mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.SpecialFunctions.Pow.Continuity

Goal state before:
pp: case inr.inr
x : ℝ
y : ℂ
h : 0 < y.re ∨ x ≠ 0
hx : x < 0
⊢ ContinuousAt (fun p ↦ ↑p.1 ^ p.2) (x, y)

Tactic:
suffices ContinuousAt (fun p => (-(p.1 : ℂ)) ^ p.2 * exp (π * I * p.2) : ℝ × ℂ → ℂ) (x, y) 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
    have A : ContinuousAt (fun p => ⟨-↑p.1, p.2⟩ : ℝ × ℂ → ℂ × ℂ) (x, y) := by fun_prop
    apply ContinuousAt.mul
    · refine (continuousAt_cpow (Or.inl ?_)).comp A
      rwa [neg_re, ofReal_re, neg_pos]
    · fun_prop

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

Goal state before:
pp: case neg
y z : ℝ
x : ℝ≥0
h : ¬x = 0
⊢ ↑x ^ (y * z) = (↑x ^ y) ^ z

Tactic:
have : x ^ y ≠ 0 := by simp [h]
      simp [← coe_rpow_of_ne_zero, h, this, NNReal.rpow_mul]

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
f : P1 →ᵃ[k] P2
p : P1
h : ⇑f.linear = ⇑(Equiv.vaddConst (f p)).symm ∘ ⇑f ∘ ⇑(Equiv.vaddConst p)
⊢ Function.Injective (⇑(Equiv.vaddConst (f p)).symm ∘ ⇑f ∘ ⇑(Equiv.vaddConst p)) ↔ Function.Injective ⇑f

Tactic:
Equiv.comp_injective,

Goal state after:
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
f : P1 →ᵃ[k] P2
p : P1
h : ⇑f.linear = ⇑(Equiv.vaddConst (f p)).symm ∘ ⇑f ∘ ⇑(Equiv.vaddConst p)
⊢ Function.Injective (⇑f ∘ ⇑(Equiv.vaddConst p)) ↔ Function.Injective ⇑f
Module: Mathlib.LinearAlgebra.AffineSpace.AffineMap

Goal state before:
pp: k : Type u_1
V1 : Type u_2
inst✝⁶ : Ring k
inst✝⁵ : AddCommGroup V1
inst✝⁴ : Module k V1
inst✝³ : LinearOrder k
inst✝² : Preorder V1
inst✝¹ : AddLeftMono V1
inst✝ : SMulPosMono k V1
p₀ p₁ : V1
h : p₁ ≤ p₀
⊢ Antitone ⇑(lineMap p₀ p₁)

Tactic:
intro x y hxy
  suffices y • (p₁ - p₀) ≤ x • (p₁ - p₀) by simpa [lineMap]
  rw [← neg_le_neg_iff, ← smul_neg, ← smul_neg]
  gcongr
  simpa

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 = f (x • (1 -ᵥ 0) +ᵥ 0)

Tactic:
simp only [vsub_eq_sub, add_zero, mul_one, vadd_eq_add, sub_zero, smul_eq_mul]

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

Goal state before:
pp: case mpr
𝕜 : Type u_1
E : Type u_2
inst✝⁴ : Semifield 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
x y z : E
⊢ (∃ a b, 0 ≤ a ∧ 0 ≤ b ∧ 0 < a + b ∧ (a / (a + b)) • y + (b / (a + b)) • z = x) → x ∈ [y -[𝕜] z]

Tactic:
· rintro ⟨a, b, ha, hb, hab, rfl⟩
    refine ⟨a / (a + b), b / (a + b), by positivity, by positivity, ?_, rfl⟩
    rw [← add_div, div_self hab.ne']

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁵ : Semiring 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : AddCommMonoid E
inst✝² : AddCommMonoid F
inst✝¹ : Module 𝕜 E
inst✝ : Module 𝕜 F
x : E
s : Set E
hs : StarConvex 𝕜 x s
f : E →ₗ[𝕜] F
y : E
hy : y ∈ s
a b : 𝕜
ha : 0 ≤ a
hb : 0 ≤ b
hab : a + b = 1
⊢ f (a • x + b • y) = a • f x + b • f y

Tactic:
rw [f.map_add, f.map_smul, f.map_smul]

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
⊢ Submodule.span k ∅ = ⊥

Tactic:
Submodule.span_empty

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
⊢ ⊥ = ⊥
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 : Set P
⊢ vectorSpan k s = ⊥ ↔ s.Subsingleton

Tactic:
refine ⟨fun h ↦ ?_, vectorSpan_of_subsingleton _⟩

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 : Set P
h : vectorSpan k s = ⊥
⊢ s.Subsingleton
Module: Mathlib.Analysis.Convex.Star

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁴ : Ring 𝕜
inst✝³ : PartialOrder 𝕜
inst✝² : AddRightMono 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
x : E
s : Set E
hs : StarConvex 𝕜 0 s
hx : x ∈ s
t : 𝕜
ht₀ : 0 ≤ t
ht₁ : t ≤ 1
⊢ t • x ∈ s

Tactic:
by simpa using hs.add_smul_mem (by simpa using hx) ht₀ ht₁

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁵ : Semiring 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : AddCommMonoid E
inst✝² : AddCommMonoid F
inst✝¹ : Module 𝕜 E
inst✝ : Module 𝕜 F
s : Set E
hs : Convex 𝕜 s
f : E →ₗ[𝕜] F
x : E
hx : x ∈ s
y : E
hy : y ∈ s
a b : 𝕜
ha : 0 ≤ a
hb : 0 ≤ b
hab : a + b = 1
⊢ f (a • x + b • y) = a • f x + b • f y

Tactic:
rw [f.map_add, f.map_smul, f.map_smul]

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
p : P
direction : Submodule k V
v : V
⊢ v ∈ (mk' p direction).direction ↔ v ∈ direction

Tactic:
mem_direction_iff_eq_vsub (mk'_nonempty _ _)

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
p : P
direction : Submodule k V
v : V
⊢ (∃ p₁ ∈ mk' p direction, ∃ p₂ ∈ mk' p direction, v = p₁ -ᵥ p₂) ↔ v ∈ direction
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 : Set P
s₁ : AffineSubspace k P
h : s ⊆ ↑s₁
p p₁ : P
hp₁ : p₁ ∈ s
v : V
hv : v ∈ vectorSpan k s
hp : p = v +ᵥ p₁
hp₁s₁ : p₁ ∈ ↑s₁
⊢ v +ᵥ p₁ ∈ ↑s₁

Tactic:
refine vadd_mem_of_mem_direction ?_ hp₁s₁

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 : Set P
s₁ : AffineSubspace k P
h : s ⊆ ↑s₁
p p₁ : P
hp₁ : p₁ ∈ s
v : V
hv : v ∈ vectorSpan k s
hp : p = v +ᵥ p₁
hp₁s₁ : p₁ ∈ ↑s₁
⊢ v ∈ s₁.direction
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 : Set P
s₁ : AffineSubspace k P
h : s ⊆ ↑s₁
p p₁ : P
hp₁ : p₁ ∈ s
v : V
hv : v ∈ vectorSpan k s
hp : p = v +ᵥ p₁
hp₁s₁ : p₁ ∈ ↑s₁
hs : ∀ ⦃x : V⦄, x ∈ vectorSpan k s → x ∈ s₁.direction
⊢ v ∈ s₁.direction

Tactic:
rw [← SetLike.mem_coe]

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 : Set P
s₁ : AffineSubspace k P
h : s ⊆ ↑s₁
p p₁ : P
hp₁ : p₁ ∈ s
v : V
hv : v ∈ vectorSpan k s
hp : p = v +ᵥ p₁
hp₁s₁ : p₁ ∈ ↑s₁
hs : ∀ ⦃x : V⦄, x ∈ vectorSpan k s → x ∈ s₁.direction
⊢ v ∈ ↑s₁.direction
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 : Set P
s₁ : AffineSubspace k P
h : s ⊆ ↑s₁
⊢ spanPoints k s ⊆ ↑s₁

Tactic:
by
  rintro p ⟨p₁, hp₁, v, hv, hp⟩
  rw [hp]
  have hp₁s₁ : p₁ ∈ (s₁ : Set P) := Set.mem_of_mem_of_subset hp₁ h
  refine vadd_mem_of_mem_direction ?_ hp₁s₁
  have hs : vectorSpan k s ≤ s₁.direction := vectorSpan_mono k h
  rw [SetLike.le_def] at hs
  rw [← SetLike.mem_coe]
  exact Set.mem_of_mem_of_subset hv hs

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

Goal state before:
pp: 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
s : Set E
h_conv : Convex 𝕜 s
⊢ 2⁻¹ • s + 2⁻¹ • s = s

Tactic:
by
  rw [← h_conv.add_smul (by norm_num) (by norm_num)]
  ring_nf
  rw [one_smul]

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✝² : Ring k
inst✝¹ : AddCommGroup V
inst✝ : Module k V
S : AffineSpace V P
⊢ Submodule.span k ∅ = ⊥

Tactic:
Submodule.span_empty

Goal state after:
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
⊢ ⊥ = ⊥
Module: Mathlib.Topology.Bornology.Absorbs

Goal state before:
pp: M : Type u_1
E : Type u_2
inst✝² : Bornology M
inst✝¹ : AddZeroClass E
inst✝ : DistribSMul M E
s₁ s₂ t : Set E
x : E
h₁ : Absorbent M s₁
h₂ : Absorbs M s₂ t
⊢ Absorbs M (s₁ + s₂) (x +ᵥ t)

Tactic:
by
  rw [← singleton_vadd]; exact (h₁ x).add h₂

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
β : Type u_5
inst✝⁶ : Semiring 𝕜
inst✝⁵ : PartialOrder 𝕜
inst✝⁴ : AddCommMonoid E
inst✝³ : AddCommMonoid β
inst✝² : PartialOrder β
inst✝¹ : Module 𝕜 E
inst✝ : Module 𝕜 β
f : E →ₗ[𝕜] β
s : Set E
hs : Convex 𝕜 s
x✝⁸ : E
x✝⁷ : x✝⁸ ∈ s
x✝⁶ : E
x✝⁵ : x✝⁶ ∈ s
x✝⁴ x✝³ : 𝕜
x✝² : 0 ≤ x✝⁴
x✝¹ : 0 ≤ x✝³
x✝ : x✝⁴ + x✝³ = 1
⊢ f (x✝⁴ • x✝⁸ + x✝³ • x✝⁶) ≤ x✝⁴ • f x✝⁸ + x✝³ • f x✝⁶

Tactic:
rw [f.map_add, f.map_smul, f.map_smul]

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
β : Type u_5
inst✝⁶ : Semiring 𝕜
inst✝⁵ : PartialOrder 𝕜
inst✝⁴ : AddCommMonoid E
inst✝³ : AddCommMonoid β
inst✝² : PartialOrder β
inst✝¹ : Module 𝕜 E
inst✝ : Module 𝕜 β
f : E →ₗ[𝕜] β
s : Set E
hs : Convex 𝕜 s
x✝⁸ : E
x✝⁷ : x✝⁸ ∈ s
x✝⁶ : E
x✝⁵ : x✝⁶ ∈ s
x✝⁴ x✝³ : 𝕜
x✝² : 0 ≤ x✝⁴
x✝¹ : 0 ≤ x✝³
x✝ : x✝⁴ + x✝³ = 1
⊢ x✝⁴ • f x✝⁸ + x✝³ • f x✝⁶ ≤ f (x✝⁴ • x✝⁸ + x✝³ • x✝⁶)

Tactic:
rw [f.map_add, f.map_smul, f.map_smul]

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

Goal state before:
pp: k : Type u_1
V : Type u_2
inst✝⁵ : Ring k
inst✝⁴ : AddCommGroup V
inst✝³ : Module k V
ι : Type u_4
G : Type u_6
inst✝² : Group G
inst✝¹ : DistribMulAction G V
inst✝ : SMulCommClass G k V
s : Finset ι
w : ι → k
p : ι → V
b : V
a : G
⊢ (s.weightedVSubOfPoint (a • p) b) w = a • (s.weightedVSubOfPoint p (a⁻¹ • b)) w

Tactic:
simp [smul_sum, smul_sub, smul_comm a (w _)]

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) = ⊤
⊢ vectorSpan k ({p} ∪ (fun v ↦ v +ᵥ p) '' s) = ⊤

Tactic:
vectorSpan_eq_span_vsub_set_right k (Set.mem_union_left _ (Set.mem_singleton _) : p ∈ _),

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) = ⊤
⊢ Submodule.span k ((fun x ↦ x -ᵥ p) '' ({p} ∪ (fun v ↦ v +ᵥ p) '' s)) = ⊤
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✝⁵ : SeminormedRing 𝕜
inst✝⁴ : AddGroup E
inst✝³ : SMul 𝕜 E
p✝ : Seminorm 𝕜 E
x : E
r✝ : ℝ
inst✝² : SMul R ℝ
inst✝¹ : SMul R ℝ≥0
inst✝ : IsScalarTower R ℝ≥0 ℝ
r : R
p : Seminorm 𝕜 E
x✝¹ : 𝕜
x✝ : E
⊢ r • p (x✝¹ • x✝) = ‖x✝¹‖ * r • p x✝

Tactic:
simp only [← smul_one_smul ℝ≥0 r (_ : ℝ), NNReal.smul_def, smul_eq_mul]

Goal state after:
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✝⁵ : SeminormedRing 𝕜
inst✝⁴ : AddGroup E
inst✝³ : SMul 𝕜 E
p✝ : Seminorm 𝕜 E
x : E
r✝ : ℝ
inst✝² : SMul R ℝ
inst✝¹ : SMul R ℝ≥0
inst✝ : IsScalarTower R ℝ≥0 ℝ
r : R
p : Seminorm 𝕜 E
x✝¹ : 𝕜
x✝ : E
⊢ ↑(r • 1) * p (x✝¹ • x✝) = ‖x✝¹‖ * (↑(r • 1) * 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
ι₂ : Type u_5
s₂ : Finset ι₂
e : ι₂ ↪ ι
w : ι → k
p : ι → P
⊢ (affineCombination k (map e s₂) p) w = (affineCombination k s₂ (p ∘ ⇑e)) (w ∘ ⇑e)

Tactic:
by
  simp_rw [affineCombination_apply, weightedVSubOfPoint_map]

Goal state after:
Module: Mathlib.Analysis.Seminorm

Goal state before:
pp: 𝕜 : Type u_3
E : Type u_7
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
ι : Sort u_12
p : ι → Seminorm 𝕜 E
hp : BddAbove (range p)
⊢ ⇑(⨆ i, p i) = ⨆ i, ⇑(p i)

Tactic:
← sSup_range,

Goal state after:
𝕜 : Type u_3
E : Type u_7
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
ι : Sort u_12
p : ι → Seminorm 𝕜 E
hp : BddAbove (range p)
⊢ ⇑(sSup (range fun i ↦ p i)) = ⨆ i, ⇑(p i)
Module: Mathlib.LinearAlgebra.AffineSpace.Centroid

Goal state before:
pp: k : Type u_1
inst✝² : DivisionRing k
ι : Type u_4
s : Finset ι
inst✝¹ : CharZero k
inst✝ : Fintype ι
n : ℕ
h : #s = n + 1
⊢ ∑ i, centroidWeightsIndicator k s i = 1

Tactic:
by
  rw [sum_centroidWeightsIndicator]
  exact s.sum_centroidWeights_eq_one_of_card_eq_add_one k h

Goal state after:
Module: Mathlib.Analysis.Seminorm

Goal state before:
pp: 𝕜 : Type u_3
E : Type u_7
inst✝² : SeminormedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p : Seminorm 𝕜 E
r : ℝ
a : 𝕜
ha : ‖a‖ ≤ 1
x y : E
hy : y ∈ p.closedBall 0 r
hx : (fun x ↦ a • x) y = x
⊢ x ∈ p.closedBall 0 r

Tactic:
rw [mem_closedBall_zero, ← hx, map_smul_eq_mul]

Goal state after:
𝕜 : Type u_3
E : Type u_7
inst✝² : SeminormedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p : Seminorm 𝕜 E
r : ℝ
a : 𝕜
ha : ‖a‖ ≤ 1
x y : E
hy : y ∈ p.closedBall 0 r
hx : (fun x ↦ a • x) y = x
⊢ ‖a‖ * p y ≤ r
Module: Mathlib.LinearAlgebra.AffineSpace.Basis

Goal state before:
pp: ι : Type u_1
k : Type u_2
inst✝ : Ring k
w : ι →₀ k
⊢ w ∈ finsuppAffineCoords ι k ↔ (Finsupp.linearCombination k 1) w = 1

Tactic:
by
  simp [finsuppAffineCoords]

Goal state after:
Module: Mathlib.Analysis.Seminorm

Goal state before:
pp: 𝕜 : Type u_3
E : Type u_7
inst✝² : NormedDivisionRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p : Seminorm 𝕜 E
k : 𝕜
r : ℝ
x y : E
hy : y ∈ p.closedBall 0 r
h : (fun x ↦ k • x) y = x
⊢ ‖k‖ * p y ≤ ‖k‖ * r

Tactic:
Seminorm.mem_closedBall_zero

Goal state after:
𝕜 : Type u_3
E : Type u_7
inst✝² : NormedDivisionRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p : Seminorm 𝕜 E
k : 𝕜
r : ℝ
x y : E
hy : p y ≤ r
h : (fun x ↦ k • x) y = x
⊢ ‖k‖ * p y ≤ ‖k‖ * r
Module: Mathlib.Analysis.Seminorm

Goal state before:
pp: 𝕜 : Type u_3
E : Type u_7
inst✝² : NormedDivisionRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p : Seminorm 𝕜 E
k : 𝕜
r : ℝ
hk : 0 < ‖k‖
x : E
⊢ x ∈ p.closedBall 0 (‖k‖ * r) → ∃ y ∈ p.closedBall 0 r, k • y = x

Tactic:
Seminorm.mem_closedBall_zero

Goal state after:
𝕜 : Type u_3
E : Type u_7
inst✝² : NormedDivisionRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p : Seminorm 𝕜 E
k : 𝕜
r : ℝ
hk : 0 < ‖k‖
x : E
⊢ p x ≤ ‖k‖ * r → ∃ y ∈ p.closedBall 0 r, k • y = x
Module: Mathlib.Analysis.Seminorm

Goal state before:
pp: 𝕜 : Type u_3
E : Type u_7
inst✝² : NormedDivisionRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p : Seminorm 𝕜 E
r : ℝ
x : E
hpr : p x < r
⊢ Absorbent 𝕜 (p.ball x r)

Tactic:
refine (p.absorbent_ball_zero <| sub_pos.2 hpr).mono fun y hy => ?_

Goal state after:
𝕜 : Type u_3
E : Type u_7
inst✝² : NormedDivisionRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p : Seminorm 𝕜 E
r : ℝ
x : E
hpr : p x < r
y : E
hy : y ∈ p.ball 0 (r - p x)
⊢ y ∈ p.ball x r
Module: Mathlib.Analysis.Seminorm

Goal state before:
pp: case inr
𝕜 : Type u_3
E : Type u_7
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
ι : Sort u_12
p : ι → Seminorm 𝕜 E
hp : BddAbove (range p)
e : E
r : ℝ
hr : 0 < r
h✝ : Nonempty ι
x : E
this : BddAbove (range fun i ↦ (p i) (x - e))
⊢ x ∈ (⨆ i, p i).closedBall e r ↔ x ∈ ⋂ i, (p i).closedBall e r

Tactic:
simp only [mem_closedBall, mem_iInter, Seminorm.iSup_apply hp, ciSup_le_iff this]

Goal state after:
Module: Mathlib.Analysis.Seminorm

Goal state before:
pp: 𝕜 : Type u_3
E : Type u_7
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : AddCommGroup E
inst✝² : Module 𝕜 E
inst✝¹ : TopologicalSpace E
inst✝ : ContinuousConstSMul 𝕜 E
p : Seminorm 𝕜 E
r : ℝ
hp : p.closedBall 0 r ∈ 𝓝 0
ε : ℝ
hε : ε > 0
k : 𝕜
hk₀ : 0 < ‖k‖
hk : ‖k‖ * r < ε
⊢ p.closedBall 0 ε ∈ 𝓝 0

Tactic:
grw [← hk]

Goal state after:
𝕜 : Type u_3
E : Type u_7
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : AddCommGroup E
inst✝² : Module 𝕜 E
inst✝¹ : TopologicalSpace E
inst✝ : ContinuousConstSMul 𝕜 E
p : Seminorm 𝕜 E
r : ℝ
hp : p.closedBall 0 r ∈ 𝓝 0
ε : ℝ
hε : ε > 0
k : 𝕜
hk₀ : 0 < ‖k‖
hk : ‖k‖ * r < ε
⊢ p.closedBall 0 (‖k‖ * r) ∈ 𝓝 0
Module: Mathlib.LinearAlgebra.AffineSpace.Independent

Goal state before:
pp: case e_6
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
ι : Type u_4
ι2 : Type u_5
f : ι2 ↪ ι
p : ι → P
ha : AffineIndependent k p
fs : Finset ι2
w : ι2 → k
hw : ∑ i ∈ fs, w i = 0
hs : (fs.weightedVSub (p ∘ ⇑f)) w = 0
i0 : ι2
hi0 : i0 ∈ fs
fs' : Finset ι := map f fs
w' : ι → k := fun i ↦ if h : ∃ i2, f i2 = i then w h.choose else 0
hw' : ∀ (i2 : ι2), w' (f i2) = w i2
hw's : ∑ i ∈ fs', w' i = 0
i : ι2
⊢ (w' ∘ ⇑f) i = w i

Tactic:
simp_all only [comp_apply]

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

Goal state before:
pp: E : Type u_1
inst✝⁴ : AddCommGroup E
inst✝³ : Module ℝ E
inst✝² : TopologicalSpace E
inst✝¹ : ContinuousAdd E
inst✝ : ContinuousSMul ℝ E
a b c d : E
⊢ (Path.segment a b).add (Path.segment c d) = Path.segment (a + c) (b + d)

Tactic:
ext
  simp [lineMap_apply_module, add_add_add_comm]

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

Goal state before:
pp: k : Type u_1
V : Type u_2
P : Type u_5
inst✝³ : Ring k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
m n : ℕ
s : Simplex k P m
e : Fin (m + 1) ≃ Fin (n + 1)
⊢ (s.reindex e).reindex e.symm = s

Tactic:
← reindex_trans,

Goal state after:
k : Type u_1
V : Type u_2
P : Type u_5
inst✝³ : Ring k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
m n : ℕ
s : Simplex k P m
e : Fin (m + 1) ≃ Fin (n + 1)
⊢ s.reindex (e.trans e.symm) = s
Module: Mathlib.LinearAlgebra.AffineSpace.Simplex.Basic

Goal state before:
pp: k : Type u_1
V : Type u_2
P : Type u_5
inst✝³ : Ring k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
m n : ℕ
s : Simplex k P m
e : Fin (n + 1) ≃ Fin (m + 1)
⊢ (s.reindex e.symm).reindex e = s

Tactic:
← reindex_trans,

Goal state after:
k : Type u_1
V : Type u_2
P : Type u_5
inst✝³ : Ring k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
m n : ℕ
s : Simplex k P m
e : Fin (n + 1) ≃ Fin (m + 1)
⊢ s.reindex (e.symm.trans e) = s
Module: Mathlib.Analysis.Convex.StdSimplex

Goal state before:
pp: case refine_2
𝕜 : Type u_2
ι : Type u_1
inst✝³ : Semiring 𝕜
inst✝² : PartialOrder 𝕜
inst✝¹ : Fintype ι
inst✝ : IsOrderedRing 𝕜
f : ι → 𝕜
hf : f ∈ stdSimplex 𝕜 ι
g : ι → 𝕜
hg : g ∈ stdSimplex 𝕜 ι
a b : 𝕜
ha : 0 ≤ a
hb : 0 ≤ b
hab : a + b = 1
⊢ ∑ x, a • f x + ∑ x, b • g x = 1

Tactic:
← Finset.smul_sum,

Goal state after:
case refine_2
𝕜 : Type u_2
ι : Type u_1
inst✝³ : Semiring 𝕜
inst✝² : PartialOrder 𝕜
inst✝¹ : Fintype ι
inst✝ : IsOrderedRing 𝕜
f : ι → 𝕜
hf : f ∈ stdSimplex 𝕜 ι
g : ι → 𝕜
hg : g ∈ stdSimplex 𝕜 ι
a b : 𝕜
ha : 0 ≤ a
hb : 0 ≤ b
hab : a + b = 1
⊢ a • ∑ x, f x + ∑ x, b • g x = 1
Module: Mathlib.Analysis.Convex.StdSimplex

Goal state before:
pp: case refine_2
𝕜 : Type u_2
ι : Type u_1
inst✝³ : Semiring 𝕜
inst✝² : PartialOrder 𝕜
inst✝¹ : Fintype ι
inst✝ : IsOrderedRing 𝕜
f : ι → 𝕜
hf : f ∈ stdSimplex 𝕜 ι
g : ι → 𝕜
hg : g ∈ stdSimplex 𝕜 ι
a b : 𝕜
ha : 0 ≤ a
hb : 0 ≤ b
hab : a + b = 1
⊢ a • ∑ x, f x + ∑ x, b • g x = 1

Tactic:
← Finset.smul_sum,

Goal state after:
case refine_2
𝕜 : Type u_2
ι : Type u_1
inst✝³ : Semiring 𝕜
inst✝² : PartialOrder 𝕜
inst✝¹ : Fintype ι
inst✝ : IsOrderedRing 𝕜
f : ι → 𝕜
hf : f ∈ stdSimplex 𝕜 ι
g : ι → 𝕜
hg : g ∈ stdSimplex 𝕜 ι
a b : 𝕜
ha : 0 ≤ a
hb : 0 ≤ b
hab : a + b = 1
⊢ a • ∑ x, f x + b • ∑ x, g x = 1
Module: Mathlib.LinearAlgebra.AffineSpace.Independent

Goal state before:
pp: k : Type u_1
V : Type u_2
P : Type u_3
inst✝⁴ : Ring k
inst✝³ : AddCommGroup V
inst✝² : Module k V
inst✝¹ : AffineSpace V P
ι : Type u_4
inst✝ : Nontrivial k
p : ι → P
ha : AffineIndependent k p
s₁ s₂ : Set ι
i : ι
his₁ : i ∈ s₁
his₂ : i ∉ s₂
h₁ : s₁.Nontrivial
j : ι
hj : j ∈ s₁
hne : j ≠ i
he : vectorSpan k (p '' s₁) = vectorSpan k (p '' s₂)
fs : Finset ↑s₂
w : ↑s₂ → k
hw : ∑ i ∈ fs, w i = 0
hs : p i -ᵥ p j = (fs.weightedVSub fun x ↦ p ↑x) w
w' : ι → k := extend Subtype.val w 0
hw' : ∑ t ∈ map (Embedding.subtype fun x ↦ x ∈ s₂) fs, w' t = 0
hs' : p i -ᵥ p j = ((map (Embedding.subtype fun x ↦ x ∈ s₂) fs).weightedVSub p) w'
fs' : Finset ι := insert i (insert j (map (Embedding.subtype fun x ↦ x ∈ s₂) fs))
⊢ False

Tactic:
have hfsfs' : fs.map (Embedding.subtype _) ⊆ fs' := by grind

Goal state after:
k : Type u_1
V : Type u_2
P : Type u_3
inst✝⁴ : Ring k
inst✝³ : AddCommGroup V
inst✝² : Module k V
inst✝¹ : AffineSpace V P
ι : Type u_4
inst✝ : Nontrivial k
p : ι → P
ha : AffineIndependent k p
s₁ s₂ : Set ι
i : ι
his₁ : i ∈ s₁
his₂ : i ∉ s₂
h₁ : s₁.Nontrivial
j : ι
hj : j ∈ s₁
hne : j ≠ i
he : vectorSpan k (p '' s₁) = vectorSpan k (p '' s₂)
fs : Finset ↑s₂
w : ↑s₂ → k
hw : ∑ i ∈ fs, w i = 0
hs : p i -ᵥ p j = (fs.weightedVSub fun x ↦ p ↑x) w
w' : ι → k := extend Subtype.val w 0
hw' : ∑ t ∈ map (Embedding.subtype fun x ↦ x ∈ s₂) fs, w' t = 0
hs' : p i -ᵥ p j = ((map (Embedding.subtype fun x ↦ x ∈ s₂) fs).weightedVSub p) w'
fs' : Finset ι := insert i (insert j (map (Embedding.subtype fun x ↦ x ∈ s₂) fs))
hfsfs' : map (Embedding.subtype fun x ↦ x ∈ s₂) fs ⊆ fs'
⊢ False
Module: Mathlib.Analysis.Convex.Combination

Goal state before:
pp: R : Type u_1
E : Type u_3
ι : Type u_5
inst✝² : Field R
inst✝¹ : AddCommGroup E
inst✝ : Module R E
s : Finset ι
w₁ w₂ : ι → R
z : ι → E
hw₁ : ∑ i ∈ s, w₁ i = 1
hw₂ : ∑ i ∈ s, w₂ i = 1
a b : R
hab : a + b = 1
hw : ∑ i ∈ s, (a * w₁ i + b * w₂ i) = 1
⊢ a • s.centerMass w₁ z + b • s.centerMass w₂ z = s.centerMass (fun i ↦ a * w₁ i + b * w₂ i) z

Tactic:
simp only [Finset.centerMass_eq_of_sum_1, smul_sum, sum_add_distrib, add_smul, mul_smul, *]

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

Goal state before:
pp: case h₁
k : Type u_1
V : Type u_2
P : Type u_4
inst✝⁷ : Ring k
inst✝⁶ : AddCommGroup V
inst✝⁵ : Module k V
inst✝⁴ : AffineSpace V P
inst✝³ : LinearOrder k
inst✝² : IsOrderedAddMonoid k
inst✝¹ : ZeroLEOneClass k
n : ℕ
inst✝ : NeZero n
s : Simplex k P n
w : Fin (n + 1) → k
hw1 : ∑ i, w i = 1
hp : (affineCombination k univ s.points) w ∈ s.closedInterior
hp' : (affineCombination k univ s.points) w ∈ ↑(affineSpan k (Set.range s.points))
⊢ (affineCombination k univ s.points) w ∈ s.interior ∪ ⋃ i, (s.faceOpposite i).closedInterior

Tactic:
Set.mem_union,

Goal state after:
case h₁
k : Type u_1
V : Type u_2
P : Type u_4
inst✝⁷ : Ring k
inst✝⁶ : AddCommGroup V
inst✝⁵ : Module k V
inst✝⁴ : AffineSpace V P
inst✝³ : LinearOrder k
inst✝² : IsOrderedAddMonoid k
inst✝¹ : ZeroLEOneClass k
n : ℕ
inst✝ : NeZero n
s : Simplex k P n
w : Fin (n + 1) → k
hw1 : ∑ i, w i = 1
hp : (affineCombination k univ s.points) w ∈ s.closedInterior
hp' : (affineCombination k univ s.points) w ∈ ↑(affineSpan k (Set.range s.points))
⊢ (affineCombination k univ s.points) w ∈ s.interior ∨
    (affineCombination k univ s.points) w ∈ ⋃ i, (s.faceOpposite i).closedInterior