop (𝓝 (s * Gamma s))
hs : ⌊1 - s.re⌋₊ = m + 1
hsne : s ≠ 0
⊢ Tendsto s.GammaSeq atTop (𝓝 (Gamma s))

Tactic:
]

Goal state after:
case succ.inr
m : ℕ
s : ℂ
IH : Tendsto (s + 1).GammaSeq atTop (𝓝 (s * Gamma s))
hs : ⌊1 - s.re⌋₊ = m + 1
hsne : s ≠ 0
⊢ Tendsto s.GammaSeq atTop (𝓝 (Gamma s))
Module: Mathlib.Analysis.Normed.Affine.ContinuousAffineMap

Goal state before:
pp: 𝕜 : Type u_1
V : Type u_3
W : Type u_4
W₂ : Type u_5
inst✝⁶ : SeminormedAddCommGroup V
inst✝⁵ : SeminormedAddCommGroup W
inst✝⁴ : SeminormedAddCommGroup W₂
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedSpace 𝕜 V
inst✝¹ : NormedSpace 𝕜 W
inst✝ : NormedSpace 𝕜 W₂
f : V →ᴬ[𝕜] W
g : W₂ →ᴬ[𝕜] V
⊢ ‖(f.comp g) 0‖ ≤ ‖f‖ * ‖g‖ + ‖f 0‖ ∧ ‖(f.comp g).contLinear‖ ≤ ‖f‖ * ‖g‖ + ‖f 0‖

Tactic:
]

Goal state after:
𝕜 : Type u_1
V : Type u_3
W : Type u_4
W₂ : Type u_5
inst✝⁶ : SeminormedAddCommGroup V
inst✝⁵ : SeminormedAddCommGroup W
inst✝⁴ : SeminormedAddCommGroup W₂
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedSpace 𝕜 V
inst✝¹ : NormedSpace 𝕜 W
inst✝ : NormedSpace 𝕜 W₂
f : V →ᴬ[𝕜] W
g : W₂ →ᴬ[𝕜] V
⊢ ‖(f.comp g) 0‖ ≤ ‖f‖ * ‖g‖ + ‖f 0‖ ∧ ‖(f.comp g).contLinear‖ ≤ ‖f‖ * ‖g‖ + ‖f 0‖
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Beta

Goal state before:
pp: case succ.inr
m : ℕ
s : ℂ
IH : Tendsto (s + 1).GammaSeq atTop (𝓝 (s * Gamma s))
hs : ⌊1 - s.re⌋₊ = m + 1
hsne : s ≠ 0
this : Tendsto s.GammaSeq atTop (𝓝 (Gamma s))
⊢ Tendsto s.GammaSeq atTop (𝓝 (Gamma s))

pp: m : ℕ
s : ℂ
IH : Tendsto (s + 1).GammaSeq atTop (𝓝 (s * Gamma s))
hs : ⌊1 - s.re⌋₊ = m + 1
hsne : s ≠ 0
this : Tendsto (fun n ↦ s.GammaSeq n * (↑n / (↑n + 1 + s))) atTop (𝓝 (Gamma s * 1))
⊢ Tendsto (fun n ↦ ↑n / (↑n + 1 + s)) atTop (𝓝 1)

Tactic:
]

Goal state after:
case succ.inr
m : ℕ
s : ℂ
IH : Tendsto (s + 1).GammaSeq atTop (𝓝 (s * Gamma s))
hs : ⌊1 - s.re⌋₊ = m + 1
hsne : s ≠ 0
this : Tendsto s.GammaSeq atTop (𝓝 (Gamma s))
⊢ Tendsto s.GammaSeq atTop (𝓝 (Gamma s))

m : ℕ
s : ℂ
IH : Tendsto (s + 1).GammaSeq atTop (𝓝 (s * Gamma s))
hs : ⌊1 - s.re⌋₊ = m + 1
hsne : s ≠ 0
this : Tendsto (fun n ↦ s.GammaSeq n * (↑n / (↑n + 1 + s))) atTop (𝓝 (Gamma s * 1))
⊢ Tendsto (fun n ↦ ↑n / (↑n + 1 + s)) atTop (𝓝 1)
Module: Mathlib.Analysis.Normed.Affine.ContinuousAffineMap

Goal state before:
pp: 𝕜 : Type u_1
V : Type u_3
W : Type u_4
W₂ : Type u_5
inst✝⁶ : SeminormedAddCommGroup V
inst✝⁵ : SeminormedAddCommGroup W
inst✝⁴ : SeminormedAddCommGroup W₂
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedSpace 𝕜 V
inst✝¹ : NormedSpace 𝕜 W
inst✝ : NormedSpace 𝕜 W₂
f : V →ᴬ[𝕜] W
g : W₂ →ᴬ[𝕜] V
⊢ ‖(⇑f.contLinear + Function.const V (f 0)) (g 0)‖ = ‖f.contLinear (g 0) + (⇑f.contLinear + Function.const V (f 0)) 0‖

Tactic:
]

Goal state after:
𝕜 : Type u_1
V : Type u_3
W : Type u_4
W₂ : Type u_5
inst✝⁶ : SeminormedAddCommGroup V
inst✝⁵ : SeminormedAddCommGroup W
inst✝⁴ : SeminormedAddCommGroup W₂
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedSpace 𝕜 V
inst✝¹ : NormedSpace 𝕜 W
inst✝ : NormedSpace 𝕜 W₂
f : V →ᴬ[𝕜] W
g : W₂ →ᴬ[𝕜] V
⊢ ‖(⇑f.contLinear + Function.const V (f 0)) (g 0)‖ = ‖f.contLinear (g 0) + (⇑f.contLinear + Function.const V (f 0)) 0‖
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
hs : Convex k s
hc : 0 ≤ c
hp : s ∈ 𝓝 p
hv : ∃ᶠ (a : k × V) in atTop ×ˢ 𝓝 v, a.1 • a.2 +ᵥ c • v +ᵥ p ∈ s
⊢ c • v +ᵥ p ∈ s

Tactic:
]

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 : Set V
c : k
v p : V
hs : Convex k s
hc : 0 ≤ c
hp : s ∈ 𝓝 p
hv : ∃ᶠ (a : k × V) in atTop ×ˢ 𝓝 v, a.1 • a.2 +ᵥ c • v +ᵥ p ∈ s
⊢ c • v +ᵥ p ∈ s
Module: Mathlib.Analysis.Normed.Affine.ContinuousAffineMap

Goal state before:
pp: 𝕜 : Type u_1
V : Type u_3
W : Type u_4
W₂ : Type u_5
inst✝⁶ : SeminormedAddCommGroup V
inst✝⁵ : SeminormedAddCommGroup W
inst✝⁴ : SeminormedAddCommGroup W₂
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedSpace 𝕜 V
inst✝¹ : NormedSpace 𝕜 W
inst✝ : NormedSpace 𝕜 W₂
f : V →ᴬ[𝕜] W
g : W₂ →ᴬ[𝕜] V
⊢ ‖f.contLinear‖ * ‖g 0‖ + ‖f 0‖ ≤ ‖f.contLinear‖ * ‖g 0‖ + ‖f 0‖

Tactic:
]

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

Goal state before:
pp: z : ℂ
n : ℕ
hn : n ≠ 0
aux : ∀ (a b c d : ℂ), a * b * (c * d) = a * c * (b * d)
⊢ ↑n ^ z * ↑n ^ (1 - z) * ↑n ! ^ 2 /
      ((∏ j ∈ Finset.range (n + 1), (z + ↑j)) * ∏ j ∈ Finset.range (n + 1), (1 - z + ↑j)) =
    ↑n / (↑n + 1 - z) * (1 / (z * ∏ j ∈ Finset.range n, (1 - z ^ 2 / (↑j + 1) ^ 2)))

Tactic:
]

Goal state after:
z : ℂ
n : ℕ
hn : n ≠ 0
aux : ∀ (a b c d : ℂ), a * b * (c * d) = a * c * (b * d)
⊢ ↑n ^ z * ↑n ^ (1 - z) * ↑n ! ^ 2 /
      ((∏ j ∈ Finset.range (n + 1), (z + ↑j)) * ∏ j ∈ Finset.range (n + 1), (1 - z + ↑j)) =
    ↑n / (↑n + 1 - z) * (1 / (z * ∏ j ∈ Finset.range n, (1 - z ^ 2 / (↑j + 1) ^ 2)))
Module: Mathlib.Analysis.Normed.Affine.ContinuousAffineMap

Goal state before:
pp: 𝕜 : Type u_1
V : Type u_3
W : Type u_4
W₂ : Type u_5
inst✝⁶ : SeminormedAddCommGroup V
inst✝⁵ : SeminormedAddCommGroup W
inst✝⁴ : SeminormedAddCommGroup W₂
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedSpace 𝕜 V
inst✝¹ : NormedSpace 𝕜 W
inst✝ : NormedSpace 𝕜 W₂
f : V →ᴬ[𝕜] W
g : W₂ →ᴬ[𝕜] V
⊢ ‖f‖ * ‖g‖ + ‖f 0‖ ≤ ‖f‖ * ‖g‖ + ‖f 0‖

Tactic:
]

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

Goal state before:
pp: z : ℂ
n : ℕ
hn : n ≠ 0
aux : ∀ (a b c d : ℂ), a * b * (c * d) = a * c * (b * d)
⊢ ↑n = ↑n

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.Normed.Affine.ContinuousAffineMap

Goal state before:
pp: 𝕜 : Type u_1
V : Type u_3
W : Type u_4
W₂ : Type u_5
inst✝⁶ : SeminormedAddCommGroup V
inst✝⁵ : SeminormedAddCommGroup W
inst✝⁴ : SeminormedAddCommGroup W₂
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedSpace 𝕜 V
inst✝¹ : NormedSpace 𝕜 W
inst✝ : NormedSpace 𝕜 W₂
f : V →ᴬ[𝕜] W
g : W₂ →ᴬ[𝕜] V
⊢ ‖f‖ * ‖g‖ ≤ ‖f‖ * ‖g‖

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.Normed.Affine.ContinuousAffineMap

Goal state before:
pp: 𝕜 : Type u_1
V : Type u_3
W : Type u_4
W₂ : Type u_5
inst✝⁶ : SeminormedAddCommGroup V
inst✝⁵ : SeminormedAddCommGroup W
inst✝⁴ : SeminormedAddCommGroup W₂
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedSpace 𝕜 V
inst✝¹ : NormedSpace 𝕜 W
inst✝ : NormedSpace 𝕜 W₂
f : V →ᴬ[𝕜] W
g : W₂ →ᴬ[𝕜] V
⊢ 0 ≤ ‖f 0‖

Tactic:
]

Goal state after:
𝕜 : Type u_1
V : Type u_3
W : Type u_4
W₂ : Type u_5
inst✝⁶ : SeminormedAddCommGroup V
inst✝⁵ : SeminormedAddCommGroup W
inst✝⁴ : SeminormedAddCommGroup W₂
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedSpace 𝕜 V
inst✝¹ : NormedSpace 𝕜 W
inst✝ : NormedSpace 𝕜 W₂
f : V →ᴬ[𝕜] W
g : W₂ →ᴬ[𝕜] V
⊢ 0 ≤ ‖f 0‖
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Beta

Goal state before:
pp: z : ℂ
n : ℕ
hn : n ≠ 0
aux : ∀ (a b c d : ℂ), a * b * (c * d) = a * c * (b * d)
this : ↑n ^ z * ↑n ^ (1 - z) = ↑n
⊢ ↑n * ↑n ! ^ 2 / ((∏ x ∈ Finset.range n, (z + ↑(x + 1)) * (1 - z + ↑x)) * (z * (↑n + 1 - z))) =
    ↑n / (↑n + 1 - z) * (1 / (z * ∏ j ∈ Finset.range n, (1 - z ^ 2 / (↑j + 1) ^ 2)))

Tactic:
]

Goal state after:
z : ℂ
n : ℕ
hn : n ≠ 0
aux : ∀ (a b c d : ℂ), a * b * (c * d) = a * c * (b * d)
this : ↑n ^ z * ↑n ^ (1 - z) = ↑n
⊢ ↑n * ↑n ! ^ 2 / ((∏ x ∈ Finset.range n, (z + ↑(x + 1)) * (1 - z + ↑x)) * (z * (↑n + 1 - z))) =
    ↑n / (↑n + 1 - z) * (1 / (z * ∏ j ∈ Finset.range n, (1 - z ^ 2 / (↑j + 1) ^ 2)))
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Beta

Goal state before:
pp: z : ℂ
n : ℕ
hn : n ≠ 0
aux : ∀ (a b c d : ℂ), a * b * (c * d) = a * c * (b * d)
this : ↑n ^ z * ↑n ^ (1 - z) = ↑n
j : ℕ
⊢ j.succ ≠ 0

Tactic:
]

Goal state after:
z : ℂ
n : ℕ
hn : n ≠ 0
aux : ∀ (a b c d : ℂ), a * b * (c * d) = a * c * (b * d)
this : ↑n ^ z * ↑n ^ (1 - z) = ↑n
j : ℕ
⊢ j.succ ≠ 0
Module: Mathlib.Analysis.Normed.Affine.ContinuousAffineMap

Goal state before:
pp: 𝕜 : Type u_1
R : Type u_2
V : Type u_3
W : Type u_4
inst✝⁸ : SeminormedAddCommGroup V
inst✝⁷ : SeminormedAddCommGroup W
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedSpace 𝕜 V
inst✝⁴ : NormedSpace 𝕜 W
inst✝³ : Ring R
inst✝² : Module R W
inst✝¹ : ContinuousConstSMul R W
inst✝ : SMulCommClass 𝕜 R W
p : W × (V →L[𝕜] W)
⊢ p.2 = p.2

Tactic:
]

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
e : PE ≃ᵢ PE
hx : e x = x
right✝ : e y = y
⊢ dist x (e z) + dist x z = dist x (e z) + dist x z

Tactic:
]

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
e : PE ≃ᵢ PE
hx : e x = x
right✝ : e y = y
⊢ dist x z + dist x z = dist x z + dist x z

Tactic:
]

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
e : PE ≃ᵢ PE
⊢ 2 * dist (e z) z = 2 * dist (e z) z

Tactic:
]

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

Goal state before:
pp: z : ℂ
n : ℕ
hn : n ≠ 0
aux : ∀ (a b c d : ℂ), a * b * (c * d) = a * c * (b * d)
this✝ : ↑n ^ z * ↑n ^ (1 - z) = ↑n
this : ∀ (j : ℕ), (z + ↑(j + 1)) * (1 - z + ↑j) = ↑((j + 1) ^ 2) * (1 - z ^ 2 / (↑j + 1) ^ 2)
⊢ ↑n / (↑n + 1 - z) / ((∏ x ∈ Finset.range n, (1 - z ^ 2 / (↑x + 1) ^ 2)) * z) =
    ↑n / (↑n + 1 - z) / ((∏ j ∈ Finset.range n, (1 - z ^ 2 / (↑j + 1) ^ 2)) * z)

pp: case hb
z : ℂ
n : ℕ
hn : n ≠ 0
aux : ∀ (a b c d : ℂ), a * b * (c * d) = a * c * (b * d)
this✝ : ↑n ^ z * ↑n ^ (1 - z) = ↑n
this : ∀ (j : ℕ), (z + ↑(j + 1)) * (1 - z + ↑j) = ↑((j + 1) ^ 2) * (1 - z ^ 2 / (↑j + 1) ^ 2)
⊢ ↑n ! ^ 2 ≠ 0

Tactic:
]

Goal state after:
case hb
z : ℂ
n : ℕ
hn : n ≠ 0
aux : ∀ (a b c d : ℂ), a * b * (c * d) = a * c * (b * d)
this✝ : ↑n ^ z * ↑n ^ (1 - z) = ↑n
this : ∀ (j : ℕ), (z + ↑(j + 1)) * (1 - z + ↑j) = ↑((j + 1) ^ 2) * (1 - z ^ 2 / (↑j + 1) ^ 2)
⊢ ↑n ! ^ 2 ≠ 0