mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.InnerProductSpace.TwoDim

Goal state before:
pp: case e'_3
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : Fact (finrank ℝ E = 2)
o : Orientation ℝ E (Fin 2)
φ : E ≃ₗᵢ[ℝ] E
hφ : 0 < LinearMap.det ↑φ.toLinearEquiv
x : E
⊢ (map (Fin 2) φ.toLinearEquiv) o = o

Tactic:
← o.map_eq_iff_det_pos φ.toLinearEquiv

Goal state after:
case e'_3
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : Fact (finrank ℝ E = 2)
o : Orientation ℝ E (Fin 2)
φ : E ≃ₗᵢ[ℝ] E
hφ : (map (Fin 2) φ.toLinearEquiv) o = o
x : E
⊢ (map (Fin 2) φ.toLinearEquiv) o = o

case e'_3
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : Fact (finrank ℝ E = 2)
o : Orientation ℝ E (Fin 2)
φ : E ≃ₗᵢ[ℝ] E
hφ : 0 < LinearMap.det ↑φ.toLinearEquiv
x : E
⊢ Fintype.card (Fin 2) = finrank ℝ E
Module: Mathlib.Analysis.InnerProductSpace.TwoDim

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : Fact (finrank ℝ E = 2)
o : Orientation ℝ E (Fin 2)
x : E
⊢ (o.kahler x) x = ↑‖x‖ ^ 2

Tactic:
simp [kahler_apply_apply]

Goal state after:
Module: Mathlib.Analysis.MellinTransform

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f g : ℝ → E
s : ℂ
hf : MellinConvergent f s
hg : MellinConvergent g s
⊢ MellinConvergent (fun t ↦ f t + g t) s

Tactic:
simpa only [MellinConvergent, smul_add] using! hf.add hg

Goal state after:
Module: Mathlib.Analysis.MellinTransform

Goal state before:
pp: f : ℝ → ℝ
hfc : AEStronglyMeasurable f (volume.restrict (Ioi 0))
a s : ℝ
hf : f =O[atTop] fun x ↦ x ^ (-a)
hs : s < a
d : ℝ
hd' : ∃ a_1, ∀ (b : ℝ), a_1 ≤ b → ‖f b‖ ≤ d * ‖b ^ (-a)‖
⊢ ∃ c, 0 < c ∧ IntegrableOn (fun t ↦ t ^ (s - 1) * f t) (Ioi c) volume

Tactic:
obtain ⟨e, he⟩ := hd'

Goal state after:
f : ℝ → ℝ
hfc : AEStronglyMeasurable f (volume.restrict (Ioi 0))
a s : ℝ
hf : f =O[atTop] fun x ↦ x ^ (-a)
hs : s < a
d e : ℝ
he : ∀ (b : ℝ), e ≤ b → ‖f b‖ ≤ d * ‖b ^ (-a)‖
⊢ ∃ c, 0 < c ∧ IntegrableOn (fun t ↦ t ^ (s - 1) * f t) (Ioi c) volume
Module: Mathlib.Analysis.SpecialFunctions.Complex.LogBounds

Goal state before:
pp: n : ℕ
z : ℂ
hz : ‖z‖ < 1
help : IntervalIntegrable (fun t ↦ t ^ n * (1 - ‖z‖)⁻¹) MeasureTheory.volume 0 1
f : ℂ → ℂ := fun z ↦ log (1 + z) - logTaylor (n + 1) z
f' : ℂ → ℂ := fun z ↦ (-z) ^ n * (1 + z)⁻¹
⊢ ∀ t ∈ Set.Icc 0 1, HasDerivAt f (f' (0 + ↑t * z)) (0 + ↑t * z)

Tactic:
intro t

Goal state after:
n : ℕ
z : ℂ
hz : ‖z‖ < 1
help : IntervalIntegrable (fun t ↦ t ^ n * (1 - ‖z‖)⁻¹) MeasureTheory.volume 0 1
f : ℂ → ℂ := fun z ↦ log (1 + z) - logTaylor (n + 1) z
f' : ℂ → ℂ := fun z ↦ (-z) ^ n * (1 + z)⁻¹
t : ℝ
⊢ t ∈ Set.Icc 0 1 → HasDerivAt f (f' (0 + ↑t * z)) (0 + ↑t * z)
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Deriv

Goal state before:
pp: s : ℂ
hs : ∀ (m : ℕ), s ≠ -↑m
⊢ DifferentiableAt ℂ Gamma s

Tactic:
suffices ∀ (n : ℕ) (s : ℂ) (hsre : -n < s.re) (hs : ∀ m : ℕ, s ≠ -m), DifferentiableAt ℂ _ s from
    this (⌊-s.re⌋₊ + 1) s (by grind [Nat.lt_floor_add_one (-s.re)]) hs
  intro n s hsre hs
  induction n generalizing s with
  | zero =>
    -- Case `n = 0`: use relation to `gammaIntegral`
    replace hsre : 0 < s.re := by simpa using hsre
    have : IsOpen {s : ℂ | 0 < s.re} := continuous_re.isOpen_preimage _ isOpen_Ioi
    apply (hasDerivAt_GammaIntegral (by simpa using hsre)).differentiableAt.congr_of_eventuallyEq
    filter_upwards [this.mem_nhds hsre] with a using Gamma_eq_integral
  | succ n IH =>
    -- Induction step: use recurrence relation
    have hsne : s ≠ 0 := by grind [hs 0]
    specialize IH (s + 1) (by grind [add_re, one_re]) (fun m ↦ by grind [hs (m + 1)])
    have := IH.comp s (show DifferentiableAt ℂ (fun s ↦ s + 1) s by fun_prop)
    apply (this.fun_div differentiableAt_id hsne).congr_of_eventuallyEq
    filter_upwards [isOpen_ne.mem_nhds hsne] using by grind

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

Goal state before:
pp: case pos
m : ℕ
hs : -↑m ≠ 0
⊢ 0 = Gamma (-↑m) + -↑m * 0

pp: case pos
m : ℕ
hs : -↑m ≠ 0
⊢ 1 ≤ m

Tactic:
Gamma_neg_nat_eq_zero,

Goal state after:
case pos
m : ℕ
hs : -↑m ≠ 0
⊢ 0 = 0 + -↑m * 0

case pos
m : ℕ
hs : -↑m ≠ 0
⊢ 1 ≤ m
Module: Mathlib.Analysis.SpecialFunctions.Complex.LogBounds

Goal state before:
pp: g : ℝ → ℝ
t : ℝ
hg : Tendsto (fun x ↦ x * g x) atTop (𝓝 t)
⊢ Tendsto (fun x ↦ x * log (1 + g x)) atTop (𝓝 t)

Tactic:
have hg0 := tendsto_zero_of_isBoundedUnder_smul_of_tendsto_cobounded
    hg.norm.isBoundedUnder_le (tendsto_id'.mpr (by simp))
  rw [← tendsto_ofReal_iff] at hg ⊢
  push_cast at hg ⊢
  apply (Complex.tendsto_mul_log_one_add_of_tendsto hg).congr'
  filter_upwards [hg0.eventually_const_le (show (-1 : ℝ) < 0 by simp)] with x hg1
  rw [Complex.ofReal_log (by linarith), Complex.ofReal_add, Complex.ofReal_one]

Goal state after:
Module: Mathlib.Analysis.MellinTransform

Goal state before:
pp: s : ℂ
hs : 0 < s.re
aux1 : -1 < (s - 1).re
⊢ s ≠ 0

Tactic:
contrapose! hs

Goal state after:
s : ℂ
aux1 : -1 < (s - 1).re
hs : s = 0
⊢ s.re ≤ 0
Module: Mathlib.Analysis.Normed.Affine.AsymptoticCone

Goal state before:
pp: V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : NormedSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
v : V
hv : v ≠ 0
p : P
⊢ Tendsto (fun x ↦ dist (id x) p) (asymptoticNhds ℝ P v) atTop

Tactic:
asymptoticNhds_eq_smul_vadd v p,

Goal state after:
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : NormedSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
v : V
hv : v ≠ 0
p : P
⊢ Tendsto (fun x ↦ dist (id x) p) (atTop • 𝓝 v +ᵥ pure p) atTop
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Beta

Goal state before:
pp: case succ
n : ℕ
IH : ∀ {u : ℂ}, 0 < u.re → u.betaIntegral (↑n + 1) = ↑n ! / ∏ j ∈ Finset.range (n + 1), (u + ↑j)
u : ℂ
hu : 0 < u.re
⊢ u.betaIntegral (↑(n + 1) + 1) = ↑(n + 1)! / ∏ j ∈ Finset.range (n + 1 + 1), (u + ↑j)

Tactic:
have := betaIntegral_recurrence hu (?_ : 0 < re n.succ)
    swap; · rw [← ofReal_natCast, ofReal_re]; positivity
    rw [mul_comm u _, ← eq_div_iff] at this
    swap; · contrapose! hu; rw [hu, zero_re]
    rw [this, Finset.prod_range_succ', Nat.cast_succ, IH]
    swap; · rw [add_re, one_re]; positivity
    rw [Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one, Nat.cast_zero, add_zero, ←
      mul_div_assoc, ← div_div]
    congr 3 with j : 1
    push_cast; abel

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

Goal state before:
pp: k : Type u_1
V : Type u_2
inst✝⁹ : Field k
inst✝⁸ : LinearOrder k
inst✝⁷ : IsStrictOrderedRing k
inst✝⁶ : TopologicalSpace k
inst✝⁵ : OrderTopology k
inst✝⁴ : AddCommGroup V
inst✝³ : Module k V
inst✝² : TopologicalSpace V
inst✝¹ : IsTopologicalAddGroup V
inst✝ : ContinuousSMul k V
s : Set V
c : k
v p : V
hs₁ : StarConvex k p s
hs₂ : IsClosed[inst✝²] s
hc : 0 ≤ c
hv : ∃ᶠ (p : V) in asymptoticNhds k V v, p ∈ s
⊢ ∃ᶠ (x : k × V) in atTop ×ˢ 𝓝 v, c • x.2 +ᵥ p ∈ s

Tactic:
asymptoticNhds_eq_smul_vadd v p,

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 (p : V) in atTop • 𝓝 v +ᵥ pure p, 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 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
⊢ ‖(Ioc 0 ↑n).indicator (fun x ↦ ↑((1 - x / ↑n) ^ n) * ↑x ^ (s - 1)) x‖ ≤ rexp (-x) * x ^ (s.re - 1)

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
⊢ ‖(Ioc 0 ↑n).indicator (fun x ↦ ↑((1 - x / ↑n) ^ n) * ↑x ^ (s - 1)) x‖ ≤ rexp (-x) * x ^ (s.re - 1)

Tactic:
· rw [indicator_of_notMem (notMem_Ioc_of_gt hxn), norm_zero,
        mul_nonneg_iff_right_nonneg_of_pos (exp_pos _)]
      positivity

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
⊢ ‖(Ioc 0 ↑n).indicator (fun x ↦ ↑((1 - x / ↑n) ^ n) * ↑x ^ (s - 1)) x‖ ≤ rexp (-x) * x ^ (s.re - 1)
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
⊢ ‖f‖ = max ‖f 0‖ ‖f.contLinear‖

Tactic:
norm_def

Goal state after:
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‖
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‖ * ‖g‖ + ‖f 0‖

Tactic:
f.norm_contLinear_le,

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‖ * ‖g 0‖ + ‖f 0‖ ≤ ‖f‖ * ‖g‖ + ‖f 0‖
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.contLinear‖ ≤ ‖f‖ * ‖g‖

Tactic:
f.norm_contLinear_le,

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‖ * ‖g.contLinear‖ ≤ ‖f‖ * ‖g‖
Module: Mathlib.LinearAlgebra.AffineSpace.Ceva

Goal state before:
pp: 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
⊢ ∑ x ∈ insert (↑i) (fs i), (↑(fs i)).indicator (w i) x = ∑ j ∈ fs i, w i j

Tactic:
by_cases hi : (i : ι) ∈ fs i

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 ∈ insert (↑i) (fs i), (↑(fs i)).indicator (w i) x = ∑ j ∈ fs i, w i j

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
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 ! ^ 2 / ((∏ x ∈ Finset.range n, ↑((x + 1) ^ 2) * (1 - z ^ 2 / (↑x + 1) ^ 2)) * (z * (↑n + 1 - z))) =
    ↑n / (↑n + 1 - z) * (1 / (z * ∏ j ∈ Finset.range n, (1 - z ^ 2 / (↑j + 1) ^ 2)))

Tactic:
rw [Finset.prod_mul_distrib, ← Nat.cast_prod, Finset.prod_pow,
    Finset.prod_range_add_one_eq_factorial, Nat.cast_pow,
    (by intros; ring : ∀ a b c d : ℂ, a * b * (c * d) = a * (d * (b * c))), ← div_div,
    mul_div_cancel_right₀, ← div_div, mul_comm z _, mul_one_div]

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
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Beta

Goal state before:
pp: case pos
z : ℂ
pi_ne : ↑π ≠ 0
k : ℤ
hk : z = -↑k
⊢ Gamma z * Gamma (1 - z) = 0

Tactic:
hk

Goal state after:
case pos
z : ℂ
pi_ne : ↑π ≠ 0
k : ℤ
hk : z = -↑k
⊢ Gamma (-↑k) * Gamma (1 - -↑k) = 0
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Beta

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

Tactic:
contrapose! hs

Goal state after:
s : ℂ
m : ℕ
hs : s = -↑m
⊢ s.re ≤ 0
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 (tsze R M) (n + 1)) fun x_1 ↦ x).snd)
    (e • x.snd - ∑ i ∈ Finset.range 1, ((expSeries 𝕜 (tsze R M) i) fun x_1 ↦ x).snd)

Tactic:
snd_expSeries_of_smul_comm _ _ hx

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 ↦ (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)
Module: Mathlib.LinearAlgebra.AffineSpace.Ceva

Goal state before:
pp: case pos
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
⊢ ∏ i, r i = ∏ i, (1 - r i)

Tactic:
have hi1 : c (i + 1) * r (i + 1) = 1 := by simpa [add_assoc, hw'i0] using hc2 (i + 1)

Goal state after:
case pos
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
⊢ ∏ i, r i = ∏ i, (1 - r i)
Module: Mathlib.Analysis.Normed.Algebra.TrivSqZeroExt

Goal state before:
pp: 𝕜 : Type u_1
S : Type u_2
R : Type u_3
M : Type u_4
inst✝¹¹ : SeminormedCommRing S
inst✝¹⁰ : SeminormedRing R
inst✝⁹ : SeminormedAddCommGroup M
inst✝⁸ : Algebra S R
inst✝⁷ : Module S M
inst✝⁶ : IsBoundedSMul S R
inst✝⁵ : IsBoundedSMul S M
inst✝⁴ : Module R M
inst✝³ : IsBoundedSMul R M
inst✝² : Module Rᵐᵒᵖ M
inst✝¹ : IsBoundedSMul Rᵐᵒᵖ M
inst✝ : SMulCommClass R Rᵐᵒᵖ M
r₁ : R
m₁ : M
r₂ : R
m₂ : M
⊢ ‖(r₁, m₁) * (r₂, m₂)‖ ≤ ‖(r₁, m₁)‖ * ‖(r₂, m₂)‖

Tactic:
norm_def

Goal state after:
𝕜 : Type u_1
S : Type u_2
R : Type u_3
M : Type u_4
inst✝¹¹ : SeminormedCommRing S
inst✝¹⁰ : SeminormedRing R
inst✝⁹ : SeminormedAddCommGroup M
inst✝⁸ : Algebra S R
inst✝⁷ : Module S M
inst✝⁶ : IsBoundedSMul S R
inst✝⁵ : IsBoundedSMul S M
inst✝⁴ : Module R M
inst✝³ : IsBoundedSMul R M
inst✝² : Module Rᵐᵒᵖ M
inst✝¹ : IsBoundedSMul Rᵐᵒᵖ M
inst✝ : SMulCommClass R Rᵐᵒᵖ M
r₁ : R
m₁ : M
r₂ : R
m₂ : M
⊢ ‖((r₁, m₁) * (r₂, m₂)).fst‖ + ‖((r₁, m₁) * (r₂, m₂)).snd‖ ≤
    (‖fst (r₁, m₁)‖ + ‖snd (r₁, m₁)‖) * (‖fst (r₂, m₂)‖ + ‖snd (r₂, m₂)‖)
Module: Mathlib.Analysis.Quaternion

Goal state before:
pp: a : ℍ
⊢ normSq a = ‖a‖ * ‖a‖

Tactic:
← inner_self,

Goal state after:
a : ℍ
⊢ ⟪a, a⟫ = ‖a‖ * ‖a‖
Module: Mathlib.Analysis.Normed.Unbundled.FiniteExtension

Goal state before:
pp: K : Type u_1
L : Type u_2
inst✝² : NormedField K
inst✝¹ : Field L
inst✝ : Algebra K L
hfd : FiniteDimensional K L
hna : IsNonarchimedean norm
h1 : LinearIndepOn K id {1}
ι : Type u_2 := { x // x ∈ h1.extend ⋯ }
B : Basis ι K L := Basis.extend h1
hfin : Fintype ι := FiniteDimensional.fintypeBasisIndex B
hem : Nonempty ι
h1L : 1 ∈ h1.extend ⋯
hB1 : B ⟨1, h1L⟩ = 1
g : L → ℝ := B.norm
hg0 : g 0 = 0
hg_nonneg : ∀ (x : L), 0 ≤ g x
hg_ext : ∀ (x : K), g ((algebraMap K L) x) = ‖x‖
hg_na : IsNonarchimedean g
hg_add : ∀ (a b : L), g (a + b) ≤ g a + g b
hg_neg : ∀ (a : L), g (-a) = g a
w✝¹ : ℝ
w✝ : 0 < w✝¹
hg_bdd : ∀ (x y : L), B.norm (x * y) ≤ w✝¹ * B.norm x * B.norm y
hg_mul : ∀ (k : K) (y : L), g ((algebraMap K L) k * y) = g ((algebraMap K L) k) * g y
f : RingSeminorm L := seminormFromBounded hg0 hg_nonneg hg_bdd hg_add hg_neg
hf_na : IsNonarchimedean ⇑f
hf_1 : f 1 ≤ 1
hf_ext : ∀ (x : K), f ((algebraMap K L) x) = ‖x‖
F' : RingSeminorm L := smoothingSeminorm f hf_1 hf_na
hF' : F' = smoothingSeminorm f hf_1 hf_na
hF'_ext : ∀ (k : K), F' ((algebraMap K L) k) = ‖k‖
hF'_1 : F' 1 = 1
hF'_0 : F' ≠ 0
k : K
y : L
⊢ (F'.toRingNorm hF'_0).toFun (k • y) = ‖k‖ * (F'.toRingNorm hF'_0).toFun y

Tactic:
have hk : ∀ y : L, f (algebraMap K L k * y) = f (algebraMap K L k) * f y :=
          seminormFromBounded_of_mul_is_mul hg_nonneg hg_bdd (hg_mul k)

Goal state after:
K : Type u_1
L : Type u_2
inst✝² : NormedField K
inst✝¹ : Field L
inst✝ : Algebra K L
hfd : FiniteDimensional K L
hna : IsNonarchimedean norm
h1 : LinearIndepOn K id {1}
ι : Type u_2 := { x // x ∈ h1.extend ⋯ }
B : Basis ι K L := Basis.extend h1
hfin : Fintype ι := FiniteDimensional.fintypeBasisIndex B
hem : Nonempty ι
h1L : 1 ∈ h1.extend ⋯
hB1 : B ⟨1, h1L⟩ = 1
g : L → ℝ := B.norm
hg0 : g 0 = 0
hg_nonneg : ∀ (x : L), 0 ≤ g x
hg_ext : ∀ (x : K), g ((algebraMap K L) x) = ‖x‖
hg_na : IsNonarchimedean g
hg_add : ∀ (a b : L), g (a + b) ≤ g a + g b
hg_neg : ∀ (a : L), g (-a) = g a
w✝¹ : ℝ
w✝ : 0 < w✝¹
hg_bdd : ∀ (x y : L), B.norm (x * y) ≤ w✝¹ * B.norm x * B.norm y
hg_mul : ∀ (k : K) (y : L), g ((algebraMap K L) k * y) = g ((algebraMap K L) k) * g y
f : RingSeminorm L := seminormFromBounded hg0 hg_nonneg hg_bdd hg_add hg_neg
hf_na : IsNonarchimedean ⇑f
hf_1 : f 1 ≤ 1
hf_ext : ∀ (x : K), f ((algebraMap K L) x) = ‖x‖
F' : RingSeminorm L := smoothingSeminorm f hf_1 hf_na
hF' : F' = smoothingSeminorm f hf_1 hf_na
hF'_ext : ∀ (k : K), F' ((algebraMap K L) k) = ‖k‖
hF'_1 : F' 1 = 1
hF'_0 : F' ≠ 0
k : K
y : L
hk : ∀ (y : L), f ((algebraMap K L) k * y) = f ((algebraMap K L) k) * f y
⊢ (F'.toRingNorm hF'_0).toFun (k • y) = ‖k‖ * (F'.toRingNorm hF'_0).toFun y
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 ^ 1

Tactic:
rpow_le_rpow_left_iff hx

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
⊢ ↑(s (ψ n)) / ↑(ψ n) ≤ 1