mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle

Goal state before:
pp: case inl
θ ψ : Angle
hθs : θ.toReal ∈ Set.Ioo 0 π
hψs : ψ.toReal ∈ Set.Ioo 0 π
hsa : (θ + ψ).sign ≠ 1
this : ↑(θ + ψ).toReal = ↑(θ.toReal + ψ.toReal)
h : (θ + ψ).toReal ≤ 0
hk : (θ + ψ).toReal - (θ.toReal + ψ.toReal) = 2 * π * ↑(-1)
⊢ |(θ + ψ).toReal| = 2 * π - (|θ.toReal| + |ψ.toReal|)

Tactic:
grind [abs_of_nonpos]

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Pow.Real

Goal state before:
pp: x : ℝ
hx : 0 ≤ x
y z : ℝ
h : y - z ≠ 0
⊢ x ^ (y - z) = x ^ y / x ^ z

Tactic:
by
  simp only [sub_eq_add_neg] at h ⊢
  simp only [rpow_add' hx h, rpow_neg hx, div_eq_mul_inv]

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

Goal state before:
pp: θ ψ : Angle
hθs : θ.toReal ∈ Set.Ioo 0 π
hψs : ψ.toReal ∈ Set.Ioo 0 π
k : ℤ
hk : π - (θ.toReal + ψ.toReal) = 2 * π * ↑k
h : θ + ψ = ↑π
⊢ 2 * π * ↑k ∈ Set.Ioo (2 * π * ↑(0 - 1)) (2 * π * ↑(0 + 1))

Tactic:
grind

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

Goal state before:
pp: case inr
θ ψ : Angle
hθs : θ.toReal ∈ Set.Ioo 0 π
hψs : ψ.toReal ∈ Set.Ioo 0 π
h : θ + ψ = ↑π
hk : π - (θ.toReal + ψ.toReal) = 2 * π * ↑0
⊢ π = 2 * π - (|θ.toReal| + |ψ.toReal|)

Tactic:
grind

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Pow.Real

Goal state before:
pp: case inl
x z : ℝ
h : 0 ≤ x
h₂ : 0 ≤ z
h₁ : x ≤ x
⊢ x ^ z ≤ x ^ z

Tactic:
rfl

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Pow.NNReal

Goal state before:
pp: ι : Type u_1
l : List ι
f : ι → ℝ≥0
r : ℝ
⊢ (List.map (fun x ↦ f x ^ r) l).prod = (List.map ((fun x ↦ x ^ r) ∘ f) l).prod

Tactic:
rfl

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Pow.NNReal

Goal state before:
pp: ι : Type u_1
l : List ι
f : ι → ℝ
hl : ∀ i ∈ l, 0 ≤ f i
r : ℝ
⊢ (List.map (fun x ↦ f x ^ r) l).prod = (List.map ((fun x ↦ x ^ r) ∘ f) l).prod

Tactic:
rfl

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Pow.NNReal

Goal state before:
pp: y : ℝ
hy : 0 < y
⊢ (StrictMono.orderIsoOfRightInverse (fun x ↦ x ^ y) ⋯ (fun x ↦ x ^ (1 / y)) ⋯).symm =
    StrictMono.orderIsoOfRightInverse (fun x ↦ x ^ (1 / y)) ⋯ (fun x ↦ x ^ y) ⋯

Tactic:
rfl

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Pow.Real

Goal state before:
pp: x y z : ℝ
hx0 : 0 < x
hx1 : x < 1
hyz : z < y
⊢ x ^ y < x ^ z

Tactic:
rpow_def_of_pos hx0

Goal state after:
x y z : ℝ
hx0 : 0 < x
hx1 : x < 1
hyz : z < y
⊢ rexp (log x * y) < x ^ z
Module: Mathlib.Analysis.SpecialFunctions.Pow.Real

Goal state before:
pp: x y z : ℝ
hx0 : 0 < x
hx1 : x < 1
hyz : z < y
⊢ rexp (log x * y) < x ^ z

Tactic:
rpow_def_of_pos hx0

Goal state after:
x y z : ℝ
hx0 : 0 < x
hx1 : x < 1
hyz : z < y
⊢ rexp (log x * y) < rexp (log x * z)
Module: Mathlib.Analysis.SpecialFunctions.Pow.Real

Goal state before:
pp: x y z : ℝ
hx0 : 0 < x
hx1 : x < 1
hyz : z < y
⊢ rexp (log x * y) < rexp (log x * z)

Tactic:
rpow_def_of_pos hx0

Goal state after:
x y z : ℝ
hx0 : 0 < x
hx1 : x < 1
hyz : z < y
⊢ rexp (log x * y) < rexp (log x * z)
Module: Mathlib.Analysis.SpecialFunctions.Pow.Real

Goal state before:
pp: x y z : ℝ
hx0 : 0 < x
hx1 : x ≤ 1
hyz : z ≤ y
⊢ x ^ y ≤ x ^ z

Tactic:
rpow_def_of_pos hx0

Goal state after:
x y z : ℝ
hx0 : 0 < x
hx1 : x ≤ 1
hyz : z ≤ y
⊢ rexp (log x * y) ≤ x ^ z
Module: Mathlib.Analysis.SpecialFunctions.Pow.Real

Goal state before:
pp: x y z : ℝ
hx0 : 0 < x
hx1 : x ≤ 1
hyz : z ≤ y
⊢ rexp (log x * y) ≤ x ^ z

Tactic:
rpow_def_of_pos hx0

Goal state after:
x y z : ℝ
hx0 : 0 < x
hx1 : x ≤ 1
hyz : z ≤ y
⊢ rexp (log x * y) ≤ rexp (log x * z)
Module: Mathlib.Analysis.SpecialFunctions.Pow.Real

Goal state before:
pp: x y z : ℝ
hx0 : 0 < x
hx1 : x ≤ 1
hyz : z ≤ y
⊢ rexp (log x * y) ≤ rexp (log x * z)

Tactic:
rpow_def_of_pos hx0

Goal state after:
x y z : ℝ
hx0 : 0 < x
hx1 : x ≤ 1
hyz : z ≤ y
⊢ rexp (log x * y) ≤ rexp (log x * z)
Module: Mathlib.Analysis.SpecialFunctions.Pow.Continuity

Goal state before:
pp: x : ℝ
y : ℂ
h : 0 < y.re ∨ x ≠ 0
hx : x < 0
this : ContinuousAt (fun p ↦ (-↑p.1) ^ p.2 * cexp (↑π * I * p.2)) (x, y)
⊢ ContinuousAt (fun p ↦ ↑p.1 ^ p.2) (x, y)

Tactic:
by
      refine this.congr (eventually_of_mem (prod_mem_nhds (Iio_mem_nhds hx) univ_mem) ?_)
      exact fun p hp => (ofReal_cpow_of_nonpos (le_of_lt hp.1) p.2).symm

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Pow.Real

Goal state before:
pp: x : ℝ
hx : x ≠ 0
y : ℝ
hy : y ∈ {y | 0 ≤ y}
z : ℝ
hz : z ∈ {y | 0 ≤ y}
hyz : y ^ x = z ^ x
⊢ y ^ 1 = z

Tactic:
← rpow_one z,

Goal state after:
x : ℝ
hx : x ≠ 0
y : ℝ
hy : y ∈ {y | 0 ≤ y}
z : ℝ
hz : z ∈ {y | 0 ≤ y}
hyz : y ^ x = z ^ x
⊢ y ^ 1 = z ^ 1
Module: Mathlib.Analysis.SpecialFunctions.Pow.Continuity

Goal state before:
pp: case inr.inr
y : ℝ
x : ℝ≥0∞
hy : y < 0
z : ℝ
hz : y = -z
z_pos : 0 < z
⊢ ContinuousAt (fun a ↦ (a ^ z)⁻¹) x

Tactic:
exact continuous_inv.continuousAt.comp (continuousAt_rpow_const_of_pos z_pos)

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Pow.NNReal

Goal state before:
pp: x y : ℝ≥0∞
hx : x ≠ ∞
hy : y ≠ ∞
z : ℝ
⊢ (x * y) ^ z = x ^ z * y ^ z

Tactic:
by simp [*, mul_rpow_eq_ite]

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Pow.NNReal

Goal state before:
pp: x y : ℝ≥0∞
hx : x ≠ 0
hy : y ≠ 0
z : ℝ
⊢ (x * y) ^ z = x ^ z * y ^ z

Tactic:
by simp [*, mul_rpow_eq_ite]

Goal state after:
Module: Mathlib.LinearAlgebra.Ray

Goal state before:
pp: case inl
R : Type u_1
inst✝⁶ : CommRing R
inst✝⁵ : PartialOrder R
inst✝⁴ : IsStrictOrderedRing R
M : Type u_2
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : IsDomain R
inst✝ : IsTorsionFree R M
r : R
hr : r < 0
⊢ 0 = 0

Tactic:
rfl

Goal state after:
Module: Mathlib.LinearAlgebra.Ray

Goal state before:
pp: case inr.inr
R : Type u_1
inst✝⁶ : CommRing R
inst✝⁵ : PartialOrder R
inst✝⁴ : IsStrictOrderedRing R
M : Type u_2
inst✝³ : AddCommGroup M
inst✝² : Module R M
x : M
inst✝¹ : IsDomain R
inst✝ : IsTorsionFree R M
r : R
hr : r < 0
r₁ r₂ : R
hr₁ : 0 < r₁
hr₂ : 0 < r₂
h : r₁ • x = r₂ • r • x
⊢ x = 0

Tactic:
rw [← sub_eq_zero, smul_smul, ← sub_smul, smul_eq_zero] at h
    refine h.resolve_left (ne_of_gt <| sub_pos.2 ?_)
    exact (mul_neg_of_pos_of_neg hr₂ hr).trans hr₁

Goal state after:
Module: Mathlib.LinearAlgebra.Ray

Goal state before:
pp: R : Type u_1
inst✝⁶ : CommRing R
inst✝⁵ : PartialOrder R
inst✝⁴ : IsStrictOrderedRing R
M : Type u_2
N : Type u_3
inst✝³ : AddCommGroup M
inst✝² : AddCommGroup N
inst✝¹ : Module R M
inst✝ : Module R N
f : M ≃ₗ[R] N
v : Ray R M
⊢ (map f) (-v) = -(map f) v

Tactic:
induction v using Module.Ray.ind with

Goal state after:
Module: Mathlib.LinearAlgebra.Ray

Goal state before:
pp: R : Type u_1
inst✝⁵ : CommRing R
inst✝⁴ : LinearOrder R
inst✝³ : IsStrictOrderedRing R
M : Type u_2
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : IsTorsionFree R M
u : Rˣ
v : Ray R M
⊢ u • v = v ↔ 0 < ↑u

Tactic:
induction v using Module.Ray.ind with

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

Goal state before:
pp: k : Type u_1
V1 : Type u_2
P1 : Type u_3
V2 : Type u_4
P2 : Type u_5
V3 : Type u_6
P3 : Type u_7
V4 : Type u_8
P4 : Type u_9
inst✝¹⁵ : Ring k
inst✝¹⁴ : AddCommGroup V1
inst✝¹³ : Module k V1
inst✝¹² : AffineSpace V1 P1
inst✝¹¹ : AddCommGroup V2
inst✝¹⁰ : Module k V2
inst✝⁹ : AffineSpace V2 P2
inst✝⁸ : AddCommGroup V3
inst✝⁷ : Module k V3
inst✝⁶ : AffineSpace V3 P3
inst✝⁵ : AddCommGroup V4
inst✝⁴ : Module k V4
inst✝³ : AffineSpace V4 P4
R : Type u_10
inst✝² : Monoid R
inst✝¹ : DistribMulAction R V2
inst✝ : SMulCommClass k R V2
c : R
f : P1 →ᵃ[k] V2
p : P1
v : V1
⊢ (c • ⇑f) (v +ᵥ p) = (c • f.linear) v +ᵥ (c • ⇑f) p

Tactic:
by simp [smul_add]

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

Goal state before:
pp: k : Type u_1
V1 : Type u_2
P1 : Type u_3
V2 : Type u_4
P2 : Type u_5
V3 : Type u_6
P3 : Type u_7
V4 : Type u_8
P4 : Type u_9
inst✝¹² : Ring k
inst✝¹¹ : AddCommGroup V1
inst✝¹⁰ : Module k V1
inst✝⁹ : AffineSpace V1 P1
inst✝⁸ : AddCommGroup V2
inst✝⁷ : Module k V2
inst✝⁶ : AffineSpace V2 P2
inst✝⁵ : AddCommGroup V3
inst✝⁴ : Module k V3
inst✝³ : AffineSpace V3 P3
inst✝² : AddCommGroup V4
inst✝¹ : Module k V4
inst✝ : AffineSpace V4 P4
f : P2 →ᵃ[k] P3
g : P1 →ᵃ[k] P2
p : P1
v : V1
⊢ f.linear (g.linear v) +ᵥ f (g p) = (f.linear ∘ₗ g.linear) v +ᵥ (⇑f ∘ ⇑g) p

Tactic:
rfl

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

Goal state before:
pp: k : Type u_1
V1 : Type u_2
P1 : Type u_3
V2 : Type u_4
P2 : Type u_5
inst✝⁶ : Ring k
inst✝⁵ : AddCommGroup V1
inst✝⁴ : Module k V1
inst✝³ : AffineSpace V1 P1
inst✝² : AddCommGroup V2
inst✝¹ : Module k V2
inst✝ : AffineSpace V2 P2
s t : Set P1
f : P1 →ᵃ[k] P2
v : V2
⊢ (∃ a ∈ s, ∃ a_1 ∈ t, f.linear (a -ᵥ a_1) = v) ↔ ∃ x, (∃ x_1 ∈ s, ∃ y ∈ t, x_1 -ᵥ y = x) ∧ f.linear x = v

Tactic:
grind

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

Goal state before:
pp: k : Type u_10
inst✝² : Field k
inst✝¹ : LinearOrder k
inst✝ : IsStrictOrderedRing k
f : k →ᵃ[k] k
a b x : k
⊢ f x = x • (f 1 -ᵥ f 0) +ᵥ f 0

Tactic:
rw [← f.linearMap_vsub, ← f.linear.map_smul, ← f.map_vadd]

Goal state after:
k : Type u_10
inst✝² : Field k
inst✝¹ : LinearOrder k
inst✝ : IsStrictOrderedRing k
f : k →ᵃ[k] k
a b x : k
⊢ f x = f (x • (1 -ᵥ 0) +ᵥ 0)
Module: Mathlib.LinearAlgebra.Ray

Goal state before:
pp: R : Type u_1
inst✝⁵ : CommRing R
inst✝⁴ : LinearOrder R
inst✝³ : IsStrictOrderedRing R
M : Type u_2
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : IsTorsionFree R M
x y : M
⊢ SameRay R x y ∨ x ≠ 0 ∧ y ≠ 0 ∧ SameRay R x (-y) ↔ ¬LinearIndependent R ![x, y]

Tactic:
by
  rw [← sameRay_or_sameRay_neg_iff_not_linearIndependent]
  by_cases hx : x = 0; · simp [hx]
  by_cases hy : y = 0 <;> simp [hx, hy]

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

Goal state before:
pp: k : Type u_2
V1 : Type u_3
P1 : Type u_4
inst✝⁵ : Ring k
inst✝⁴ : AddCommGroup V1
inst✝³ : AffineSpace V1 P1
inst✝² : Module k V1
ι : Type u_9
φv : ι → Type u_10
inst✝¹ : (i : ι) → AddCommGroup (φv i)
inst✝ : (i : ι) → Module k (φv i)
p✝ : P1
x✝ : ι
⊢ (pi fun x ↦ 0) p✝ x✝ = 0 p✝ x✝

Tactic:
rfl

Goal state after:
Module: Mathlib.Analysis.Convex.Segment

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁵ : Ring 𝕜
inst✝⁴ : LinearOrder 𝕜
inst✝³ : IsStrictOrderedRing 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
inst✝ : Invertible 2
x y : E
⊢ midpoint 𝕜 x y ∈ openSegment 𝕜 x y

Tactic:
openSegment_eq_image_lineMap

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝⁵ : Ring 𝕜
inst✝⁴ : LinearOrder 𝕜
inst✝³ : IsStrictOrderedRing 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
inst✝ : Invertible 2
x y : E
⊢ midpoint 𝕜 x y ∈ ⇑(AffineMap.lineMap x y) '' Ioo 0 1
Module: Mathlib.Analysis.Convex.Segment

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁶ : Ring 𝕜
inst✝⁵ : LinearOrder 𝕜
inst✝⁴ : IsStrictOrderedRing 𝕜
inst✝³ : AddCommGroup E
inst✝² : Module 𝕜 E
x y : E
inst✝¹ : DenselyOrdered 𝕜
inst✝ : IsTorsionFree 𝕜 E
⊢ y ∈ openSegment 𝕜 y x ↔ x = y

Tactic:
left_mem_openSegment_iff,

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝⁶ : Ring 𝕜
inst✝⁵ : LinearOrder 𝕜
inst✝⁴ : IsStrictOrderedRing 𝕜
inst✝³ : AddCommGroup E
inst✝² : Module 𝕜 E
x y : E
inst✝¹ : DenselyOrdered 𝕜
inst✝ : IsTorsionFree 𝕜 E
⊢ y = x ↔ x = y
Module: Mathlib.Analysis.Convex.Segment

Goal state before:
pp: case inr.inl
𝕜 : Type u_1
E : Type u_2
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
x y : E
a : 𝕜
h₀ : 0 < a
h₁ : a < 1
⊢ (lineMap x y) a = (lineMap x y) a

Tactic:
rfl

Goal state after:
Module: Mathlib.Analysis.Convex.Segment

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁶ : Semiring 𝕜
inst✝⁵ : PartialOrder 𝕜
inst✝⁴ : AddCommMonoid E
inst✝³ : LinearOrder E
inst✝² : IsOrderedAddMonoid E
inst✝¹ : Module 𝕜 E
inst✝ : PosSMulMono 𝕜 E
x y : E
⊢ [x -[𝕜] y] ⊆ uIcc x y

Tactic:
by
  rcases le_total x y with h | h
  · rw [uIcc_of_le h]
    exact segment_subset_Icc h
  · rw [uIcc_of_ge h, segment_symm]
    exact segment_subset_Icc h

Goal state after:
Module: Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Defs

Goal state before:
pp: k : Type u_1
V : Type u_2
P : Type u_3
inst✝³ : Ring k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
s : AffineSubspace k P
h : (↑s).Nonempty
⊢ ↑{ carrier := ↑s -ᵥ ↑s, add_mem' := ⋯, zero_mem' := ⋯ } ⊆ ↑(vectorSpan k ↑s)

Tactic:
AddSubmonoid.coe_set_mk

Goal state after:
k : Type u_1
V : Type u_2
P : Type u_3
inst✝³ : Ring k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
s : AffineSubspace k P
h : (↑s).Nonempty
⊢ ↑{ carrier := ↑s -ᵥ ↑s, add_mem' := ⋯ } ⊆ ↑(vectorSpan k ↑s)
Module: Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Defs

Goal state before:
pp: k : Type u_1
V : Type u_2
inst✝² : Ring k
inst✝¹ : AddCommGroup V
inst✝ : Module k V
s : Set V
x✝ : V
⊢ (∃ x ∈ vectorSpan k s, ∃ y ∈ s, x + y = x✝) ↔ ∃ p₁ ∈ s, ∃ v ∈ vectorSpan k s, x✝ = v + p₁

Tactic:
grind

Goal state after:
Module: Mathlib.Analysis.Convex.Hull

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝³ : Semiring 𝕜
inst✝² : PartialOrder 𝕜
inst✝¹ : AddCommMonoid E
inst✝ : Module 𝕜 E
s : Set E
hs : Convex 𝕜 s
x : E
hx : x ∉ (convexHull 𝕜) (s \ {x})
h : s \ {x} = (convexHull 𝕜) (s \ {x})
⊢ Convex 𝕜 (s \ {x})

Tactic:
by
    rw [h]
    exact convex_convexHull 𝕜 _

Goal state after:
Module: Mathlib.Analysis.Convex.Hull

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝³ : Semiring 𝕜
inst✝² : PartialOrder 𝕜
inst✝¹ : AddCommMonoid E
inst✝ : Module 𝕜 E
s : Set E
hs : Convex 𝕜 s
x : E
⊢ Convex 𝕜 (s \ {x}) ↔ x ∉ (convexHull 𝕜) (s \ {x})

Tactic:
by
  constructor
  · rintro hsx hx
    rw [hsx.convexHull_eq] at hx
    exact hx.2 (mem_singleton _)
  rintro hx
  suffices h : s \ {x} = convexHull 𝕜 (s \ {x}) by
    rw [h]
    exact convex_convexHull 𝕜 _
  exact
    Subset.antisymm (subset_convexHull 𝕜 _) fun y hy =>
      ⟨convexHull_min sdiff_subset hs hy, by
        rintro (rfl : y = x)
        exact hx hy⟩

Goal state after:
Module: Mathlib.Topology.Bornology.Absorbs

Goal state before:
pp: G₀ : Type u_1
α : Type u_2
inst✝² : GroupWithZero G₀
inst✝¹ : Bornology G₀
inst✝ : MulAction G₀ α
s t : Set α
c : G₀
hc : c ≠ 0
⊢ t ⊆ (fun x ↦ c⁻¹ • x) ⁻¹' s ↔ MapsTo (fun x ↦ c⁻¹ • x) t s

Tactic:
rfl

Goal state after:
Module: Mathlib.Analysis.LocallyConvex.Basic

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_3
inst✝¹ : SeminormedRing 𝕜
inst✝ : SMul 𝕜 E
A B : Set E
⊢ (∃ i, ∀ ⦃x : 𝕜⦄, x ∈ norm ⁻¹' Ici i → B ⊆ x • A) ↔ ∃ r, ∀ (c : 𝕜), r ≤ ‖c‖ → B ⊆ c • A

Tactic:
rfl

Goal state after:
Module: Mathlib.Analysis.LocallyConvex.Basic

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_3
inst✝² : NormedDivisionRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Set E
⊢ (∀ᶠ (c : 𝕜) in Bornology.cobounded 𝕜, MapsTo (fun x ↦ c⁻¹ • x) t s) ↔
    ∀ᶠ (c : 𝕜) in (Bornology.cobounded 𝕜)⁻¹, MapsTo (fun x ↦ c • x) t s

Tactic:
rfl

Goal state after:
Module: Mathlib.Analysis.LocallyConvex.Basic

Goal state before:
pp: 𝕜 : Type u_1
𝕝 : Type u_2
E : Type u_3
inst✝⁷ : NormedDivisionRing 𝕜
inst✝⁶ : AddCommGroup E
inst✝⁵ : Module 𝕜 E
s : Set E
inst✝⁴ : NormedRing 𝕝
inst✝³ : Module 𝕜 𝕝
inst✝² : NormSMulClass 𝕜 𝕝
inst✝¹ : SMulWithZero 𝕝 E
inst✝ : IsScalarTower 𝕜 𝕝 E
b : 𝕜
hs : Balanced 𝕝 s
a : 𝕝
h : ‖a‖ ≤ ‖b‖
⊢ a • s ⊆ b • s

Tactic:
obtain rfl | hb := eq_or_ne b 0
  · rw [norm_zero, norm_le_zero_iff] at h
    simp only [h, ← image_smul, zero_smul, Subset.rfl]
  · calc
      a • s = b • (b⁻¹ • a) • s := by rw [smul_assoc, smul_inv_smul₀ hb]
      _ ⊆ b • s := smul_set_mono <| hs _ <| by
        rw [norm_smul, norm_inv, ← div_eq_inv_mul]
        exact div_le_one_of_le₀ h (norm_nonneg _)

Goal state after:
Module: Mathlib.Analysis.LocallyConvex.Basic

Goal state before:
pp: case inl
𝕜 : Type u_1
E : Type u_3
inst✝⁴ : NormedField 𝕜
inst✝³ : AddCommGroup E
inst✝² : Module 𝕜 E
A : Set E
inst✝¹ : TopologicalSpace E
inst✝ : ContinuousSMul 𝕜 E
hA : Balanced 𝕜 A
ha : ‖0‖ ≤ 1
⊢ 0 ⊆ insert 0 (interior A)

pp: case inl
𝕜 : Type u_1
E : Type u_3
inst✝⁴ : NormedField 𝕜
inst✝³ : AddCommGroup E
inst✝² : Module 𝕜 E
A : Set E
inst✝¹ : TopologicalSpace E
inst✝ : ContinuousSMul 𝕜 E
hA : Balanced 𝕜 A
ha : ‖0‖ ≤ 1
⊢ (insert 0 (interior A)).Nonempty

Tactic:
exacts [subset_union_left, ⟨0, Or.inl rfl⟩]

Goal state after:
Module: Mathlib.Analysis.LocallyConvex.Basic

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_3
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
s : Set E
inst✝ : PartialOrder 𝕜
hs : Balanced 𝕜 s
⊢ Balanced 𝕜 ((convexHull 𝕜) s)

Tactic:
by
  suffices Convex 𝕜 { x | ∀ a : 𝕜, ‖a‖ ≤ 1 → a • x ∈ convexHull 𝕜 s } by
    rw [balanced_iff_smul_mem] at hs ⊢
    refine fun a ha x hx => convexHull_min ?_ this hx a ha
    exact fun y hy a ha => subset_convexHull 𝕜 s (hs ha hy)
  intro x hx y hy u v hu hv huv a ha
  rw [smul_add, ← smul_comm u, ← smul_comm v]
  exact convex_convexHull 𝕜 s (hx a ha) (hy a ha) hu hv huv

Goal state after:
Module: Mathlib.Analysis.LocallyConvex.BalancedCoreHull

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁴ : NormedDivisionRing 𝕜
inst✝³ : AddCommGroup E
inst✝² : Module 𝕜 E
inst✝¹ : TopologicalSpace E
inst✝ : ContinuousConstSMul 𝕜 E
s : Set E
hs : IsOpen[inst✝¹] s
hzero : 0 ∈ s
⊢ ∀ (i : 𝕜), ‖i‖ ≤ 1 ∧ i ≠ 0 → ∃ i', ∃ (_ : ‖i'‖ ≤ 1), i • s ⊆ i' • s

Tactic:
grind

Goal state after:
Module: Mathlib.Analysis.LocallyConvex.Bounded

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_3
inst✝⁵ : NormedField 𝕜
inst✝⁴ : AddCommGroup E
inst✝³ : Module 𝕜 E
inst✝² : TopologicalSpace E
inst✝¹ : ContinuousSMul 𝕜 E
inst✝ : ContinuousAdd E
s : Set E
hs : IsVonNBounded 𝕜 s
x : E
⊢ IsVonNBounded 𝕜 ({x} +ᵥ s)

Tactic:
exact IsVonNBounded.add (isVonNBounded_singleton x) hs

Goal state after:
Module: Mathlib.Analysis.Convex.Function

Goal state before:
pp: 𝕜 : Type u_1
β : Type u_5
inst✝⁴ : Semiring 𝕜
inst✝³ : PartialOrder 𝕜
inst✝² : AddCommMonoid β
inst✝¹ : PartialOrder β
inst✝ : SMul 𝕜 β
s : Set β
hs : Convex 𝕜 s
x✝ : β
a✝⁵ : x✝ ∈ s
y✝ : β
a✝⁴ : y✝ ∈ s
a✝³ b✝ : 𝕜
a✝² : 0 ≤ a✝³
a✝¹ : 0 ≤ b✝
a✝ : a✝³ + b✝ = 1
⊢ _root_.id (a✝³ • x✝ + b✝ • y✝) ≤ a✝³ • _root_.id x✝ + b✝ • _root_.id y✝

Tactic:
rfl

Goal state after:
Module: Mathlib.Analysis.Convex.Function

Goal state before:
pp: 𝕜 : Type u_1
β : Type u_5
inst✝⁴ : Semiring 𝕜
inst✝³ : PartialOrder 𝕜
inst✝² : AddCommMonoid β
inst✝¹ : PartialOrder β
inst✝ : SMul 𝕜 β
s : Set β
hs : Convex 𝕜 s
x✝ : β
a✝⁵ : x✝ ∈ s
y✝ : β
a✝⁴ : y✝ ∈ s
a✝³ b✝ : 𝕜
a✝² : 0 ≤ a✝³
a✝¹ : 0 ≤ b✝
a✝ : a✝³ + b✝ = 1
⊢ a✝³ • _root_.id x✝ + b✝ • _root_.id y✝ ≤ _root_.id (a✝³ • x✝ + b✝ • y✝)

Tactic:
rfl

Goal state after:
Module: Mathlib.Analysis.Convex.Strict

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_3
inst✝⁵ : Ring 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : TopologicalSpace E
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
s : Set E
x y : E
inst✝ : AddRightMono 𝕜
h : StrictConvex 𝕜 s
hx : x ∈ s
hy : y ∈ s
hxy : x ≠ y
t : 𝕜
ht₀ : 0 < t
ht₁ : t < 1
⊢ x + t • (y - x) ∈ interior s

Tactic:
apply h.openSegment_subset hx hy hxy
  rw [openSegment_eq_image']
  exact mem_image_of_mem _ ⟨ht₀, ht₁⟩

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

Goal state before:
pp: k : Type u_1
V : Type u_2
P : Type u_3
inst✝² : Ring k
inst✝¹ : AddCommGroup V
inst✝ : Module k V
S : AffineSpace V P
ι : Type u_4
s : Finset ι
w₁ w₂ : ι → k
hw : ∀ i ∈ s, w₁ i = w₂ i
p₁ p₂ : ι → P
hp : ∀ i ∈ s, p₁ i = p₂ i
b : P
⊢ (s.weightedVSubOfPoint p₁ b) w₁ = (s.weightedVSubOfPoint p₂ b) w₂

Tactic:
by
  simp_rw [weightedVSubOfPoint_apply]
  refine sum_congr rfl fun i hi => ?_
  rw [hw i hi, hp i hi]

Goal state after:
Module: Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Basic

Goal state before:
pp: case convert_1
k : Type u_1
V : Type u_2
P : Type u_3
inst✝³ : Ring k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
s : Set V
p : P
h : Submodule.span k (range Subtype.val) = ⊤
⊢ Submodule.span k (range Subtype.val) ≤ Submodule.span k ((fun x ↦ x -ᵥ p) '' ({p} ∪ (fun v ↦ v +ᵥ p) '' s))

Tactic:
apply Submodule.span_mono

Goal state after:
case convert_1
k : Type u_1
V : Type u_2
P : Type u_3
inst✝³ : Ring k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
s : Set V
p : P
h : Submodule.span k (range Subtype.val) = ⊤
⊢ range Subtype.val ⊆ (fun x ↦ x -ᵥ p) '' ({p} ∪ (fun v ↦ v +ᵥ p) '' s)
Module: Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Basic

Goal state before:
pp: k : Type u_1
V : Type u_2
P : Type u_3
inst✝³ : Ring k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
s : Set V
p : P
h : Submodule.span k (range Subtype.val) = ⊤
⊢ affineSpan k ({p} ∪ (fun v ↦ v +ᵥ p) '' s) = ⊤

Tactic:
by
  convert!
    ext_of_direction_eq _
      ⟨p, mem_affineSpan k (Set.mem_union_left _ (Set.mem_singleton _)), mem_top k V p⟩
  rw [direction_affineSpan, direction_top,
    vectorSpan_eq_span_vsub_set_right k (Set.mem_union_left _ (Set.mem_singleton _) : p ∈ _),
    eq_top_iff, ← h]
  apply Submodule.span_mono
  rintro v ⟨v', rfl⟩
  use (v' : V) +ᵥ p
  simp

Goal state after:
Module: Mathlib.Analysis.Seminorm

Goal state before:
pp: R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
ι : Type u_11
inst✝² : SeminormedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
f : E → ℝ
add_le : ∀ (x y : E), f (x + y) ≤ f x + f y
smul : ∀ (a : 𝕜) (x : E), f (a • x) = ‖a‖ * f x
x : E
⊢ f (-x) = f x

Tactic:
rw [← neg_one_smul 𝕜, smul, norm_neg, ← smul, one_smul]

Goal state after:
Module: Mathlib.Analysis.Seminorm

Goal state before:
pp: case inl
𝕜 : Type u_3
E : Type u_7
ι : Type u_11
inst✝² : SeminormedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p : ι → Seminorm 𝕜 E
x : E
⊢ (∅.sup p) x = 0

Tactic:
rfl

Goal state after:
Module: Mathlib.Analysis.Seminorm

Goal state before:
pp: 𝕜 : Type u_3
E : Type u_7
ι : Type u_11
inst✝² : SeminormedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p : ι → Seminorm 𝕜 E
s : Finset ι
C : ℝ≥0
x : E
⊢ (s.sup (C • p)) x = C • (s.sup p) x

Tactic:
finset_sup_apply,

Goal state after:
𝕜 : Type u_3
E : Type u_7
ι : Type u_11
inst✝² : SeminormedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p : ι → Seminorm 𝕜 E
s : Finset ι
C : ℝ≥0
x : E
⊢ ↑(s.sup fun i ↦ NNReal.mk (((C • p) i) x) ⋯) = C • (s.sup p) x
Module: Mathlib.LinearAlgebra.AffineSpace.Combination

Goal state before:
pp: k : Type u_1
V : Type u_2
P : Type u_3
inst✝² : Ring k
inst✝¹ : AddCommGroup V
inst✝ : Module k V
S : AffineSpace V P
ι : Type u_4
s : Finset ι
w : ι → k
p₁ p₂ : ι → P
⊢ ∑ i ∈ s, w i • (p₁ i -ᵥ p₂ i) =
    (s.weightedVSubOfPoint p₁ (Classical.choice ⋯)) w - (s.weightedVSubOfPoint p₂ (Classical.choice ⋯)) w

Tactic:
exact s.sum_smul_vsub_eq_weightedVSubOfPoint_sub _ _ _ _

Goal state after:
Module: Mathlib.Analysis.Seminorm

Goal state before:
pp: 𝕜 : Type u_3
E : Type u_7
ι : Type u_11
inst✝² : SeminormedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p : ι → Seminorm 𝕜 E
s : Finset ι
x : E
a : ℝ≥0
h : ∀ i ∈ s, (p i) x ≤ ↑a
⊢ (s.sup p) x ≤ ↑a

Tactic:
finset_sup_apply,

Goal state after:
𝕜 : Type u_3
E : Type u_7
ι : Type u_11
inst✝² : SeminormedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p : ι → Seminorm 𝕜 E
s : Finset ι
x : E
a : ℝ≥0
h : ∀ i ∈ s, (p i) x ≤ ↑a
⊢ ↑(s.sup fun i ↦ NNReal.mk ((p i) x) ⋯) ≤ ↑a
Module: Mathlib.Analysis.Seminorm

Goal state before:
pp: 𝕜 : Type u_3
E : Type u_7
ι : Type u_11
inst✝² : SeminormedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p : ι → Seminorm 𝕜 E
s : Finset ι
x : E
a : ℝ≥0
ha : 0 < ↑a
h : ∀ i ∈ s, (p i) x < ↑a
⊢ (s.sup p) x < ↑a

Tactic:
finset_sup_apply,

Goal state after:
𝕜 : Type u_3
E : Type u_7
ι : Type u_11
inst✝² : SeminormedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p : ι → Seminorm 𝕜 E
s : Finset ι
x : E
a : ℝ≥0
ha : 0 < ↑a
h : ∀ i ∈ s, (p i) x < ↑a
⊢ ↑(s.sup fun i ↦ NNReal.mk ((p i) x) ⋯) < ↑a