mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.SpecialFunctions.Complex.LogBounds

Goal state before:
pp: g : ℝ → ℂ
t : ℂ
hg : Tendsto (fun x ↦ ↑x * g x) atTop (𝓝 t)
⊢ (fun x ↦ (↑x * g x) ^ 2 * ↑x⁻¹) =O[atTop] fun x ↦ ↑x⁻¹

Tactic:
simpa using

Goal state after:
g : ℝ → ℂ
t : ℂ
hg : Tendsto (fun x ↦ ↑x * g x) atTop (𝓝 t)
⊢ (fun x ↦ (↑x * g x) ^ 2 * (↑x)⁻¹) =O[atTop] fun x ↦ (↑x)⁻¹
Module: Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup

Goal state before:
pp: case e'_4
s t a b : ℝ
hs : 0 < s
ht : 0 < t
ha : 0 < a
hb : 0 < b
hab : a + b = 1
f : ℝ → ℝ → ℝ → ℝ := fun c u x ↦ rexp (-c * x) * x ^ (c * (u - 1))
e : (1 / a).HolderConjugate (1 / b)
hab' : b = 1 - a
hst : 0 < a * s + b * t
posf : ∀ (c u x : ℝ), x ∈ Ioi 0 → 0 ≤ f c u x
posf' : ∀ (c u : ℝ), ∀ᵐ (x : ℝ) ∂volume.restrict (Ioi 0), 0 ≤ f c u x
fpow : ∀ {c x : ℝ}, 0 < c → ∀ (u : ℝ), 0 < x → rexp (-x) * x ^ (u - 1) = f c u x ^ (1 / c)
f_mem_Lp : ∀ {c u : ℝ}, 0 < c → 0 < u → MemLp (f c u) (ENNReal.ofReal (1 / c)) (volume.restrict (Ioi 0))
⊢ (∫ (x : ℝ) in Ioi 0, rexp (-x) * x ^ (s - 1)) ^ a * (∫ (x : ℝ) in Ioi 0, rexp (-x) * x ^ (t - 1)) ^ b =
    (∫ (a_1 : ℝ) in Ioi 0, f a s a_1 ^ (1 / a)) ^ (1 / (1 / a)) *
      (∫ (a : ℝ) in Ioi 0, f b t a ^ (1 / b)) ^ (1 / (1 / b))

Tactic:
one_div_one_div,

Goal state after:
case e'_4
s t a b : ℝ
hs : 0 < s
ht : 0 < t
ha : 0 < a
hb : 0 < b
hab : a + b = 1
f : ℝ → ℝ → ℝ → ℝ := fun c u x ↦ rexp (-c * x) * x ^ (c * (u - 1))
e : (1 / a).HolderConjugate (1 / b)
hab' : b = 1 - a
hst : 0 < a * s + b * t
posf : ∀ (c u x : ℝ), x ∈ Ioi 0 → 0 ≤ f c u x
posf' : ∀ (c u : ℝ), ∀ᵐ (x : ℝ) ∂volume.restrict (Ioi 0), 0 ≤ f c u x
fpow : ∀ {c x : ℝ}, 0 < c → ∀ (u : ℝ), 0 < x → rexp (-x) * x ^ (u - 1) = f c u x ^ (1 / c)
f_mem_Lp : ∀ {c u : ℝ}, 0 < c → 0 < u → MemLp (f c u) (ENNReal.ofReal (1 / c)) (volume.restrict (Ioi 0))
⊢ (∫ (x : ℝ) in Ioi 0, rexp (-x) * x ^ (s - 1)) ^ a * (∫ (x : ℝ) in Ioi 0, rexp (-x) * x ^ (t - 1)) ^ b =
    (∫ (a_1 : ℝ) in Ioi 0, f a s a_1 ^ (1 / a)) ^ a * (∫ (a : ℝ) in Ioi 0, f b t a ^ (1 / b)) ^ (1 / (1 / b))
Module: Mathlib.Analysis.MellinTransform

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

Tactic:
refine
        le_add_of_nonneg_of_le (rpow_pos_of_pos ht _).le (rpow_le_rpow_of_exponent_ge ht h.le ?_)

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

Goal state before:
pp: u v : ℂ
⊢ v.betaIntegral u = u.betaIntegral v

Tactic:
simpa using

Goal state after:
u v : ℂ
⊢ ∫ (x : ℝ) in 0..1, ↑x ^ (v + -1) * (-↑x + 1) ^ (u + -1) = ∫ (x : ℝ) in 0..1, ↑x ^ (u + -1) * (-↑x + 1) ^ (v + -1)
Module: Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup

Goal state before:
pp: f : ℝ → ℝ
x : ℝ
n : ℕ
hf_conv : ConvexOn ℝ (Ioi 0) f
hf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + log y
hn : n ≠ 0
hx : 0 < x
hx' : x ≤ 1
hn' : 0 < ↑n
this : f ↑n + x * log ↑n = (1 - x) * f ↑n + x * f (↑n + 1)
⊢ f ((1 - x) * ↑n + x * (↑n + 1)) ≤ (1 - x) * f ↑n + x * f (↑n + 1)

Tactic:
simpa using

Goal state after:
f : ℝ → ℝ
x : ℝ
n : ℕ
hf_conv : ConvexOn ℝ (Ioi 0) f
hf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + log y
hn : n ≠ 0
hx : 0 < x
hx' : x ≤ 1
hn' : 0 < ↑n
this : f ↑n + x * log ↑n = (1 - x) * f ↑n + x * f (↑n + 1)
⊢ f ((1 - x) * ↑n + x * (↑n + 1)) ≤ (1 - x) * f ↑n + x * f (↑n + 1)
Module: Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup

Goal state before:
pp: x : ℝ
hx : x ∈ Icc 1 2
hmin : IsMinOn Γ (Icc 1 2) x
⊢ Γ (3 / 2) < Γ 1 ∧ Γ (3 / 2) < Γ 2 ∧ Γ x ≤ Γ (3 / 2)

Tactic:
simpa using

Goal state after:
x : ℝ
hx : x ∈ Icc 1 2
hmin : IsMinOn Γ (Icc 1 2) x
⊢ Γ x ≤ Γ (3 / 2)
Module: Mathlib.Analysis.Normed.Affine.AsymptoticCone

Goal state before:
pp: V : Type u_1
P : Type u_2
inst✝⁴ : NormedAddCommGroup V
inst✝³ : NormedSpace ℝ V
inst✝² : MetricSpace P
inst✝¹ : NormedAddTorsor V P
inst✝ : FiniteDimensional ℝ V
s : Set P
p : P
hs : ∀ i ∈ Metric.sphere 0 1, s ∈ asymptoticNhds ℝ P i
⊢ s ∈ cobounded P

Tactic:
asymptoticNhds_eq_smul_vadd _ p,

Goal state after:
V : Type u_1
P : Type u_2
inst✝⁴ : NormedAddCommGroup V
inst✝³ : NormedSpace ℝ V
inst✝² : MetricSpace P
inst✝¹ : NormedAddTorsor V P
inst✝ : FiniteDimensional ℝ V
s : Set P
p : P
hs : ∀ i ∈ Metric.sphere 0 1, s ∈ atTop • 𝓝 i +ᵥ pure p
⊢ s ∈ cobounded P
Module: Mathlib.Topology.Algebra.AsymptoticCone

Goal state before:
pp: k : Type u_1
V : Type u_2
P : Type u_3
inst✝¹⁰ : Field k
inst✝⁹ : LinearOrder k
inst✝⁸ : AddCommGroup V
inst✝⁷ : Module k V
inst✝⁶ : AddTorsor V P
inst✝⁵ : TopologicalSpace V
inst✝⁴ : TopologicalSpace k
inst✝³ : OrderTopology k
inst✝² : IsStrictOrderedRing k
inst✝¹ : IsTopologicalAddGroup V
inst✝ : ContinuousSMul k V
⊢ asymptoticNhds k P 0 = ⊤

Tactic:
have ⟨p⟩ : Nonempty P := inferInstance
  rw [← asymptoticNhds_vadd_pure 0 p, asymptoticNhds_zero', vadd_pure]
  exact (Equiv.vaddConst p).surjective.filter_map_top

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

Goal state before:
pp: k : Type u_1
V : Type u_2
P : Type u_3
inst✝¹⁰ : Field k
inst✝⁹ : LinearOrder k
inst✝⁸ : AddCommGroup V
inst✝⁷ : Module k V
inst✝⁶ : AddTorsor V P
inst✝⁵ : TopologicalSpace V
inst✝⁴ : TopologicalSpace k
inst✝³ : OrderTopology k
inst✝² : IsStrictOrderedRing k
inst✝¹ : IsTopologicalAddGroup V
inst✝ : ContinuousSMul k V
v : V
c : k
hc : 0 < c
p : P
⊢ asymptoticNhds k P (c • v) = asymptoticNhds k P v

Tactic:
asymptoticNhds_eq_smul_vadd _ p,

Goal state after:
k : Type u_1
V : Type u_2
P : Type u_3
inst✝¹⁰ : Field k
inst✝⁹ : LinearOrder k
inst✝⁸ : AddCommGroup V
inst✝⁷ : Module k V
inst✝⁶ : AddTorsor V P
inst✝⁵ : TopologicalSpace V
inst✝⁴ : TopologicalSpace k
inst✝³ : OrderTopology k
inst✝² : IsStrictOrderedRing k
inst✝¹ : IsTopologicalAddGroup V
inst✝ : ContinuousSMul k V
v : V
c : k
hc : 0 < c
p : P
⊢ atTop • 𝓝 (c • v) +ᵥ pure p = atTop • 𝓝 v +ᵥ pure p
Module: Mathlib.Topology.Algebra.AsymptoticCone

Goal state before:
pp: case a
k : Type u_1
V : Type u_2
P : Type u_3
inst✝¹⁰ : Field k
inst✝⁹ : LinearOrder k
inst✝⁸ : AddCommGroup V
inst✝⁷ : Module k V
inst✝⁶ : AddTorsor V P
inst✝⁵ : TopologicalSpace V
inst✝⁴ : TopologicalSpace k
inst✝³ : OrderTopology k
inst✝² : IsStrictOrderedRing k
inst✝¹ : IsTopologicalAddGroup V
inst✝ : ContinuousSMul k V
v : V
p : P
⊢ ((𝓝 v).bind fun v ↦ asymptoticNhds k P v) ≤ asymptoticNhds k P v

Tactic:
asymptoticNhds_eq_smul_vadd _ p,

Goal state after:
case a
k : Type u_1
V : Type u_2
P : Type u_3
inst✝¹⁰ : Field k
inst✝⁹ : LinearOrder k
inst✝⁸ : AddCommGroup V
inst✝⁷ : Module k V
inst✝⁶ : AddTorsor V P
inst✝⁵ : TopologicalSpace V
inst✝⁴ : TopologicalSpace k
inst✝³ : OrderTopology k
inst✝² : IsStrictOrderedRing k
inst✝¹ : IsTopologicalAddGroup V
inst✝ : ContinuousSMul k V
v : V
p : P
⊢ ((𝓝 v).bind fun v ↦ atTop • 𝓝 v +ᵥ pure p) ≤ atTop • 𝓝 v +ᵥ pure p
Module: Mathlib.Topology.Algebra.AsymptoticCone

Goal state before:
pp: k : Type u_1
V : Type u_2
P : Type u_3
inst✝¹² : Field k
inst✝¹¹ : LinearOrder k
inst✝¹⁰ : AddCommGroup V
inst✝⁹ : Module k V
inst✝⁸ : AddTorsor V P
inst✝⁷ : TopologicalSpace V
inst✝⁶ : TopologicalSpace k
inst✝⁵ : OrderTopology k
inst✝⁴ : IsStrictOrderedRing k
inst✝³ : IsTopologicalAddGroup V
inst✝² : ContinuousSMul k V
inst✝¹ : TopologicalSpace P
inst✝ : IsTopologicalAddTorsor P
v : V
s : Set P
h : s ∈ asymptoticNhds k P v
p : P
⊢ s ∈ (asymptoticNhds k P v).bind 𝓝

Tactic:
asymptoticNhds_eq_smul_vadd _ p,

Goal state after:
k : Type u_1
V : Type u_2
P : Type u_3
inst✝¹² : Field k
inst✝¹¹ : LinearOrder k
inst✝¹⁰ : AddCommGroup V
inst✝⁹ : Module k V
inst✝⁸ : AddTorsor V P
inst✝⁷ : TopologicalSpace V
inst✝⁶ : TopologicalSpace k
inst✝⁵ : OrderTopology k
inst✝⁴ : IsStrictOrderedRing k
inst✝³ : IsTopologicalAddGroup V
inst✝² : ContinuousSMul k V
inst✝¹ : TopologicalSpace P
inst✝ : IsTopologicalAddTorsor P
v : V
s : Set P
p : P
h : s ∈ atTop • 𝓝 v +ᵥ pure p
⊢ s ∈ (atTop • 𝓝 v +ᵥ pure p).bind 𝓝
Module: Mathlib.Analysis.Normed.Affine.Ceva

Goal state before:
pp: 𝕜 : Type u_1
V : Type u_2
P : Type u_3
inst✝⁴ : SeminormedAddCommGroup V
inst✝³ : NormedField 𝕜
inst✝² : NormedSpace 𝕜 V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
t : Triangle 𝕜 P
p : Fin 3 → P
p' : P
hp0 : ∀ (i : Fin 3), p i ≠ t.points (i + 2)
hp : ∀ (i : Fin 3), p i ∈ line[𝕜, t.points (i + 1), t.points (i + 2)]
hp' : ∀ (i : Fin 3), p' ∈ line[𝕜, t.points i, p i]
i : Fin 3
⊢ dist (p i) (t.points (i + 2)) ≠ 0

Tactic:
simpa using

Goal state after:
𝕜 : Type u_1
V : Type u_2
P : Type u_3
inst✝⁴ : SeminormedAddCommGroup V
inst✝³ : NormedField 𝕜
inst✝² : NormedSpace 𝕜 V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
t : Triangle 𝕜 P
p : Fin 3 → P
p' : P
hp0 : ∀ (i : Fin 3), p i ≠ t.points (i + 2)
hp : ∀ (i : Fin 3), p i ∈ line[𝕜, t.points (i + 1), t.points (i + 2)]
hp' : ∀ (i : Fin 3), p' ∈ line[𝕜, t.points i, p i]
i : Fin 3
⊢ ¬p i = t.points (i + 2)
Module: Mathlib.Topology.Algebra.AsymptoticCone

Goal state before:
pp: k : Type u_1
V : Type u_2
P : Type u_3
inst✝¹⁰ : Field k
inst✝⁹ : LinearOrder k
inst✝⁸ : AddCommGroup V
inst✝⁷ : Module k V
inst✝⁶ : AddTorsor V P
inst✝⁵ : TopologicalSpace V
inst✝⁴ : TopologicalSpace k
inst✝³ : OrderTopology k
inst✝² : IsStrictOrderedRing k
inst✝¹ : IsTopologicalAddGroup V
inst✝ : ContinuousSMul k V
v : V
⊢ (asymptoticNhds k V v).bind (asymptoticNhds k P) = asymptoticNhds k P v

Tactic:
refine Filter.ext' fun p => ?_

Goal state after:
k : Type u_1
V : Type u_2
P : Type u_3
inst✝¹⁰ : Field k
inst✝⁹ : LinearOrder k
inst✝⁸ : AddCommGroup V
inst✝⁷ : Module k V
inst✝⁶ : AddTorsor V P
inst✝⁵ : TopologicalSpace V
inst✝⁴ : TopologicalSpace k
inst✝³ : OrderTopology k
inst✝² : IsStrictOrderedRing k
inst✝¹ : IsTopologicalAddGroup V
inst✝ : ContinuousSMul k V
v : V
p : P → Prop
⊢ (∀ᶠ (x : P) in (asymptoticNhds k V v).bind (asymptoticNhds k P), p x) ↔ ∀ᶠ (x : P) in asymptoticNhds k P v, p x
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:
by
  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.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

Goal state after:
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
⊢ s.re ≤ 0
Module: Mathlib.LinearAlgebra.AffineSpace.Ceva

Goal state before:
pp: case pos
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
hs : s.Nonempty
fs : ↑s → Finset ι
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)]
fsx : ↑s → Finset ι := fun i ↦ insert (↑i) (fs i)
hfsx : ∀ (i : ↑s), ↑i ∈ fsx i
wx : ↑s → ι → k := fun i ↦ (↑(fs i)).indicator (w i)
i : ↑s
hi : ↑i ∈ fs i
⊢ ∑ x ∈ insert (↑i) (fs i), (↑(fs i)).indicator (w i) x = ∑ j ∈ fs i, w i j

Tactic:
simpa using

Goal state after:
case pos
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
hs : s.Nonempty
fs : ↑s → Finset ι
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)]
fsx : ↑s → Finset ι := fun i ↦ insert (↑i) (fs i)
hfsx : ∀ (i : ↑s), ↑i ∈ fsx i
wx : ↑s → ι → k := fun i ↦ (↑(fs i)).indicator (w i)
i : ↑s
hi : ↑i ∈ fs i
⊢ ∑ x ∈ fs i, (↑(fs i)).indicator (w i) x = ∑ j ∈ fs i, w i j
Module: Mathlib.LinearAlgebra.AffineSpace.Ceva

Goal state before:
pp: case neg
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
hs : s.Nonempty
fs : ↑s → Finset ι
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)]
fsx : ↑s → Finset ι := fun i ↦ insert (↑i) (fs i)
hfsx : ∀ (i : ↑s), ↑i ∈ fsx i
wx : ↑s → ι → k := fun i ↦ (↑(fs i)).indicator (w i)
i : ↑s
hi : ↑i ∉ fs i
⊢ ∑ x ∈ insert (↑i) (fs i), (↑(fs i)).indicator (w i) x = ∑ j ∈ fs i, w i j

Tactic:
simpa using

Goal state after:
case neg
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
hs : s.Nonempty
fs : ↑s → Finset ι
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)]
fsx : ↑s → Finset ι := fun i ↦ insert (↑i) (fs i)
hfsx : ∀ (i : ↑s), ↑i ∈ fsx i
wx : ↑s → ι → k := fun i ↦ (↑(fs i)).indicator (w i)
i : ↑s
hi : ↑i ∉ fs i
⊢ ∑ x ∈ fs i, (↑(fs i)).indicator (w i) x = ∑ j ∈ fs i, w i j
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Beta

Goal state before:
pp: s : ℂ
hs : ∀ (m : ℕ), s ≠ -↑m
h_im : s.im = 0
⊢ ↑s.re + ↑s.im * I = ↑s.re

Tactic:
h_im,

Goal state after:
s : ℂ
hs : ∀ (m : ℕ), s ≠ -↑m
h_im : s.im = 0
⊢ ↑s.re + ↑0 * I = ↑s.re
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
p' : P
hp' : ∀ (i : Fin 3), p' ∈ line[k, t.points i, (AffineMap.lineMap (t.points (i + 1)) (t.points (i + 2))) (r i)]
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
⊢ ∀ (i : ↑Set.univ), p' ∈ line[k, t.points ↑i, (Finset.affineCombination k Finset.univ t.points) (w i)]

Tactic:
simpa using

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
p' : P
hp' : ∀ (i : Fin 3), p' ∈ line[k, t.points i, (AffineMap.lineMap (t.points (i + 1)) (t.points (i + 2))) (r i)]
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
⊢ ∀ (a : Fin 3), p' ∈ line[k, t.points a, (AffineMap.lineMap (t.points (a + 1)) (t.points (a + 2))) (r a)]
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Beta

Goal state before:
pp: s : ℂ
m : ℕ
hs : s = -↑m
⊢ s.re ≤ 0

Tactic:
simpa using

Goal state after:
s : ℂ
m : ℕ
hs : s = -↑m
⊢ 0 ≤ ↑m
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
c : k
hc : ∀ (j : Fin (2 + 1)), c * {↑⟨i, ⋯⟩}ᶜ.indicator (w ⟨i, ⋯⟩) j = {↑⟨i, ⋯⟩}ᶜ.indicator w' j
j : Fin 3
hj : j ≠ i
⊢ c * w ⟨i, ⋯⟩ j = w' j

Tactic:
simpa using

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)]
i : Fin 3
c : k
hc : ∀ (j : Fin (2 + 1)), c * {↑⟨i, ⋯⟩}ᶜ.indicator (w ⟨i, ⋯⟩) j = {↑⟨i, ⋯⟩}ᶜ.indicator w' j
j : Fin 3
hj : j ≠ i
⊢ c * w ⟨i, ⋯⟩ j = w' j
Module: Mathlib.Analysis.Normed.Affine.Simplex

Goal state before:
pp: case refine_1
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)
h : (s.reindex e).Regular
σ : Equiv.Perm (Fin (m + 1))
x : P ≃ᵢ P
hx : (s.reindex e).points ∘ ⇑((e.symm.trans σ).trans e) = ⇑x ∘ (s.reindex e).points
i : Fin (m + 1)
⊢ (s.points ∘ ⇑σ) i = (⇑x ∘ s.points) i

Tactic:
simpa using

Goal state after:
case refine_1
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)
h : (s.reindex e).Regular
σ : Equiv.Perm (Fin (m + 1))
x : P ≃ᵢ P
hx : (s.reindex e).points ∘ ⇑((e.symm.trans σ).trans e) = ⇑x ∘ (s.reindex e).points
i : Fin (m + 1)
⊢ s.points (σ i) = x (s.points i)
Module: Mathlib.Analysis.Normed.Affine.Simplex

Goal state before:
pp: case refine_2
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)
h : s.Regular
σ : Equiv.Perm (Fin (n + 1))
x : P ≃ᵢ P
hx : s.points ∘ ⇑((e.trans σ).trans e.symm) = ⇑x ∘ s.points
i : Fin (n + 1)
⊢ ((s.reindex e).points ∘ ⇑σ) i = (⇑x ∘ (s.reindex e).points) i

Tactic:
simpa using

Goal state after:
case refine_2
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)
h : s.Regular
σ : Equiv.Perm (Fin (n + 1))
x : P ≃ᵢ P
hx : s.points ∘ ⇑((e.trans σ).trans e.symm) = ⇑x ∘ s.points
i : Fin (n + 1)
⊢ s.points (e.symm (σ i)) = x (s.points (e.symm i))
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
⊢ DifferentiableOn ℂ (fun z ↦ (Gamma (2 * z))⁻¹ * 2 ^ (2 * z - 1) / ↑√π) univ

Tactic:
refine (Differentiable.mul ?_ (differentiable_const _)).differentiableOn

Goal state after:
s : ℂ
h1 : AnalyticOnNhd ℂ (fun z ↦ (Gamma z)⁻¹ * (Gamma (z + 1 / 2))⁻¹) univ
⊢ Differentiable ℂ fun z ↦ (Gamma (2 * z))⁻¹ * 2 ^ (2 * z - 1)
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
n : ℕ
s : Simplex R P n
hr : s.Regular
⊢ s.Equilateral

Tactic:
by
  refine ⟨dist (s.points 0) (s.points 1), fun i j hij ↦ ?_⟩
  have hn : n ≠ 0 := by lia
  by_cases hi : i = 1
  · rw [hi, dist_comm]
    rcases hr (Equiv.swap 0 j) with ⟨x, hx⟩
    nth_rw 2 [← x.dist_eq]
    simp_rw [← Function.comp_apply (f := x), ← hx]
    simp only [comp_apply, Equiv.swap_apply_left]
    convert! rfl
    rw [Equiv.swap_apply_of_ne_of_ne (by simp [hn]) (by lia)]
  · rcases hr ((Equiv.swap 0 i).trans (Equiv.swap 1 j)) with ⟨x, hx⟩
    nth_rw 2 [← x.dist_eq]
    simp_rw [← Function.comp_apply (f := x), ← hx]
    simp only [Equiv.coe_trans, comp_apply, Equiv.swap_apply_left]
    convert! rfl
    · exact Equiv.swap_apply_of_ne_of_ne hi hij
    · rw [Equiv.swap_apply_of_ne_of_ne (by simp [hn]) (Ne.symm hi)]
      simp

Goal state after: