mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.LinearAlgebra.AffineSpace.Combination

Goal state before:
pp: case neg
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
s : Finset ι
w : ι → k
w' : ι → k := Function.update w j (1 - (s \ {j}).sum w)
hj : j ∉ s
⊢ (insert j s).sum w' = 1

Tactic:
simp_rw [w', Finset.sum_insert hj, Finset.sum_update_of_notMem hj, Function.update_self,
            ← Finset.erase_eq, Finset.erase_eq_of_notMem hj, sub_add_cancel]

Goal state after:
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✝ : Finite ι
q₁ q₂ : P
h : ∀ (i : ι), (b.coord i) q₁ = (b.coord i) q₂
⊢ q₁ = q₂

Tactic:
by
  cases nonempty_fintype ι
  rw [← b.affineCombination_coord_eq_self q₁, ← b.affineCombination_coord_eq_self q₂]
  simp only [h]

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

Goal state before:
pp: S : Type u_1
inst✝² : Semiring S
inst✝¹ : PartialOrder S
X : Type u_2
inst✝ : Fintype X
s t : ↑(stdSimplex S X)
h : ⇑s = ⇑t
⊢ s = t

Tactic:
ext : 1
  assumption

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

Goal state before:
pp: case refine_2
k : Type u_1
V : Type u_2
P : Type u_4
inst✝³ : Ring k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
I : Set k
m n : ℕ
s : Simplex k P n
e : Fin (n + 1) ≃ Fin (m + 1)
w : Fin (n + 1) → k
hw : ∑ i, w i = 1
hwI : ∀ (i : Fin (n + 1)), w i ∈ I
x✝ : (affineCombination k univ s.points) w ∈ Simplex.setInterior I s
⊢ (affineCombination k univ (s.points ∘ ⇑e.symm)) (w ∘ ⇑e.symm) ∈ Simplex.setInterior I (s.reindex e)

Tactic:
change Finset.univ.affineCombination k (s.reindex e).points _ ∈ _

Goal state after:
case refine_2
k : Type u_1
V : Type u_2
P : Type u_4
inst✝³ : Ring k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
I : Set k
m n : ℕ
s : Simplex k P n
e : Fin (n + 1) ≃ Fin (m + 1)
w : Fin (n + 1) → k
hw : ∑ i, w i = 1
hwI : ∀ (i : Fin (n + 1)), w i ∈ I
x✝ : (affineCombination k univ s.points) w ∈ Simplex.setInterior I s
⊢ (affineCombination k univ (s.reindex e).points) (w ∘ ⇑e.symm) ∈ Simplex.setInterior I (s.reindex e)
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✝ : Subsingleton X
s t : ↑(stdSimplex S X)
i : X
this : ∀ (u : ↑(stdSimplex S X)), u i = 1
⊢ s i = t i

Tactic:
simp [this]

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 x = 1

Tactic:
by
  obtain rfl : s = default := by subsingleton
  rfl

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

Goal state before:
pp: case pos
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
h : vectorSpan k (p '' s₁) = vectorSpan k (p '' s₂)
he : ¬s₁ = s₂
h₁ : ¬s₁.Subsingleton
h₂ : s₂.Subsingleton
⊢ s₁.Subsingleton ∧ s₂.Subsingleton

Tactic:
rw [vectorSpan_of_subsingleton _ (h₂.image _), vectorSpan_eq_bot_iff_subsingleton]
        at h

Goal state after:
case pos
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
h : (p '' s₁).Subsingleton
he : ¬s₁ = s₂
h₁ : ¬s₁.Subsingleton
h₂ : s₂.Subsingleton
⊢ s₁.Subsingleton ∧ s₂.Subsingleton
Module: Mathlib.Analysis.Convex.Combination

Goal state before:
pp: case insert
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 : ι
t : Finset ι
hi : i ∉ t
ht : (∀ i ∈ t, 0 ≤ w i) → 0 < ∑ i ∈ t, w i → (∀ i ∈ t, z i ∈ s) → t.centerMass w z ∈ s
h₀ : ∀ i_1 ∈ insert i t, 0 ≤ w i_1
hpos : 0 < ∑ i ∈ insert i t, w i
hmem : ∀ i_1 ∈ insert i t, z i_1 ∈ s
zi : z i ∈ s
hs₀ : ∀ j ∈ t, 0 ≤ w j
⊢ (insert i t).centerMass w z ∈ s

Tactic:
rw [sum_insert hi] at hpos

Goal state after:
case insert
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 : ι
t : Finset ι
hi : i ∉ t
ht : (∀ i ∈ t, 0 ≤ w i) → 0 < ∑ i ∈ t, w i → (∀ i ∈ t, z i ∈ s) → t.centerMass w z ∈ s
h₀ : ∀ i_1 ∈ insert i t, 0 ≤ w i_1
hpos : 0 < w i + ∑ x ∈ t, w x
hmem : ∀ i_1 ∈ insert i t, z i_1 ∈ s
zi : z i ∈ s
hs₀ : ∀ j ∈ t, 0 ≤ w j
⊢ (insert i t).centerMass w z ∈ s
Module: Mathlib.Analysis.Convex.Combination

Goal state before:
pp: case pos
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 : ι
t : Finset ι
hi : i ∉ t
ht : (∀ i ∈ t, 0 ≤ w i) → 0 < ∑ i ∈ t, w i → (∀ i ∈ t, z i ∈ s) → t.centerMass w z ∈ s
h₀ : ∀ i_1 ∈ insert i t, 0 ≤ w i_1
hmem : ∀ i_1 ∈ insert i t, z i_1 ∈ s
zi : z i ∈ s
hs₀ : ∀ j ∈ t, 0 ≤ w j
hsum_t : ∑ j ∈ t, w j = 0
ws : ∀ j ∈ t, w j = 0
wz : ∑ j ∈ t, w j • z j = 0
hpos : 0 < w i
⊢ 1 • z i ∈ s

Tactic:
one_smul

Goal state after:
case pos
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 : ι
t : Finset ι
hi : i ∉ t
ht : (∀ i ∈ t, 0 ≤ w i) → 0 < ∑ i ∈ t, w i → (∀ i ∈ t, z i ∈ s) → t.centerMass w z ∈ s
h₀ : ∀ i_1 ∈ insert i t, 0 ≤ w i_1
hmem : ∀ i_1 ∈ insert i t, z i_1 ∈ s
zi : z i ∈ s
hs₀ : ∀ j ∈ t, 0 ≤ w j
hsum_t : ∑ j ∈ t, w j = 0
ws : ∀ j ∈ t, w j = 0
wz : ∑ j ∈ t, w j • z j = 0
hpos : 0 < w i
⊢ z i ∈ s
Module: Mathlib.Analysis.Convex.Combination

Goal state before:
pp: case pos
R : Type u_1
E : Type u_3
inst✝⁴ : Field R
inst✝³ : AddCommGroup E
inst✝² : Module R E
s : Set E
inst✝¹ : LinearOrder R
inst✝ : IsStrictOrderedRing R
h : ∀ (t : Finset E) (w : E → R), (∀ i ∈ t, 0 ≤ w i) → ∑ i ∈ t, w i = 1 → (∀ x ∈ t, x ∈ s) → ∑ x ∈ t, w x • x ∈ s
x : E
hx : x ∈ s
y : E
hy : y ∈ s
a b : R
ha : 0 ≤ a
hb : 0 ≤ b
hab : a + b = 1
h_cases : x = y
⊢ 1 • y ∈ s

Tactic:
one_smul

Goal state after:
case pos
R : Type u_1
E : Type u_3
inst✝⁴ : Field R
inst✝³ : AddCommGroup E
inst✝² : Module R E
s : Set E
inst✝¹ : LinearOrder R
inst✝ : IsStrictOrderedRing R
h : ∀ (t : Finset E) (w : E → R), (∀ i ∈ t, 0 ≤ w i) → ∑ i ∈ t, w i = 1 → (∀ x ∈ t, x ∈ s) → ∑ x ∈ t, w x • x ∈ s
x : E
hx : x ∈ s
y : E
hy : y ∈ s
a b : R
ha : 0 ≤ a
hb : 0 ≤ b
hab : a + b = 1
h_cases : x = y
⊢ y ∈ s
Module: Mathlib.Topology.Connected.LocallyPathConnected

Goal state before:
pp: X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : LocallyPathConnectedSpace X
x y : X
hxy : y ∈ (pathComponent x)ᶜ
V : Set X
hVy : V ∈ 𝓝 y
hVc : IsPathConnected V
⊢ (pathComponent x)ᶜ ∈ 𝓝 y

Tactic:
filter_upwards [hVy] with z hz hxz

Goal state after:
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : LocallyPathConnectedSpace X
x y : X
hxy : y ∈ (pathComponent x)ᶜ
V : Set X
hVy : V ∈ 𝓝 y
hVc : IsPathConnected V
z : X
hz : z ∈ V
hxz : z ∈ pathComponent x
⊢ False
Module: Mathlib.LinearAlgebra.AffineSpace.Independent

Goal state before:
pp: case inr.refine_2.hd
k : Type u_1
V : Type u_2
P : Type u_3
inst✝³ : DivisionRing k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
s : Set P
p : P
hp : p ∈ s
b : Set V
hb₁ : b ⊆ ⇑(Equiv.vaddConst p).symm '' s
hb₂ : Submodule.span k b = vectorSpan k s
hb₃ : AffineIndependent k fun p_1 ↦ ↑p_1
hb₀ : ∀ v ∈ b, v ≠ 0
this : Submodule.span k b = Submodule.span k (insert 0 b)
⊢ (affineSpan k ({p} ∪ ⇑(Equiv.vaddConst p) '' b)).direction = (affineSpan k s).direction

Tactic:
simp only [direction_affineSpan, ← hb₂, Equiv.coe_vaddConst, Set.singleton_union,
        vectorSpan_eq_span_vsub_set_right k (Set.mem_insert p _), this]

Goal state after:
case inr.refine_2.hd
k : Type u_1
V : Type u_2
P : Type u_3
inst✝³ : DivisionRing k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
s : Set P
p : P
hp : p ∈ s
b : Set V
hb₁ : b ⊆ ⇑(Equiv.vaddConst p).symm '' s
hb₂ : Submodule.span k b = vectorSpan k s
hb₃ : AffineIndependent k fun p_1 ↦ ↑p_1
hb₀ : ∀ v ∈ b, v ≠ 0
this : Submodule.span k b = Submodule.span k (insert 0 b)
⊢ Submodule.span k ((fun x ↦ x -ᵥ p) '' insert p ((fun v ↦ v +ᵥ p) '' b)) = Submodule.span k (insert 0 b)
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
⊢ t.centerMass w z ∈ (convexHull R) s

Tactic:
rw [← centerMass_neg_left]

Goal state after:
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
⊢ t.centerMass (-w) z ∈ (convexHull R) s
Module: Mathlib.LinearAlgebra.AffineSpace.Independent

Goal state before:
pp: k : Type u_1
V : Type u_2
P : Type u_3
inst✝³ : DivisionRing k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
ι : Type u_4
p : ι → P
i : ι
ha : AffineIndependent k fun x ↦ p ↑x
hi : p i ∉ affineSpan k (p '' {x | x ≠ i})
s : Finset ι
w : ι → k
hw : ∑ i ∈ s, w i = 0
hs : (s.weightedVSub p) w = 0
s' : Finset { y // y ≠ i } := Finset.subtype (fun x ↦ x ≠ i) s
p' : { y // y ≠ i } → P := fun x ↦ p ↑x
his : i ∉ s ∨ w i = 0
w' : { y // y ≠ i } → k := fun x ↦ w ↑x
hw' : ∑ x ∈ s', w' x = 0
⊢ ({x ∈ s | x ≠ i}.weightedVSub p) w = 0

Tactic:
rw [Finset.weightedVSub_filter_of_ne, hs]

Goal state after:
case h
k : Type u_1
V : Type u_2
P : Type u_3
inst✝³ : DivisionRing k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
ι : Type u_4
p : ι → P
i : ι
ha : AffineIndependent k fun x ↦ p ↑x
hi : p i ∉ affineSpan k (p '' {x | x ≠ i})
s : Finset ι
w : ι → k
hw : ∑ i ∈ s, w i = 0
hs : (s.weightedVSub p) w = 0
s' : Finset { y // y ≠ i } := Finset.subtype (fun x ↦ x ≠ i) s
p' : { y // y ≠ i } → P := fun x ↦ p ↑x
his : i ∉ s ∨ w i = 0
w' : { y // y ≠ i } → k := fun x ↦ w ↑x
hw' : ∑ x ∈ s', w' x = 0
⊢ ∀ i_1 ∈ s, w i_1 ≠ 0 → i_1 ≠ i
Module: Mathlib.Analysis.Convex.Combination

Goal state before:
pp: case refine_1
R : Type u_1
E : Type u_3
inst✝⁴ : Field R
inst✝³ : AddCommGroup E
inst✝² : Module R E
inst✝¹ : LinearOrder R
inst✝ : IsStrictOrderedRing R
s : Finset E
⊢ ↑s ⊆ {x | ∃ w, (∀ y ∈ s, 0 ≤ w y) ∧ ∑ y ∈ s, w y = 1 ∧ s.centerMass w id = x}

Tactic:
intro x hx
    rw [Finset.mem_coe] at hx
    refine ⟨_, ?_, ?_, Finset.centerMass_ite_eq _ _ _ hx⟩
    · intros
      split_ifs
      exacts [zero_le_one, le_refl 0]
    · rw [Finset.sum_ite_eq, if_pos hx]

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

Goal state before:
pp: case neg
F : Type u_2
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
inst✝ : Nontrivial F
x✝ : F
r : ℝ
hr : 0 ≤ r
x : F
h : x ∈ closedBall 0 r
U : Set F
hU_sub : sphere 0 r ⊆ U
hU : Convex ℝ U
zero_mem : 0 ∈ U
hr₀ : ¬r = 0
x_zero : ¬x = 0
z : F := (r * ‖x‖⁻¹) • x
hz_def : z = (r * ‖x‖⁻¹) • x
hr₁ : r⁻¹ * ‖x‖ ≤ 1
hz : z ∈ U
this : 1 • x ∈ U
⊢ x ∈ U

Tactic:
one_smul

Goal state after:
case neg
F : Type u_2
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
inst✝ : Nontrivial F
x✝ : F
r : ℝ
hr : 0 ≤ r
x : F
h : x ∈ closedBall 0 r
U : Set F
hU_sub : sphere 0 r ⊆ U
hU : Convex ℝ U
zero_mem : 0 ∈ U
hr₀ : ¬r = 0
x_zero : ¬x = 0
z : F := (r * ‖x‖⁻¹) • x
hz_def : z = (r * ‖x‖⁻¹) • x
hr₁ : r⁻¹ * ‖x‖ ≤ 1
hz : z ∈ U
this : x ∈ U
⊢ x ∈ U
Module: Mathlib.Analysis.LocallyConvex.WithSeminorms

Goal state before:
pp: 𝕜 : Type u_11
E : Type u_12
ι : Type u_13
inst✝³ : NormedField 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
inst✝ : TopologicalSpace E
p : SeminormFamily 𝕜 E ι
hp : ∀ (i : ι), Continuous[inst✝, _] ⇑(p i)
s : Finset ι
r : ℝ
hr : 0 < r
⊢ (s.sup p).ball 0 r ∈ 𝓝 0

Tactic:
refine Seminorm.ball_mem_nhds ?_ hr

Goal state after:
𝕜 : Type u_11
E : Type u_12
ι : Type u_13
inst✝³ : NormedField 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
inst✝ : TopologicalSpace E
p : SeminormFamily 𝕜 E ι
hp : ∀ (i : ι), Continuous[inst✝, _] ⇑(p i)
s : Finset ι
r : ℝ
hr : 0 < r
⊢ Continuous[inst✝, _] ⇑(s.sup p)
Module: Mathlib.Analysis.LocallyConvex.WithSeminorms

Goal state before:
pp: case mp
𝕜 : Type u_2
𝕜₂ : Type u_3
E : Type u_6
F : Type u_7
ι' : Type u_10
inst✝⁷ : SeminormedRing 𝕜
inst✝⁶ : AddCommGroup E
inst✝⁵ : Module 𝕜 E
inst✝⁴ : SeminormedRing 𝕜₂
inst✝³ : AddCommGroup F
inst✝² : Module 𝕜₂ F
σ₁₂ : 𝕜 →+* 𝕜₂
inst✝¹ : RingHomIsometric σ₁₂
ι : Type u_11
inst✝ : Nonempty ι
p : Seminorm 𝕜 E
q : ι' → Seminorm 𝕜₂ F
f : E →ₛₗ[σ₁₂] F
⊢ IsBounded (fun x ↦ p) q f → ∀ (i : ι'), ∃ C, (q i).comp f ≤ C • p

Tactic:
intro h i

Goal state after:
case mp
𝕜 : Type u_2
𝕜₂ : Type u_3
E : Type u_6
F : Type u_7
ι' : Type u_10
inst✝⁷ : SeminormedRing 𝕜
inst✝⁶ : AddCommGroup E
inst✝⁵ : Module 𝕜 E
inst✝⁴ : SeminormedRing 𝕜₂
inst✝³ : AddCommGroup F
inst✝² : Module 𝕜₂ F
σ₁₂ : 𝕜 →+* 𝕜₂
inst✝¹ : RingHomIsometric σ₁₂
ι : Type u_11
inst✝ : Nonempty ι
p : Seminorm 𝕜 E
q : ι' → Seminorm 𝕜₂ F
f : E →ₛₗ[σ₁₂] F
h : IsBounded (fun x ↦ p) q f
i : ι'
⊢ ∃ C, (q i).comp f ≤ C • p
Module: Mathlib.Analysis.LocallyConvex.WithSeminorms

Goal state before:
pp: case mpr
𝕜 : Type u_2
𝕜₂ : Type u_3
E : Type u_6
F : Type u_7
ι' : Type u_10
inst✝⁷ : SeminormedRing 𝕜
inst✝⁶ : AddCommGroup E
inst✝⁵ : Module 𝕜 E
inst✝⁴ : SeminormedRing 𝕜₂
inst✝³ : AddCommGroup F
inst✝² : Module 𝕜₂ F
σ₁₂ : 𝕜 →+* 𝕜₂
inst✝¹ : RingHomIsometric σ₁₂
ι : Type u_11
inst✝ : Nonempty ι
p : Seminorm 𝕜 E
q : ι' → Seminorm 𝕜₂ F
f : E →ₛₗ[σ₁₂] F
⊢ (∀ (i : ι'), ∃ C, (q i).comp f ≤ C • p) → IsBounded (fun x ↦ p) q f

Tactic:
intro h i

Goal state after:
case mpr
𝕜 : Type u_2
𝕜₂ : Type u_3
E : Type u_6
F : Type u_7
ι' : Type u_10
inst✝⁷ : SeminormedRing 𝕜
inst✝⁶ : AddCommGroup E
inst✝⁵ : Module 𝕜 E
inst✝⁴ : SeminormedRing 𝕜₂
inst✝³ : AddCommGroup F
inst✝² : Module 𝕜₂ F
σ₁₂ : 𝕜 →+* 𝕜₂
inst✝¹ : RingHomIsometric σ₁₂
ι : Type u_11
inst✝ : Nonempty ι
p : Seminorm 𝕜 E
q : ι' → Seminorm 𝕜₂ F
f : E →ₛₗ[σ₁₂] F
h : ∀ (i : ι'), ∃ C, (q i).comp f ≤ C • p
i : ι'
⊢ ∃ s C, (q i).comp f ≤ C • s.sup fun x ↦ p
Module: Mathlib.Topology.Algebra.Module.Spaces.ContinuousLinearMap

Goal state before:
pp: R : Type u_1
𝕜₂ : Type u_3
𝕜₃ : Type u_4
E : Type u_5
F : Type u_6
G : Type u_7
inst✝¹³ : Semiring R
inst✝¹² : NormedField 𝕜₂
inst✝¹¹ : NormedField 𝕜₃
inst✝¹⁰ : AddCommGroup E
inst✝⁹ : Module R E
inst✝⁸ : TopologicalSpace E
inst✝⁷ : AddCommGroup F
inst✝⁶ : Module 𝕜₂ F
inst✝⁵ : TopologicalSpace F
inst✝⁴ : AddCommGroup G
inst✝³ : Module 𝕜₃ G
inst✝² : TopologicalSpace G
inst✝¹ : IsTopologicalAddGroup G
inst✝ : ContinuousConstSMul 𝕜₃ G
σ₁₃ : R →+* 𝕜₃
σ₂₃ : 𝕜₂ →+* 𝕜₃
f : E →SL[σ₁₃] F →SL[σ₂₃] G
x : E
y : F
⊢ (f (-x)) y = -(f x) y

Tactic:
rw [map_neg, neg_apply]

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

Goal state before:
pp: case refine_1
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
s : Finset ι
⊢ ((fun s ↦ s.sup p) s).comp LinearMap.id ≤ 1 • s.sup p

Tactic:
one_smul

Goal state after:
case refine_1
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
s : Finset ι
⊢ ((fun s ↦ s.sup p) s).comp LinearMap.id ≤ 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
i : ι
⊢ (p i).comp LinearMap.id ≤ 1 • p i

Tactic:
one_smul

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 ≤ p i
Module: Mathlib.Analysis.LocallyConvex.WithSeminorms

Goal state before:
pp: case refine_1
𝕜 : Type u_2
E : Type u_6
ι : Type u_9
inst✝⁵ : NormedField 𝕜
inst✝⁴ : AddCommGroup E
inst✝³ : Module 𝕜 E
inst✝² : Preorder ι
inst✝¹ : LocallyFiniteOrderBot ι
p : SeminormFamily 𝕜 E ι
inst✝ : TopologicalSpace E
hp : WithSeminorms p
i : ι
⊢ ((fun i ↦ (Finset.Iic i).sup p) i).comp LinearMap.id ≤ 1 • (Finset.Iic i).sup p

Tactic:
one_smul

Goal state after:
case refine_1
𝕜 : Type u_2
E : Type u_6
ι : Type u_9
inst✝⁵ : NormedField 𝕜
inst✝⁴ : AddCommGroup E
inst✝³ : Module 𝕜 E
inst✝² : Preorder ι
inst✝¹ : LocallyFiniteOrderBot ι
p : SeminormFamily 𝕜 E ι
inst✝ : TopologicalSpace E
hp : WithSeminorms p
i : ι
⊢ ((fun i ↦ (Finset.Iic i).sup p) i).comp LinearMap.id ≤ (Finset.Iic i).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✝⁵ : NormedField 𝕜
inst✝⁴ : AddCommGroup E
inst✝³ : Module 𝕜 E
inst✝² : Preorder ι
inst✝¹ : LocallyFiniteOrderBot ι
p : SeminormFamily 𝕜 E ι
inst✝ : TopologicalSpace E
hp : WithSeminorms p
i : ι
⊢ (p i).comp LinearMap.id ≤ 1 • (Finset.Iic i).sup p

Tactic:
one_smul

Goal state after:
case refine_2
𝕜 : Type u_2
E : Type u_6
ι : Type u_9
inst✝⁵ : NormedField 𝕜
inst✝⁴ : AddCommGroup E
inst✝³ : Module 𝕜 E
inst✝² : Preorder ι
inst✝¹ : LocallyFiniteOrderBot ι
p : SeminormFamily 𝕜 E ι
inst✝ : TopologicalSpace E
hp : WithSeminorms p
i : ι
⊢ (p i).comp LinearMap.id ≤ (Finset.Iic i).sup p
Module: Mathlib.Analysis.LocallyConvex.WithSeminorms

Goal state before:
pp: 𝕜 : Type u_2
E : Type u_6
ι : Type u_9
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p : SeminormFamily 𝕜 E ι
q : Seminorm 𝕜 E
s : Finset ι
ε : ℝ
ε_pos : 0 < ε
hV : (s.sup p).ball 0 ε ∈ p.basisSets
hε : id ((s.sup p).ball 0 ε) ⊆ q.ball 0 1
x✝¹ : SeminormedAddCommGroup E := (s.sup p).toSeminormedAddCommGroup
x✝ : NormedSpace 𝕜 E := { toModule := inst✝, norm_smul_le := ⋯ }
this : Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, _] ⇑q
C : ℝ
C_pos : 0 < C
hC : ∀ (x : E), q x ≤ C * ‖x‖
⊢ ∃ s C, C ≠ 0 ∧ q ≤ C • s.sup p

Tactic:
exact ⟨s, ⟨C, C_pos.le⟩, fun H ↦ C_pos.ne.symm (congr_arg NNReal.toReal H), hC⟩

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

Goal state before:
pp: case inr.refine_2
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
h✝ : Nontrivial E
this : NormedSpace ℝ E
ub : ℝ≥0
hub : ub < ‖f‖₊
x : E
hx : ‖x‖₊ = 1
hxf : ub < ‖f x‖₊
⊢ ∃ a ∈ (fun x ↦ ‖f x‖₊) '' sphere 0 1, ub < a

Tactic:
exact ⟨_, ⟨x, by simpa using! congrArg NNReal.toReal hx, rfl⟩, hxf⟩

Goal state after:
Module: Mathlib.Topology.Compactness.CompactlyCoherentSpace

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
K : Set X
hK : IsCompact K
⊢ ContinuousOn (⇑(CompactCoherentification.mk X)) K

Tactic:
by
  rw [continuousOn_iff_continuous_domRestrict]
  exact ⟨fun U hU ↦ isOpen_iff.mp hU K hK⟩

Goal state after:
Module: Mathlib.Probability.ConditionalProbability

Goal state before:
pp: case hs
Ω : Type u_1
Ω' : Type u_2
m : MeasurableSpace Ω
m' : MeasurableSpace Ω'
μ : Measure Ω
s : Set Ω
i : Ω' → Ω
hi : MeasurableEmbedding i
hs : MeasurableSet s
t : Set Ω'
ht : MeasurableSet t
hi' : μ (range i)ᶜ = 0
⊢ MeasurableSet (i ⁻¹' s ∩ t)

Tactic:
exact hi.measurable hs

Goal state after:
case hs
Ω : Type u_1
Ω' : Type u_2
m : MeasurableSpace Ω
m' : MeasurableSpace Ω'
μ : Measure Ω
s : Set Ω
i : Ω' → Ω
hi : MeasurableEmbedding i
hs : MeasurableSet s
t : Set Ω'
ht : MeasurableSet t
hi' : μ (range i)ᶜ = 0
⊢ MeasurableSet (i ⁻¹' s ∩ t)
Module: Mathlib.Probability.ConditionalProbability

Goal state before:
pp: case hs
Ω : Type u_1
Ω' : Type u_2
m : MeasurableSpace Ω
m' : MeasurableSpace Ω'
μ : Measure Ω
s : Set Ω
i : Ω' → Ω
hi : MeasurableEmbedding i
hs : MeasurableSet s
t : Set Ω'
ht : MeasurableSet t
hi' : μ (range i)ᶜ = 0
⊢ MeasurableSet (i ⁻¹' s)

Tactic:
exact hi.measurable hs

Goal state after:
Module: Mathlib.Probability.ConditionalProbability

Goal state before:
pp: case hms
Ω : Type u_1
Ω' : Type u_2
m : MeasurableSpace Ω
m' : MeasurableSpace Ω'
μ : Measure Ω
s : Set Ω
i : Ω' → Ω
hi : MeasurableEmbedding i
hs : MeasurableSet s
t : Set Ω'
ht : MeasurableSet t
hi' : μ (range i)ᶜ = 0
⊢ MeasurableSet (i ⁻¹' s)

Tactic:
exact hi.measurable hs

Goal state after:
Module: Mathlib.Order.Filter.ENNReal

Goal state before:
pp: case mpr
α : Type u_1
f : Filter α
u : α → ℝ
h₁ : IsCoboundedUnder (fun x1 x2 ↦ x1 ≤ x2) f u
h₂ : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) f u
r : ℝ≥0
⊢ (∀ y > ↑r, ∀ᶠ (a : α) in f, ENNReal.ofReal (u a) < y) → ∀ y > ↑r, ∀ᶠ (a : α) in f, u a < y

Tactic:
· rintro h x hx
    have : 0 < x := hx.trans_le' (by simp)
    filter_upwards [h (.ofReal x) (by simpa [this] using hx)] with a ha
    exact (toReal_lt_of_lt_ofReal ha).trans_le' (by simp [toReal_ofReal'])

Goal state after:
Module: Mathlib.Order.Filter.ENNReal

Goal state before:
pp: α : Type u_1
f : Filter α
inst✝ : f.NeBot
u : α → ℝ≥0∞
C : ℝ≥0
hf : ∀ᶠ (a : α) in f, u a ≤ ↑C
h₁ : IsCoboundedUnder (fun x1 x2 ↦ x1 ≤ x2) f fun a ↦ (u a).toReal
h₂ : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) f fun a ↦ (u a).toReal
⊢ ∀ᶠ (a : α) in f, ENNReal.ofReal (u a).toReal = u a

Tactic:
filter_upwards [hf] with x hx

Goal state after:
α : Type u_1
f : Filter α
inst✝ : f.NeBot
u : α → ℝ≥0∞
C : ℝ≥0
hf : ∀ᶠ (a : α) in f, u a ≤ ↑C
h₁ : IsCoboundedUnder (fun x1 x2 ↦ x1 ≤ x2) f fun a ↦ (u a).toReal
h₂ : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) f fun a ↦ (u a).toReal
x : α
hx : u x ≤ ↑C
⊢ ENNReal.ofReal (u x).toReal = u x
Module: Mathlib.MeasureTheory.Function.AEEqFun

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : MeasurableSpace α
μ : Measure α
inst✝¹ : TopologicalSpace β
inst✝ : TopologicalSpace γ
f : α →ₘ[μ] β
g : α →ₘ[μ] γ
⊢ ↑(f.pair g) =ᵐ[μ] fun x ↦ (↑f x, ↑g x)

Tactic:
rw [pair_eq_mk]

Goal state after:
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : MeasurableSpace α
μ : Measure α
inst✝¹ : TopologicalSpace β
inst✝ : TopologicalSpace γ
f : α →ₘ[μ] β
g : α →ₘ[μ] γ
⊢ ↑(mk (fun x ↦ (↑f x, ↑g x)) ⋯) =ᵐ[μ] fun x ↦ (↑f x, ↑g x)
Module: Mathlib.MeasureTheory.Function.EssSup

Goal state before:
pp: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝³ : ConditionallyCompleteLinearOrder β
f : α → β
inst✝² : TopologicalSpace β
inst✝¹ : FirstCountableTopology β
inst✝ : OrderTopology β
hf : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) (ae μ) f
⊢ μ {y | essSup f μ < f y} = 0

Tactic:
← not_le

Goal state after:
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝³ : ConditionallyCompleteLinearOrder β
f : α → β
inst✝² : TopologicalSpace β
inst✝¹ : FirstCountableTopology β
inst✝ : OrderTopology β
hf : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) (ae μ) f
⊢ μ {y | ¬f y ≤ essSup f μ} = 0