mathlib-initiative/mathlib-tactics huggingface dataset:
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
s : Set (Seminorm 𝕜 E)
H : BddAbove (DFunLike.coe '' s)
p : Seminorm 𝕜 E
hp : p ∈ s
x : E
⊢ p x ≤ (sSup s) x

Tactic:
rw [Seminorm.coe_sSup_eq' H, iSup_apply]

Goal state after:
𝕜 : Type u_3
E : Type u_7
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s : Set (Seminorm 𝕜 E)
H : BddAbove (DFunLike.coe '' s)
p : Seminorm 𝕜 E
hp : p ∈ s
x : E
⊢ p x ≤ ⨆ i, ↑i x
Module: Mathlib.LinearAlgebra.AffineSpace.Combination

Goal state before:
pp: case inl
ι : Type u_1
k : Type u_2
V : Type u_3
P : Type u_4
inst✝⁴ : Ring k
inst✝³ : AddCommGroup V
inst✝² : Module k V
inst✝¹ : AffineSpace V P
inst✝ : Nonempty ι
s : Finset ι
w : ι → k
h : ∑ i ∈ s, w i = 1
p : ι → P
hs : Subsingleton k
hnv : Subsingleton P
⊢ (Finset.affineCombination k s p) w ∈ affineSpan k (Set.range p)

Tactic:
(affineSpan_eq_top_iff_nonempty_of_subsingleton k).2 (Set.range_nonempty p)

Goal state after:
case inl
ι : Type u_1
k : Type u_2
V : Type u_3
P : Type u_4
inst✝⁴ : Ring k
inst✝³ : AddCommGroup V
inst✝² : Module k V
inst✝¹ : AffineSpace V P
inst✝ : Nonempty ι
s : Finset ι
w : ι → k
h : ∑ i ∈ s, w i = 1
p : ι → P
hs : Subsingleton k
hnv : Subsingleton P
⊢ (Finset.affineCombination k s p) w ∈ ⊤
Module: Mathlib.LinearAlgebra.AffineSpace.Pointwise

Goal state before:
pp: k : Type u_2
V : Type u_3
P : Type u_4
inst✝³ : Ring k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
v : V
s : AffineSubspace k P
⊢ (v +ᵥ s).direction = s.direction

Tactic:
by
  rw [pointwise_vadd_eq_map, map_direction]
  exact Submodule.map_id _

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

Goal state before:
pp: case mp
ι : Type u_1
k : Type u_2
V : Type u_3
P : Type u_4
inst✝⁴ : Ring k
inst✝³ : AddCommGroup V
inst✝² : Module k V
inst✝¹ : AffineSpace V P
inst✝ : Nontrivial k
p : ι → P
j : ι
q : P
⊢ q ∈ affineSpan k (Set.range p) → ∃ s w, q = (s.weightedVSubOfPoint p (p j)) w +ᵥ p j

Tactic:
intro hq

Goal state after:
case mp
ι : Type u_1
k : Type u_2
V : Type u_3
P : Type u_4
inst✝⁴ : Ring k
inst✝³ : AddCommGroup V
inst✝² : Module k V
inst✝¹ : AffineSpace V P
inst✝ : Nontrivial k
p : ι → P
j : ι
q : P
hq : q ∈ affineSpan k (Set.range p)
⊢ ∃ s w, q = (s.weightedVSubOfPoint p (p j)) w +ᵥ p j
Module: Mathlib.LinearAlgebra.AffineSpace.Independent

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
hs : ∀ v ∈ s, v ≠ 0
p₁ : P
⊢ (fun p ↦ p -ᵥ p₁) '' (({p₁} ∪ (fun v ↦ v +ᵥ p₁) '' s) \ {p₁}) = s

Tactic:
Set.union_sdiff_left,

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 V
hs : ∀ v ∈ s, v ≠ 0
p₁ : P
⊢ (fun p ↦ p -ᵥ p₁) '' ((fun v ↦ v +ᵥ p₁) '' s \ {p₁}) = s
Module: Mathlib.LinearAlgebra.AffineSpace.Basis

Goal state before:
pp: ι : Type u_1
k : Type u_5
V : Type u_6
P : Type u_7
inst✝⁴ : AddCommGroup V
inst✝³ : AffineSpace V P
inst✝² : Ring k
inst✝¹ : Module k V
b : AffineBasis ι k P
inst✝ : Fintype ι
q : P
⊢ ∑ i, (b.coord i) q = 1

Tactic:
have hq : q ∈ affineSpan k (range b) := by
    rw [b.tot]
    exact AffineSubspace.mem_top k V q
  obtain ⟨w, hw, rfl⟩ := eq_affineCombination_of_mem_affineSpan_of_fintype hq
  convert! hw
  exact b.coord_apply_combination_of_mem (Finset.mem_univ _) hw

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

Goal state before:
pp: case mpr
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
p : ι → P
⊢ (∀ (s1 s2 : Finset ι) (w1 w2 : ι → k),
      ∑ i ∈ s1, w1 i = 1 →
        ∑ i ∈ s2, w2 i = 1 →
          (affineCombination k s1 p) w1 = (affineCombination k s2 p) w2 → (↑s1).indicator w1 = (↑s2).indicator w2) →
    AffineIndependent k p

Tactic:
intro ha s w hw hs i0 hi0

Goal state after:
case mpr
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
p : ι → P
ha :
  ∀ (s1 s2 : Finset ι) (w1 w2 : ι → k),
    ∑ i ∈ s1, w1 i = 1 →
      ∑ i ∈ s2, w2 i = 1 →
        (affineCombination k s1 p) w1 = (affineCombination k s2 p) w2 → (↑s1).indicator w1 = (↑s2).indicator w2
s : Finset ι
w : ι → k
hw : ∑ i ∈ s, w i = 0
hs : (s.weightedVSub p) w = 0
i0 : ι
hi0 : i0 ∈ s
⊢ w i0 = 0
Module: Mathlib.Analysis.Seminorm

Goal state before:
pp: case refine_3
𝕜 : Type u_3
E : Type u_7
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p : Seminorm 𝕜 E
c : 𝕜
hc : 1 < ‖c‖
ε : ℝ
εpos : 0 < ε
x : E
hx : p x ≠ 0
xεpos : 0 < p x / ε
n : ℤ
hn : p x / ε ∈ Ico (‖c‖ ^ n) (‖c‖ ^ (n + 1))
cpos : 0 < ‖c‖
cnpos : 0 < ‖c ^ (n + 1)‖
⊢ ε ≤ (‖c‖ ^ n * ‖c‖)⁻¹ * p x * ‖c‖

Tactic:
mul_inv_rev,

Goal state after:
case refine_3
𝕜 : Type u_3
E : Type u_7
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p : Seminorm 𝕜 E
c : 𝕜
hc : 1 < ‖c‖
ε : ℝ
εpos : 0 < ε
x : E
hx : p x ≠ 0
xεpos : 0 < p x / ε
n : ℤ
hn : p x / ε ∈ Ico (‖c‖ ^ n) (‖c‖ ^ (n + 1))
cpos : 0 < ‖c‖
cnpos : 0 < ‖c ^ (n + 1)‖
⊢ ε ≤ ‖c‖⁻¹ * (‖c‖ ^ n)⁻¹ * p x * ‖c‖
Module: Mathlib.LinearAlgebra.AffineSpace.Independent

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
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
hs' : (fs'.weightedVSub p) w' = 0
⊢ w i0 = 0

Tactic:
rw [← ha fs' w' hw's hs' (f i0) ((Finset.mem_map' _).2 hi0), hw']

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

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
ι : Type u_4
p : ι → P
ha : AffineIndependent k p
fs : Finset ι
w : ι → k
hw : ∑ i ∈ fs, w i = 1
s : Set ι
hm : (affineCombination k fs p) w ∈ affineSpan k (p '' s)
i : ι
hifs : i ∈ fs
his : i ∉ s
fs' : Finset ι
w' : ι → k
hfs's : ↑fs' ⊆ s
hw' : ∑ i ∈ fs', w' i = 1
he : (affineCombination k fs p) w = (affineCombination k fs' p) w'
⊢ (↑fs').indicator w' i = 0

Tactic:
exact Set.indicator_of_notMem (Set.notMem_subset hfs's his) w'

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

Goal state before:
pp: case refine_2
R : Type u_1
ι : Type u_2
inst✝⁴ : Field R
inst✝³ : LinearOrder R
inst✝² : IsStrictOrderedRing R
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
w : ι → R
hw₀ : ∀ (x : ι), 0 ≤ w x
hw₁ : ∑ x, w x = 1
⊢ (∑ i, w i • fun j ↦ if i = j then 1 else 0) ∈ (convexHull R) (range fun i j ↦ if i = j then 1 else 0)

Tactic:
rw [← Finset.univ.centerMass_eq_of_sum_1 _ hw₁]

Goal state after:
case refine_2
R : Type u_1
ι : Type u_2
inst✝⁴ : Field R
inst✝³ : LinearOrder R
inst✝² : IsStrictOrderedRing R
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
w : ι → R
hw₀ : ∀ (x : ι), 0 ≤ w x
hw₁ : ∑ x, w x = 1
⊢ (Finset.univ.centerMass w fun i j ↦ if i = j then 1 else 0) ∈ (convexHull R) (range fun i j ↦ if i = j then 1 else 0)
Module: Mathlib.Analysis.Convex.StdSimplex

Goal state before:
pp: case h₁
𝕜 : Type u_1
ι : Type u_2
inst✝⁵ : Fintype ι
inst✝⁴ : TopologicalSpace 𝕜
inst✝³ : Semiring 𝕜
inst✝² : PartialOrder 𝕜
inst✝¹ : OrderClosedTopology 𝕜
inst✝ : ContinuousAdd 𝕜
⊢ ∀ (i : ι), IsClosed {f | 0 ≤ f i}

Tactic:
exact fun i ↦ isClosed_le continuous_const (continuous_apply i)

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

Goal state before:
pp: S : Type u_1
inst✝⁶ : Semiring S
inst✝⁵ : PartialOrder S
X : Type u_2
Y : Type u_3
Z : Type u_4
inst✝⁴ : Fintype X
inst✝³ : Fintype Y
inst✝² : Fintype Z
inst✝¹ : IsOrderedRing S
inst✝ : Unique X
⊢ ∀ (a : ↑(stdSimplex S X)), a = ⟨1, ⋯⟩

Tactic:
subsingleton

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

Goal state before:
pp: S : Type u_1
inst✝⁴ : Semiring S
inst✝³ : PartialOrder S
X : Type u_2
inst✝² : Fintype X
inst✝¹ : IsOrderedRing S
inst✝ : Unique X
s : ↑(stdSimplex S X)
x : X
⊢ s = default

Tactic:
subsingleton

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

Goal state before:
pp: S : Type u_1
inst✝⁸ : Semiring S
inst✝⁷ : PartialOrder S
X : Type u_2
Y : Type u_3
Z : Type u_4
inst✝⁶ : Fintype X
inst✝⁵ : Fintype Y
inst✝⁴ : Fintype Z
𝕜 : Type u_5
inst✝³ : Field 𝕜
inst✝² : LinearOrder 𝕜
inst✝¹ : IsStrictOrderedRing 𝕜
inst✝ : Nonempty X
⊢ (fun i ↦ (↑(Fintype.card X))⁻¹) ∈ stdSimplex 𝕜 X

Tactic:
by simp [stdSimplex]

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

Goal state before:
pp: R : Type u_1
E : Type u_3
ι : Type u_5
ι' : Type u_6
inst✝² : Field R
inst✝¹ : AddCommGroup E
inst✝ : Module R E
s : Finset ι
t : Finset ι'
ws : ι → R
zs : ι → E
wt : ι' → R
zt : ι' → E
hws : ∑ i ∈ s, ws i = 1
hwt : ∑ i ∈ t, wt i = 1
a b : R
hab : a + b = 1
⊢ a • ∑ i ∈ s, ws i • zs i + b • ∑ i ∈ t, wt i • zt i =
    (s.disjSum t).centerMass (Sum.elim (fun i ↦ a * ws i) fun j ↦ b * wt j) (Sum.elim zs zt)

Tactic:
smul_sum,

Goal state after:
R : Type u_1
E : Type u_3
ι : Type u_5
ι' : Type u_6
inst✝² : Field R
inst✝¹ : AddCommGroup E
inst✝ : Module R E
s : Finset ι
t : Finset ι'
ws : ι → R
zs : ι → E
wt : ι' → R
zt : ι' → E
hws : ∑ i ∈ s, ws i = 1
hwt : ∑ i ∈ t, wt i = 1
a b : R
hab : a + b = 1
⊢ ∑ x ∈ s, a • ws x • zs x + b • ∑ i ∈ t, wt i • zt i =
    (s.disjSum t).centerMass (Sum.elim (fun i ↦ a * ws i) fun j ↦ b * wt j) (Sum.elim zs zt)
Module: Mathlib.Analysis.Convex.Combination

Goal state before:
pp: R : Type u_1
E : Type u_3
ι : Type u_5
ι' : Type u_6
inst✝² : Field R
inst✝¹ : AddCommGroup E
inst✝ : Module R E
s : Finset ι
t : Finset ι'
ws : ι → R
zs : ι → E
wt : ι' → R
zt : ι' → E
hws : ∑ i ∈ s, ws i = 1
hwt : ∑ i ∈ t, wt i = 1
a b : R
hab : a + b = 1
⊢ ∑ x ∈ s, a • ws x • zs x + b • ∑ i ∈ t, wt i • zt i =
    (s.disjSum t).centerMass (Sum.elim (fun i ↦ a * ws i) fun j ↦ b * wt j) (Sum.elim zs zt)

Tactic:
smul_sum,

Goal state after:
R : Type u_1
E : Type u_3
ι : Type u_5
ι' : Type u_6
inst✝² : Field R
inst✝¹ : AddCommGroup E
inst✝ : Module R E
s : Finset ι
t : Finset ι'
ws : ι → R
zs : ι → E
wt : ι' → R
zt : ι' → E
hws : ∑ i ∈ s, ws i = 1
hwt : ∑ i ∈ t, wt i = 1
a b : R
hab : a + b = 1
⊢ ∑ x ∈ s, a • ws x • zs x + ∑ x ∈ t, b • wt x • zt x =
    (s.disjSum t).centerMass (Sum.elim (fun i ↦ a * ws i) fun j ↦ b * wt j) (Sum.elim zs zt)
Module: Mathlib.Topology.Algebra.Affine

Goal state before:
pp: R : Type u_1
V : Type u_2
P : Type u_3
W : Type u_4
Q : Type u_5
inst✝¹² : AddCommGroup V
inst✝¹¹ : TopologicalSpace V
inst✝¹⁰ : AddTorsor V P
inst✝⁹ : TopologicalSpace P
inst✝⁸ : IsTopologicalAddTorsor P
inst✝⁷ : AddCommGroup W
inst✝⁶ : TopologicalSpace W
inst✝⁵ : AddTorsor W Q
inst✝⁴ : TopologicalSpace Q
inst✝³ : IsTopologicalAddTorsor Q
inst✝² : Ring R
inst✝¹ : Module R V
inst✝ : Module R W
f : P →ᵃ[R] Q
⊢ IsOpenMap ⇑f.linear ↔ IsOpenMap ⇑f

Tactic:
by
  inhabit P
  have :
    (f.linear : V → W) =
      (Homeomorph.vaddConst <| f default).symm ∘ f ∘ (Homeomorph.vaddConst default) := by
    ext v
    simp
  rw [this]
  simp only [Homeomorph.comp_isOpenMap_iff, Homeomorph.comp_isOpenMap_iff']

Goal state after:
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
t : Finset ι
w : ι → R
z : ι → E
t' : Finset ι
ht : t ⊆ t'
h : ∀ i ∈ t', i ∉ t → w i = 0
⊢ (∑ x ∈ t', w x)⁻¹ • ∑ i ∈ t, w i • z i = t'.centerMass w z

Tactic:
smul_sum,

Goal state after:
R : Type u_1
E : Type u_3
ι : Type u_5
inst✝² : Field R
inst✝¹ : AddCommGroup E
inst✝ : Module R E
t : Finset ι
w : ι → R
z : ι → E
t' : Finset ι
ht : t ⊆ t'
h : ∀ i ∈ t', i ∉ t → w i = 0
⊢ ∑ x ∈ t, (∑ x ∈ t', w x)⁻¹ • w x • z x = t'.centerMass w z
Module: Mathlib.LinearAlgebra.AffineSpace.Simplex.Basic

Goal state before:
pp: k : Type u_1
V : Type u_2
P : Type u_4
inst✝⁶ : Ring k
inst✝⁵ : AddCommGroup V
inst✝⁴ : Module k V
inst✝³ : AffineSpace V P
inst✝² : PartialOrder k
inst✝¹ : Nontrivial k
inst✝ : ZeroLEOneClass k
n : ℕ
s : Simplex k P n
fs : Finset (Fin (n + 1))
m : ℕ
h : #fs = m + 1
j : Fin (n + 1)
hj : s.points j ∈ (s.face h).closedInterior
this : s.points j ∈ affineSpan k (s.points '' ↑fs)
⊢ j ∈ fs

Tactic:
by simpa

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

Goal state before:
pp: R : Type u_1
E : Type u_3
ι : Type u_5
inst✝⁴ : Field R
inst✝³ : AddCommGroup E
inst✝² : Module R E
s : Set E
t : Finset ι
w : ι → R
z : ι → E
inst✝¹ : LinearOrder R
inst✝ : IsStrictOrderedRing R
hs : Convex R s
⊢ (∀ i ∈ t, 0 ≤ w i) → 0 < ∑ i ∈ t, w i → (∀ i ∈ t, z i ∈ s) → t.centerMass w z ∈ s

Tactic:
induction t using Finset.induction with
  | empty => simp
  | insert i t hi ht =>
    intro h₀ hpos hmem
    have zi : z i ∈ s := hmem _ (mem_insert_self _ _)
    have hs₀ : ∀ j ∈ t, 0 ≤ w j := fun j hj => h₀ j <| mem_insert_of_mem hj
    rw [sum_insert hi] at hpos
    by_cases hsum_t : ∑ j ∈ t, w j = 0
    · have ws : ∀ j ∈ t, w j = 0 := (sum_eq_zero_iff_of_nonneg hs₀).1 hsum_t
      have wz : ∑ j ∈ t, w j • z j = 0 := sum_eq_zero fun i hi => by simp [ws i hi]
      simp only [centerMass, sum_insert hi, wz, hsum_t, add_zero]
      simp only [hsum_t, add_zero] at hpos
      rw [← mul_smul, inv_mul_cancel₀ (ne_of_gt hpos), one_smul]
      exact zi
    · rw [Finset.centerMass_insert _ _ _ hi hsum_t]
      refine convex_iff_div.1 hs zi (ht hs₀ ?_ ?_) ?_ (sum_nonneg hs₀) hpos
      · exact lt_of_le_of_ne (sum_nonneg hs₀) (Ne.symm hsum_t)
      · intro j hj
        exact hmem j (mem_insert_of_mem hj)
      · exact h₀ _ (mem_insert_self _ _)

Goal state after:
Module: Mathlib.Topology.AlexandrovDiscrete

Goal state before:
pp: α : Type u_3
inst✝ : TopologicalSpace α
| AlexandrovDiscrete α

Tactic:
tactic =>
    simp_rw +singlePass [alexandrovDiscrete_iff, compl_surjective.image_surjective.forall,
      forall_mem_image, ← compl_sUnion, isOpen_compl_iff]

Goal state after:
α : Type u_3
inst✝ : TopologicalSpace α
| ∀ (x : Set (Set α)), (∀ ⦃x_1 : Set α⦄, x_1 ∈ x → IsClosed[inst✝] x_1) → IsClosed[inst✝] (⋃₀ x)
Module: Mathlib.Analysis.Convex.Combination

Goal state before:
pp: R : Type u_1
E : Type u_3
ι : Type u_5
inst✝⁴ : Field R
inst✝³ : AddCommGroup E
inst✝² : Module R E
s : Set E
w : ι → R
z : ι → E
inst✝¹ : LinearOrder R
inst✝ : IsStrictOrderedRing R
t : Finset ι
hw₀ : ∀ i ∈ t, w i ≤ 0
hws : ∑ i ∈ t, w i < 0
hz : ∀ i ∈ t, z i ∈ s
⊢ 0 < ∑ i ∈ t, -w i

Tactic:
by simpa

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

Goal state before:
pp: E : Type u_1
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
s : Set E
hs : Convex ℝ s
x : E
x✝² : x ∈ s
y : E
x✝¹ : y ∈ s
a b : ℝ
ha : 0 ≤ a
hb : 0 ≤ b
x✝ : a + b = 1
⊢ ‖a • x‖ + ‖b • y‖ = a * ‖x‖ + b * ‖y‖

Tactic:
rw [norm_smul, norm_smul, Real.norm_of_nonneg ha, Real.norm_of_nonneg hb]

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

Goal state before:
pp: R : Type u_1
E : Type u_3
F : Type u_4
inst✝⁶ : Field R
inst✝⁵ : AddCommGroup E
inst✝⁴ : AddCommGroup F
inst✝³ : Module R E
inst✝² : Module R F
s : Set E
inst✝¹ : LinearOrder R
inst✝ : IsStrictOrderedRing R
t : Set F
x : E
y : F
ι : Type
w✝¹ : Fintype ι
w : ι → R
f : ι → E
hw₀ : ∀ (i : ι), 0 ≤ w i
hw₁ : ∑ i, w i = 1
hfs : ∀ (i : ι), f i ∈ s
hf : ∑ i, w i • f i = x
κ : Type
w✝ : Fintype κ
v : κ → R
g : κ → F
hv₀ : ∀ (i : κ), 0 ≤ v i
hv₁ : ∑ i, v i = 1
hgt : ∀ (i : κ), g i ∈ t
hg : ∑ i, v i • g i = y
⊢ 1 * ∑ j, v j = 1

Tactic:
hv₁,

Goal state after:
R : Type u_1
E : Type u_3
F : Type u_4
inst✝⁶ : Field R
inst✝⁵ : AddCommGroup E
inst✝⁴ : AddCommGroup F
inst✝³ : Module R E
inst✝² : Module R F
s : Set E
inst✝¹ : LinearOrder R
inst✝ : IsStrictOrderedRing R
t : Set F
x : E
y : F
ι : Type
w✝¹ : Fintype ι
w : ι → R
f : ι → E
hw₀ : ∀ (i : ι), 0 ≤ w i
hw₁ : ∑ i, w i = 1
hfs : ∀ (i : ι), f i ∈ s
hf : ∑ i, w i • f i = x
κ : Type
w✝ : Fintype κ
v : κ → R
g : κ → F
hv₀ : ∀ (i : κ), 0 ≤ v i
hv₁ : ∑ i, v i = 1
hgt : ∀ (i : κ), g i ∈ t
hg : ∑ i, v i • g i = y
⊢ 1 * 1 = 1
Module: Mathlib.Analysis.Convex.Combination

Goal state before:
pp: case fst
R : Type u_1
E : Type u_3
F : Type u_4
inst✝⁶ : Field R
inst✝⁵ : AddCommGroup E
inst✝⁴ : AddCommGroup F
inst✝³ : Module R E
inst✝² : Module R F
s : Set E
inst✝¹ : LinearOrder R
inst✝ : IsStrictOrderedRing R
t : Set F
x : E
y : F
ι : Type
w✝¹ : Fintype ι
w : ι → R
f : ι → E
hw₀ : ∀ (i : ι), 0 ≤ w i
hw₁ : ∑ i, w i = 1
hfs : ∀ (i : ι), f i ∈ s
hf : ∑ i, w i • f i = x
κ : Type
w✝ : Fintype κ
v : κ → R
g : κ → F
hv₀ : ∀ (i : κ), 0 ≤ v i
hv₁ : ∑ i, v i = 1
hgt : ∀ (i : κ), g i ∈ t
hg : ∑ i, v i • g i = y
h_sum : ∑ i, w i.1 * v i.2 = 1
⊢ (∑ i, v i) • ∑ j, w j • f j = x

Tactic:
hv₁,

Goal state after:
case fst
R : Type u_1
E : Type u_3
F : Type u_4
inst✝⁶ : Field R
inst✝⁵ : AddCommGroup E
inst✝⁴ : AddCommGroup F
inst✝³ : Module R E
inst✝² : Module R F
s : Set E
inst✝¹ : LinearOrder R
inst✝ : IsStrictOrderedRing R
t : Set F
x : E
y : F
ι : Type
w✝¹ : Fintype ι
w : ι → R
f : ι → E
hw₀ : ∀ (i : ι), 0 ≤ w i
hw₁ : ∑ i, w i = 1
hfs : ∀ (i : ι), f i ∈ s
hf : ∑ i, w i • f i = x
κ : Type
w✝ : Fintype κ
v : κ → R
g : κ → F
hv₀ : ∀ (i : κ), 0 ≤ v i
hv₁ : ∑ i, v i = 1
hgt : ∀ (i : κ), g i ∈ t
hg : ∑ i, v i • g i = y
h_sum : ∑ i, w i.1 * v i.2 = 1
⊢ 1 • ∑ j, w j • f j = x
Module: Mathlib.Analysis.Convex.Jensen

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
β : Type u_4
ι : Type u_5
inst✝⁹ : Field 𝕜
inst✝⁸ : LinearOrder 𝕜
inst✝⁷ : IsStrictOrderedRing 𝕜
inst✝⁶ : AddCommGroup E
inst✝⁵ : AddCommGroup β
inst✝⁴ : LinearOrder β
inst✝³ : IsOrderedAddMonoid β
inst✝² : Module 𝕜 E
inst✝¹ : Module 𝕜 β
inst✝ : IsStrictOrderedModule 𝕜 β
s : Set E
f : E → β
w : ι → 𝕜
p : ι → E
t : Finset ι
h : ConvexOn 𝕜 s f
hw₀ : ∀ i ∈ t, 0 ≤ w i
hw₁ : 0 < ∑ i ∈ t, w i
hp : ∀ i ∈ t, p i ∈ s
y : E := t.centerMass w p
hw' : 0 < ∑ i ∈ t with w i ≠ 0, w i
⊢ ∑ i ∈ t with w i ≠ 0, w i • f y ≤ ∑ i ∈ t with w i ≠ 0, w i • (f ∘ p) i

Tactic:
← sum_smul,

Goal state after:
𝕜 : Type u_1
E : Type u_2
β : Type u_4
ι : Type u_5
inst✝⁹ : Field 𝕜
inst✝⁸ : LinearOrder 𝕜
inst✝⁷ : IsStrictOrderedRing 𝕜
inst✝⁶ : AddCommGroup E
inst✝⁵ : AddCommGroup β
inst✝⁴ : LinearOrder β
inst✝³ : IsOrderedAddMonoid β
inst✝² : Module 𝕜 E
inst✝¹ : Module 𝕜 β
inst✝ : IsStrictOrderedModule 𝕜 β
s : Set E
f : E → β
w : ι → 𝕜
p : ι → E
t : Finset ι
h : ConvexOn 𝕜 s f
hw₀ : ∀ i ∈ t, 0 ≤ w i
hw₁ : 0 < ∑ i ∈ t, w i
hp : ∀ i ∈ t, p i ∈ s
y : E := t.centerMass w p
hw' : 0 < ∑ i ∈ t with w i ≠ 0, w i
⊢ (∑ i ∈ t with w i ≠ 0, w i) • f y ≤ ∑ i ∈ t with w i ≠ 0, w i • (f ∘ p) i
Module: Mathlib.Analysis.Convex.Combination

Goal state before:
pp: case mpr
𝕜 : Type u_1
V : Type u_2
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsOrderedRing 𝕜
inst✝¹ : AddCommGroup V
inst✝ : Module 𝕜 V
n : ℕ
s : Simplex 𝕜 V n
w : Fin (n + 1) → 𝕜
hw1 : ∑ i, w i = 1
h : (Finset.affineCombination 𝕜 Finset.univ s.points) w ∈ s.closedInterior
⊢ ∃ s_1 w_1,
    (∀ i ∈ s_1, 0 ≤ w_1 i) ∧
      s_1.sum w_1 = 1 ∧
        (Finset.affineCombination 𝕜 s_1 s.points) w_1 = (Finset.affineCombination 𝕜 Finset.univ s.points) w

Tactic:
rw [affineCombination_mem_closedInterior_iff hw1] at h

Goal state after:
case mpr
𝕜 : Type u_1
V : Type u_2
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsOrderedRing 𝕜
inst✝¹ : AddCommGroup V
inst✝ : Module 𝕜 V
n : ℕ
s : Simplex 𝕜 V n
w : Fin (n + 1) → 𝕜
hw1 : ∑ i, w i = 1
h : ∀ (i : Fin (n + 1)), w i ∈ Icc 0 1
⊢ ∃ s_1 w_1,
    (∀ i ∈ s_1, 0 ≤ w_1 i) ∧
      s_1.sum w_1 = 1 ∧
        (Finset.affineCombination 𝕜 s_1 s.points) w_1 = (Finset.affineCombination 𝕜 Finset.univ s.points) w
Module: Mathlib.Analysis.Normed.Operator.Basic

Goal state before:
pp: 𝕜 : Type u_1
𝕜₂ : Type u_2
E : Type u_4
F : Type u_5
𝓕 : Type u_8
inst✝⁸ : SeminormedAddCommGroup E
inst✝⁷ : SeminormedAddCommGroup F
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NontriviallyNormedField 𝕜₂
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedSpace 𝕜₂ F
σ₁₂ : 𝕜 →+* 𝕜₂
inst✝² : FunLike 𝓕 E F
inst✝¹ : SemilinearMapClass 𝓕 σ₁₂ E F
inst✝ : RingHomSurjective σ₁₂
f : 𝓕
r : ℝ
hr : 0 < r
⊢ ‖0‖ < r

Tactic:
by simpa

Goal state after:
Module: Mathlib.Topology.Algebra.Module.Spaces.UniformConvergenceCLM

Goal state before:
pp: 𝕜₂ : Type u_2
inst✝¹⁶ : NormedField 𝕜₂
E : Type u_3
F : Type u_4
G : Type u_5
inst✝¹⁵ : AddCommGroup E
inst✝¹⁴ : TopologicalSpace E
inst✝¹³ : AddCommGroup F
inst✝¹² : Module 𝕜₂ F
𝕜₁ : Type u_6
inst✝¹¹ : NontriviallyNormedField 𝕜₁
σ : 𝕜₁ →+* 𝕜₂
inst✝¹⁰ : Module 𝕜₁ E
inst✝⁹ : AddCommGroup G
𝕜₃ : Type u_7
inst✝⁸ : NormedField 𝕜₃
inst✝⁷ : Module 𝕜₃ G
τ : 𝕜₃ →+* 𝕜₂
inst✝⁶ : RingHomSurjective τ
inst✝⁵ : TopologicalSpace F
inst✝⁴ : IsTopologicalAddGroup F
inst✝³ : ContinuousConstSMul 𝕜₂ F
inst✝² : TopologicalSpace G
inst✝¹ : IsTopologicalAddGroup G
inst✝ : ContinuousConstSMul 𝕜₃ G
𝔖 : Set (Set E)
h𝔖 : ∀ s ∈ 𝔖, IsVonNBounded 𝕜₁ s
B : G →ₛₗ[τ] E →SLᵤ[σ, 𝔖] F
hB : Continuous[instTopologicalSpaceProd, inst✝⁵] fun p ↦ (B p.1) p.2
S : Set E
hS : S ∈ 𝔖
U : Set F
hU : U ∈ 𝓝 0
V : Set G
hV : V ∈ 𝓝 0
W : Set E
hW : W ∈ 𝓝 0
hVW : V ×ˢ W ⊆ (fun p ↦ (B p.1) p.2) ⁻¹' U
c : 𝕜₁
hc : MapsTo (fun x ↦ c • x) S W
c_ne : c ≠ 0
d : 𝕜₃
hd : τ d = σ c
⊢ σ c ≠ 0

Tactic:
map_ne_zero σ

Goal state after:
𝕜₂ : Type u_2
inst✝¹⁶ : NormedField 𝕜₂
E : Type u_3
F : Type u_4
G : Type u_5
inst✝¹⁵ : AddCommGroup E
inst✝¹⁴ : TopologicalSpace E
inst✝¹³ : AddCommGroup F
inst✝¹² : Module 𝕜₂ F
𝕜₁ : Type u_6
inst✝¹¹ : NontriviallyNormedField 𝕜₁
σ : 𝕜₁ →+* 𝕜₂
inst✝¹⁰ : Module 𝕜₁ E
inst✝⁹ : AddCommGroup G
𝕜₃ : Type u_7
inst✝⁸ : NormedField 𝕜₃
inst✝⁷ : Module 𝕜₃ G
τ : 𝕜₃ →+* 𝕜₂
inst✝⁶ : RingHomSurjective τ
inst✝⁵ : TopologicalSpace F
inst✝⁴ : IsTopologicalAddGroup F
inst✝³ : ContinuousConstSMul 𝕜₂ F
inst✝² : TopologicalSpace G
inst✝¹ : IsTopologicalAddGroup G
inst✝ : ContinuousConstSMul 𝕜₃ G
𝔖 : Set (Set E)
h𝔖 : ∀ s ∈ 𝔖, IsVonNBounded 𝕜₁ s
B : G →ₛₗ[τ] E →SLᵤ[σ, 𝔖] F
hB : Continuous[instTopologicalSpaceProd, inst✝⁵] fun p ↦ (B p.1) p.2
S : Set E
hS : S ∈ 𝔖
U : Set F
hU : U ∈ 𝓝 0
V : Set G
hV : V ∈ 𝓝 0
W : Set E
hW : W ∈ 𝓝 0
hVW : V ×ˢ W ⊆ (fun p ↦ (B p.1) p.2) ⁻¹' U
c : 𝕜₁
hc : MapsTo (fun x ↦ c • x) S W
c_ne : c ≠ 0
d : 𝕜₃
hd : τ d = σ c
⊢ c ≠ 0
Module: Mathlib.Analysis.LocallyConvex.WithSeminorms

Goal state before:
pp: case refine_2
𝕜 : Type u_2
E : Type u_6
ι : Type u_9
inst✝³ : NormedField 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
p : SeminormFamily 𝕜 E ι
inst✝ : TopologicalSpace E
hp : WithSeminorms p
i : ι
⊢ ∃ s C, (p i).comp LinearMap.id ≤ C • s.sup fun s ↦ s.sup p

Tactic:
refine ⟨{{i}}, 1, ?_⟩

Goal state after:
case refine_2
𝕜 : Type u_2
E : Type u_6
ι : Type u_9
inst✝³ : NormedField 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
p : SeminormFamily 𝕜 E ι
inst✝ : TopologicalSpace E
hp : WithSeminorms p
i : ι
⊢ (p i).comp LinearMap.id ≤ 1 • {{i}}.sup fun s ↦ s.sup p
Module: Mathlib.Analysis.LocallyConvex.WithSeminorms

Goal state before:
pp: case refine_2
𝕜 : Type u_2
E : Type u_6
ι : Type u_9
inst✝³ : NormedField 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
p : SeminormFamily 𝕜 E ι
inst✝ : TopologicalSpace E
hp : WithSeminorms p
⊢ Seminorm.IsBounded (fun s ↦ s.sup p) p LinearMap.id

Tactic:
intro i
    refine ⟨{{i}}, 1, ?_⟩
    rw [Finset.sup_singleton, Finset.sup_singleton, one_smul]
    rfl

Goal state after:
Module: Mathlib.Analysis.Normed.Operator.NNNorm

Goal state before:
pp: 𝕜 : Type u_1
𝕜₂ : Type u_2
E : Type u_4
F : Type u_5
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : SeminormedAddCommGroup F
inst✝⁵ : DenselyNormedField 𝕜
inst✝⁴ : NontriviallyNormedField 𝕜₂
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedSpace 𝕜₂ F
σ₁₂ : 𝕜 →+* 𝕜₂
inst✝¹ : RingHomIsometric σ₁₂
inst✝ : NormedAlgebra ℝ 𝕜
f : E →SL[σ₁₂] F
r : ℝ≥0
hr✝ : r < ‖f‖₊
x : E
hlt : ‖x‖₊ < 1
hr : r < ‖f x‖₊
hx0 : 0 < ‖x‖₊
⊢ r < 1⁻¹ * ‖f x‖₊

Tactic:
by simpa

Goal state after:
Module: Mathlib.Order.Filter.Germ.Basic

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
l : Filter α
f g h : α → β
M : Type u_5
N : Type u_6
R : Type u_7
inst✝² : Monoid M
inst✝¹ : AddMonoid N
inst✝ : DistribMulAction M N
c : M
⊢ c • 0 = 0

Tactic:
simp only [← coe_zero, ← coe_smul, smul_zero]

Goal state after: