mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Complex.ValueDistribution.LogCounting.Basic

Goal state before:
pp: E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : ProperSpace E
f₁ f₂ : locallyFinsupp E ℤ
r : ℝ
h : 0 ≤ f₂ - f₁
hr : 1 ≤ r
⊢ 0 ≤ logCounting f₂ r - logCounting f₁ r

Tactic:
simpa using

Goal state after:
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : ProperSpace E
f₁ f₂ : locallyFinsupp E ℤ
r : ℝ
h : 0 ≤ f₂ - f₁
hr : 1 ≤ r
⊢ logCounting f₁ r ≤ logCounting f₂ r
Module: Mathlib.Analysis.Complex.ValueDistribution.LogCounting.Basic

Goal state before:
pp: case pos
𝕜 : Type u_1
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : ProperSpace 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
U : Set 𝕜
f g : 𝕜 → E
a : WithTop E
a₀ : E
h : a = ⊤
⊢ ℝ → ℝ

Tactic:
exact (divisor f univ)⁻.logCounting

Goal state after:
Module: Mathlib.Analysis.Complex.ValueDistribution.LogCounting.Asymptotic

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : ProperSpace E
D : locallyFinsupp E ℤ
h : 0 ≤ D
hO : logCounting D =O[atTop] log
C : ℝ
hC : ∀ᶠ (x : ℝ) in atTop, ‖logCounting D x‖ ≤ C * ‖log x‖
N : ℕ
hCN : max C 0 < ↑N
hCN' : C < ↑N
hInf : (support D).Infinite
t : Finset E
htsub : ↑t ⊆ support D
htcard : t.card = N
D' : locallyFinsupp E ℤ := ∑ z ∈ t, single z 1
hD' : D' = ∑ z ∈ t, single z 1
w : E
hw : w ∈ t
h₁ : D w ≠ 0
⊢ 0 ≤ D w

Tactic:
simpa using

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : ProperSpace E
D : locallyFinsupp E ℤ
h : 0 ≤ D
hO : logCounting D =O[atTop] log
C : ℝ
hC : ∀ᶠ (x : ℝ) in atTop, ‖logCounting D x‖ ≤ C * ‖log x‖
N : ℕ
hCN : max C 0 < ↑N
hCN' : C < ↑N
hInf : (support D).Infinite
t : Finset E
htsub : ↑t ⊆ support D
htcard : t.card = N
D' : locallyFinsupp E ℤ := ∑ z ∈ t, single z 1
hD' : D' = ∑ z ∈ t, single z 1
w : E
hw : w ∈ t
h₁ : D w ≠ 0
⊢ 0 ≤ D w
Module: Mathlib.Analysis.Complex.ValueDistribution.LogCounting.Asymptotic

Goal state before:
pp: case neg
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : ProperSpace E
D : locallyFinsupp E ℤ
h : 0 ≤ D
hO : logCounting D =O[atTop] log
C : ℝ
hC : ∀ᶠ (x : ℝ) in atTop, ‖logCounting D x‖ ≤ C * ‖log x‖
N : ℕ
hCN : max C 0 < ↑N
hCN' : C < ↑N
hInf : (support D).Infinite
t : Finset E
htsub : ↑t ⊆ support D
htcard : t.card = N
D' : locallyFinsupp E ℤ := ∑ z ∈ t, single z 1
hD' : D' = ∑ z ∈ t, single z 1
w : E
hw : w ∉ t
⊢ (if w ∈ t then 1 else 0) ≤ D w

Tactic:
simpa using

Goal state after:
case neg
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : ProperSpace E
D : locallyFinsupp E ℤ
h : 0 ≤ D
hO : logCounting D =O[atTop] log
C : ℝ
hC : ∀ᶠ (x : ℝ) in atTop, ‖logCounting D x‖ ≤ C * ‖log x‖
N : ℕ
hCN : max C 0 < ↑N
hCN' : C < ↑N
hInf : (support D).Infinite
t : Finset E
htsub : ↑t ⊆ support D
htcard : t.card = N
D' : locallyFinsupp E ℤ := ∑ z ∈ t, single z 1
hD' : D' = ∑ z ∈ t, single z 1
w : E
hw : w ∉ t
⊢ 0 ≤ D w
Module: Mathlib.Analysis.Complex.ValueDistribution.LogCounting.Basic

Goal state before:
pp: case pos
𝕜 : Type u_1
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : ProperSpace 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
e : WithTop E
h : e = ⊤
⊢ MonotoneOn (logCounting f e) (Ioi 0)

Tactic:
simpa using

Goal state after:
case pos
𝕜 : Type u_1
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : ProperSpace 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
e : WithTop E
h : e = ⊤
⊢ MonotoneOn (locallyFinsuppWithin.logCounting (divisor f univ)⁻) (Ioi 0)
Module: Mathlib.Analysis.Complex.ValueDistribution.LogCounting.Basic

Goal state before:
pp: case neg
𝕜 : Type u_1
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : ProperSpace 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
e : WithTop E
h : ¬e = ⊤
⊢ MonotoneOn (logCounting f e) (Ioi 0)

Tactic:
simpa using

Goal state after:
case neg
𝕜 : Type u_1
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : ProperSpace 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
e : WithTop E
h : ¬e = ⊤
⊢ MonotoneOn (locallyFinsuppWithin.logCounting (divisor (fun x ↦ f x - e.untop₀) univ)⁺) (Ioi 0)
Module: Mathlib.Analysis.Complex.ValueDistribution.LogCounting.Basic

Goal state before:
pp: case pos
𝕜 : Type u_1
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : ProperSpace 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
r : ℝ
f : 𝕜 → E
e : WithTop E
hr : 1 ≤ r
h : e = ⊤
⊢ 0 ≤ logCounting f e r

Tactic:
simp [logCounting, h, locallyFinsuppWithin.logCounting_nonneg
      (negPart_nonneg (divisor f univ)) hr]

Goal state after:
Module: Mathlib.Analysis.Complex.ValueDistribution.LogCounting.Asymptotic

Goal state before:
pp: case a
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : ProperSpace E
D : locallyFinsupp E ℤ
h : 0 ≤ D
hO : logCounting D =O[atTop] log
C : ℝ
hC : ∀ᶠ (x : ℝ) in atTop, ‖logCounting D x‖ ≤ C * ‖log x‖
N : ℕ
hCN : max C 0 < ↑N
hCN' : C < ↑N
hInf : (support D).Infinite
t : Finset E
htsub : ↑t ⊆ support D
htcard : t.card = N
D' : locallyFinsupp E ℤ := ∑ z ∈ t, single z 1
hD' : D' = ∑ z ∈ t, single z 1
hle : D' ≤ D
R₀ : ℝ
hR₀ : ∀ z ∈ t, ‖z‖ ≤ R₀
K : ℝ := ∑ z ∈ t, log ‖z‖
hK : K = ∑ z ∈ t, log ‖z‖
r : ℝ
hr : R₀ ≤ r
z : E
hz : z ∈ t
⊢ logCounting (single z 1) r = log r - log ‖z‖

Tactic:
simpa using

Goal state after:
case a
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : ProperSpace E
D : locallyFinsupp E ℤ
h : 0 ≤ D
hO : logCounting D =O[atTop] log
C : ℝ
hC : ∀ᶠ (x : ℝ) in atTop, ‖logCounting D x‖ ≤ C * ‖log x‖
N : ℕ
hCN : max C 0 < ↑N
hCN' : C < ↑N
hInf : (support D).Infinite
t : Finset E
htsub : ↑t ⊆ support D
htcard : t.card = N
D' : locallyFinsupp E ℤ := ∑ z ∈ t, single z 1
hD' : D' = ∑ z ∈ t, single z 1
hle : D' ≤ D
R₀ : ℝ
hR₀ : ∀ z ∈ t, ‖z‖ ≤ R₀
K : ℝ := ∑ z ∈ t, log ‖z‖
hK : K = ∑ z ∈ t, log ‖z‖
r : ℝ
hr : R₀ ≤ r
z : E
hz : z ∈ t
⊢ logCounting (single z 1) r = log r - log ‖z‖
Module: Mathlib.Analysis.Complex.ValueDistribution.LogCounting.Basic

Goal state before:
pp: case e_a.hf
R : ℝ
f : ℂ → ℂ
h : Meromorphic f
hR : R ≠ 0
h₁f : MeromorphicOn f (closedBall 0 |R|)
⊢ MeromorphicOn f (closedBall 0 |R|)

pp: case e_a.hz
R : ℝ
f : ℂ → ℂ
h : Meromorphic f
hR : R ≠ 0
h₁f : MeromorphicOn f (closedBall 0 |R|)
⊢ 0 ∈ closedBall 0 |R|

pp: case e_a.hf
R : ℝ
f : ℂ → ℂ
h : Meromorphic f
hR : R ≠ 0
h₁f : MeromorphicOn f (closedBall 0 |R|)
⊢ MeromorphicOn f univ

pp: case e_a.hz
R : ℝ
f : ℂ → ℂ
h : Meromorphic f
hR : R ≠ 0
h₁f : MeromorphicOn f (closedBall 0 |R|)
⊢ 0 ∈ univ

Tactic:
all_goals aesop

Goal state after:
Module: Mathlib.Analysis.ConstantSpeed

Goal state before:
pp: case mpr
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : ℝ → E
s : Set ℝ
h : eVariationOn f s = 0
x : ℝ
x✝¹ : x ∈ s
y : ℝ
x✝ : y ∈ s
⊢ eVariationOn f (s ∩ Icc x y) = 0

Tactic:
simpa using

Goal state after:
case mpr
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : ℝ → E
s : Set ℝ
h : eVariationOn f s = 0
x : ℝ
x✝¹ : x ∈ s
y : ℝ
x✝ : y ∈ s
⊢ eVariationOn f (s ∩ Icc x y) = 0
Module: Mathlib.Combinatorics.Hall.Finite

Goal state before:
pp: ι : Type u
α : Type v
inst✝¹ : DecidableEq α
t : ι → Finset α
inst✝ : Fintype ι
x : ι
a : α
s' : Finset ↑{x' | x' ≠ x}
this : DecidableEq ι
ha : s'.Nonempty → image (fun z ↦ ↑z) s' ≠ univ → #s' < #((image (fun z ↦ ↑z) s').biUnion t)
he : s'.Nonempty
h : image (fun z ↦ ↑z) s' = univ
⊢ False

Tactic:
simpa using

Goal state after:
ι : Type u
α : Type v
inst✝¹ : DecidableEq α
t : ι → Finset α
inst✝ : Fintype ι
x : ι
a : α
s' : Finset ↑{x' | x' ≠ x}
this : DecidableEq ι
ha : s'.Nonempty → image (fun z ↦ ↑z) s' ≠ univ → #s' < #((image (fun z ↦ ↑z) s').biUnion t)
he : s'.Nonempty
h : image (fun z ↦ ↑z) s' = univ
⊢ False
Module: Mathlib.Analysis.Convex.BetweenList

Goal state before:
pp: case cons.refine_2.refine_1.cons.inl.cons
R : Type u_1
V : Type u_2
P : Type u_4
inst✝⁵ : Ring R
inst✝⁴ : PartialOrder R
inst✝³ : AddCommGroup V
inst✝² : Module R V
inst✝¹ : AddTorsor V P
inst✝ : IsOrderedRing R
head head3 : P
tail : List P
x✝ :
  (Pairwise (Sbtw R head) (head :: head3 :: tail) ∧ Triplewise (Sbtw R) (head :: head3 :: tail)) ∧
    ∀ (a : P), head :: head :: head3 :: tail ≠ [a, a]
hp : (∀ a' ∈ head3 :: tail, Sbtw R head head a') ∧ Pairwise (Sbtw R head) (head3 :: tail)
ht : Triplewise (Sbtw R) (head :: head3 :: tail)
ha : ∀ (a : P), head :: head :: head3 :: tail ≠ [a, a]
ht' : List.Wbtw R (head :: head3 :: tail)
ih : Pairwise (fun x1 x2 ↦ x1 ≠ x2) (head :: head3 :: tail) ↔ ∀ (a : P), head :: head3 :: tail ≠ [a, a]
ha' : head ∈ head :: head3 :: tail
⊢ False

Tactic:
simpa using

Goal state after:
case cons.refine_2.refine_1.cons.inl.cons
R : Type u_1
V : Type u_2
P : Type u_4
inst✝⁵ : Ring R
inst✝⁴ : PartialOrder R
inst✝³ : AddCommGroup V
inst✝² : Module R V
inst✝¹ : AddTorsor V P
inst✝ : IsOrderedRing R
head head3 : P
tail : List P
x✝ :
  (Pairwise (Sbtw R head) (head :: head3 :: tail) ∧ Triplewise (Sbtw R) (head :: head3 :: tail)) ∧
    ∀ (a : P), head :: head :: head3 :: tail ≠ [a, a]
hp : (∀ a' ∈ head3 :: tail, Sbtw R head head a') ∧ Pairwise (Sbtw R head) (head3 :: tail)
ht : Triplewise (Sbtw R) (head :: head3 :: tail)
ha : ∀ (a : P), head :: head :: head3 :: tail ≠ [a, a]
ht' : List.Wbtw R (head :: head3 :: tail)
ih : Pairwise (fun x1 x2 ↦ x1 ≠ x2) (head :: head3 :: tail) ↔ ∀ (a : P), head :: head3 :: tail ≠ [a, a]
ha' : head ∈ head :: head3 :: tail
⊢ False
Module: Mathlib.Analysis.Convex.Approximation

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
s : Set E
φ : E → ℝ
inst✝⁸ : RCLike 𝕜
inst✝⁷ : TopologicalSpace E
inst✝⁶ : AddCommGroup E
inst✝⁵ : Module ℝ E
inst✝⁴ : Module 𝕜 E
inst✝³ : IsScalarTower ℝ 𝕜 E
inst✝² : IsTopologicalAddGroup E
inst✝¹ : ContinuousSMul 𝕜 E
inst✝ : LocallyConvexSpace ℝ E
x : E
a : ℝ
hx : x ∈ s
hax : a < φ x
hsc : IsClosed s
hφc : LowerSemicontinuousOn φ s
hφcv : ConvexOn ℝ s φ
⊢ ∃ l c, s.domRestrict (⇑re ∘ ⇑l) + const (↑s) c ≤ s.domRestrict φ ∧ re (l x) + c = a

Tactic:
let A := { p : E × 𝕜 | p.1 ∈ s ∧ φ p.1 ≤ re p.2 }

Goal state after:
𝕜 : Type u_1
E : Type u_2
s : Set E
φ : E → ℝ
inst✝⁸ : RCLike 𝕜
inst✝⁷ : TopologicalSpace E
inst✝⁶ : AddCommGroup E
inst✝⁵ : Module ℝ E
inst✝⁴ : Module 𝕜 E
inst✝³ : IsScalarTower ℝ 𝕜 E
inst✝² : IsTopologicalAddGroup E
inst✝¹ : ContinuousSMul 𝕜 E
inst✝ : LocallyConvexSpace ℝ E
x : E
a : ℝ
hx : x ∈ s
hax : a < φ x
hsc : IsClosed s
hφc : LowerSemicontinuousOn φ s
hφcv : ConvexOn ℝ s φ
A : Set (E × 𝕜) := {p | p.1 ∈ s ∧ φ p.1 ≤ re p.2}
⊢ ∃ l c, s.domRestrict (⇑re ∘ ⇑l) + const (↑s) c ≤ s.domRestrict φ ∧ re (l x) + c = a
Module: Mathlib.Analysis.Convex.BetweenList

Goal state before:
pp: case cons.refine_2.refine_1.cons.inr
R : Type u_1
V : Type u_2
P : Type u_4
inst✝⁵ : Ring R
inst✝⁴ : PartialOrder R
inst✝³ : AddCommGroup V
inst✝² : Module R V
inst✝¹ : AddTorsor V P
inst✝ : IsOrderedRing R
head head2 : P
tail : List P
x✝ :
  (Pairwise (Sbtw R head) (head2 :: tail) ∧ Triplewise (Sbtw R) (head2 :: tail)) ∧
    ∀ (a : P), head :: head2 :: tail ≠ [a, a]
hp : (∀ a' ∈ tail, Sbtw R head head2 a') ∧ Pairwise (Sbtw R head) tail
ht : Triplewise (Sbtw R) (head2 :: tail)
ha : ∀ (a : P), head :: head2 :: tail ≠ [a, a]
ht' : List.Wbtw R (head2 :: tail)
ih : Pairwise (fun x1 x2 ↦ x1 ≠ x2) (head2 :: tail) ↔ ∀ (a : P), head2 :: tail ≠ [a, a]
ha' : head ∈ head2 :: tail
hat : head ∈ tail
⊢ False

Tactic:
simpa using

Goal state after:
case cons.refine_2.refine_1.cons.inr
R : Type u_1
V : Type u_2
P : Type u_4
inst✝⁵ : Ring R
inst✝⁴ : PartialOrder R
inst✝³ : AddCommGroup V
inst✝² : Module R V
inst✝¹ : AddTorsor V P
inst✝ : IsOrderedRing R
head head2 : P
tail : List P
x✝ :
  (Pairwise (Sbtw R head) (head2 :: tail) ∧ Triplewise (Sbtw R) (head2 :: tail)) ∧
    ∀ (a : P), head :: head2 :: tail ≠ [a, a]
hp : (∀ a' ∈ tail, Sbtw R head head2 a') ∧ Pairwise (Sbtw R head) tail
ht : Triplewise (Sbtw R) (head2 :: tail)
ha : ∀ (a : P), head :: head2 :: tail ≠ [a, a]
ht' : List.Wbtw R (head2 :: tail)
ih : Pairwise (fun x1 x2 ↦ x1 ≠ x2) (head2 :: tail) ↔ ∀ (a : P), head2 :: tail ≠ [a, a]
ha' : head ∈ head2 :: tail
hat : head ∈ tail
⊢ False
Module: Mathlib.Combinatorics.Hall.Finite

Goal state before:
pp: ι : Type u
α : Type v
inst✝¹ : DecidableEq α
t : ι → Finset α
inst✝ : Fintype ι
n : ℕ
hn : Fintype.card ι = n + 1
ht : ∀ (s : Finset ι), #s ≤ #(s.biUnion t)
ih :
  ∀ {ι' : Type u} [inst : Fintype ι'] (t' : ι' → Finset α),
    Fintype.card ι' ≤ n →
      (∀ (s' : Finset ι'), #s' ≤ #(s'.biUnion t')) → ∃ f, Function.Injective f ∧ ∀ (x : ι'), f x ∈ t' x
ha : ∀ (s : Finset ι), s.Nonempty → s ≠ univ → #s < #(s.biUnion t)
this✝ : Nonempty ι
this : DecidableEq ι
x✝ : ι := Classical.arbitrary ι
y : α
hy : y ∈ t x✝
ι' : Set ι := {x' | x' ≠ x✝}
t' : ↑ι' → Finset α := fun x' ↦ (t ↑x').erase y
card_ι' : Fintype.card ↑ι' = n
f' : ↑ι' → α
hfinj : Function.Injective f'
hfr : ∀ (x : ↑ι'), f' x ∈ t' x
z₁ z₂ : ι
x : ↑ι'
h : y = f' x
⊢ False

Tactic:
simpa using

Goal state after:
ι : Type u
α : Type v
inst✝¹ : DecidableEq α
t : ι → Finset α
inst✝ : Fintype ι
n : ℕ
hn : Fintype.card ι = n + 1
ht : ∀ (s : Finset ι), #s ≤ #(s.biUnion t)
ih :
  ∀ {ι' : Type u} [inst : Fintype ι'] (t' : ι' → Finset α),
    Fintype.card ι' ≤ n →
      (∀ (s' : Finset ι'), #s' ≤ #(s'.biUnion t')) → ∃ f, Function.Injective f ∧ ∀ (x : ι'), f x ∈ t' x
ha : ∀ (s : Finset ι), s.Nonempty → s ≠ univ → #s < #(s.biUnion t)
this✝ : Nonempty ι
this : DecidableEq ι
x✝ : ι := Classical.arbitrary ι
y : α
hy : y ∈ t x✝
ι' : Set ι := {x' | x' ≠ x✝}
t' : ↑ι' → Finset α := fun x' ↦ (t ↑x').erase y
card_ι' : Fintype.card ↑ι' = n
f' : ↑ι' → α
hfinj : Function.Injective f'
hfr : ∀ (x : ↑ι'), f' x ∈ t' x
z₁ z₂ : ι
x : ↑ι'
h : y = f' x
⊢ False
Module: Mathlib.Analysis.Convex.Approximation

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
s : Set E
φ : E → ℝ
inst✝⁸ : RCLike 𝕜
inst✝⁷ : TopologicalSpace E
inst✝⁶ : AddCommGroup E
inst✝⁵ : Module ℝ E
inst✝⁴ : Module 𝕜 E
inst✝³ : IsScalarTower ℝ 𝕜 E
inst✝² : IsTopologicalAddGroup E
inst✝¹ : ContinuousSMul 𝕜 E
inst✝ : LocallyConvexSpace ℝ E
x : E
a : ℝ
hx : x ∈ s
hax : a < φ x
hsc : IsClosed s
hφc : LowerSemicontinuousOn φ s
hφcv : ConvexOn ℝ s φ
A : Set (E × 𝕜) := {p | p.1 ∈ s ∧ φ p.1 ≤ re p.2}
L : StrongDual 𝕜 (E × 𝕜)
b : ℝ
hLb : re (L (x, ↑a)) < b ∧ ∀ b_1 ∈ {p | p.1 ∈ s ∧ φ p.1 ≤ re p.2}, b < re (L b_1)
u : E →L[𝕜] 𝕜 := L ∘SL inl 𝕜 E 𝕜
c : ℝ := (re (L (0, 1)))⁻¹
z : ↑s
hv : ∀ (v : 𝕜), v * L (0, 1) = L (0, v)
w : E
h : w ∈ s
hw :
  re (((L ∘SL inl 𝕜 E 𝕜).coprod (L ∘SL inr 𝕜 E 𝕜)) (x, ↑a)) <
    re (((L ∘SL inl 𝕜 E 𝕜).coprod (L ∘SL inr 𝕜 E 𝕜)) (w, ↑(φ w)))
⊢ re (L (x, 0)) + re (L (0, 1)) * a < re (L (w, 0)) + re (L (0, 1)) * φ w

Tactic:
simpa using

Goal state after:
𝕜 : Type u_1
E : Type u_2
s : Set E
φ : E → ℝ
inst✝⁸ : RCLike 𝕜
inst✝⁷ : TopologicalSpace E
inst✝⁶ : AddCommGroup E
inst✝⁵ : Module ℝ E
inst✝⁴ : Module 𝕜 E
inst✝³ : IsScalarTower ℝ 𝕜 E
inst✝² : IsTopologicalAddGroup E
inst✝¹ : ContinuousSMul 𝕜 E
inst✝ : LocallyConvexSpace ℝ E
x : E
a : ℝ
hx : x ∈ s
hax : a < φ x
hsc : IsClosed s
hφc : LowerSemicontinuousOn φ s
hφcv : ConvexOn ℝ s φ
A : Set (E × 𝕜) := {p | p.1 ∈ s ∧ φ p.1 ≤ re p.2}
L : StrongDual 𝕜 (E × 𝕜)
b : ℝ
hLb : re (L (x, ↑a)) < b ∧ ∀ b_1 ∈ {p | p.1 ∈ s ∧ φ p.1 ≤ re p.2}, b < re (L b_1)
u : E →L[𝕜] 𝕜 := L ∘SL inl 𝕜 E 𝕜
c : ℝ := (re (L (0, 1)))⁻¹
z : ↑s
hv : ∀ (v : 𝕜), v * L (0, 1) = L (0, v)
w : E
h : w ∈ s
hw :
  re (((L ∘SL inl 𝕜 E 𝕜).coprod (L ∘SL inr 𝕜 E 𝕜)) (x, ↑a)) <
    re (((L ∘SL inl 𝕜 E 𝕜).coprod (L ∘SL inr 𝕜 E 𝕜)) (w, ↑(φ w)))
⊢ re (L (x, 0)) + re (L (0, 1)) * a < re (L (w, 0)) + re (L (0, 1)) * φ w
Module: Mathlib.Analysis.Convex.Approximation

Goal state before:
pp: case refine_1
𝕜 : Type u_1
E : Type u_2
s : Set E
φ : E → ℝ
inst✝⁸ : RCLike 𝕜
inst✝⁷ : TopologicalSpace E
inst✝⁶ : AddCommGroup E
inst✝⁵ : Module ℝ E
inst✝⁴ : Module 𝕜 E
inst✝³ : IsScalarTower ℝ 𝕜 E
inst✝² : IsTopologicalAddGroup E
inst✝¹ : ContinuousSMul 𝕜 E
inst✝ : LocallyConvexSpace ℝ E
x : E
a : ℝ
hx : x ∈ s
hax : a < φ x
hsc : IsClosed s
hφc : LowerSemicontinuousOn φ s
hφcv : ConvexOn ℝ s φ
A : Set (E × 𝕜) := {p | p.1 ∈ s ∧ φ p.1 ≤ re p.2}
L : StrongDual 𝕜 (E × 𝕜)
b : ℝ
hLb : re (L (x, ↑a)) < b ∧ ∀ b_1 ∈ {p | p.1 ∈ s ∧ φ p.1 ≤ re p.2}, b < re (L b_1)
u : E →L[𝕜] 𝕜 := L ∘SL inl 𝕜 E 𝕜
c : ℝ := (re (L (0, 1)))⁻¹
z : ↑s
hv : ∀ (v : 𝕜), v * L (0, 1) = L (0, v)
hine : ∀ {w : E}, w ∈ s → re (L (x, 0)) + re (L (0, 1)) * a < re (L (w, 0)) + re (L (0, 1)) * φ w
hc : 0 < c
⊢ (s.domRestrict (⇑re ∘ ⇑(-c • u)) + const (↑s) (c * re (u x) + a)) z ≤ s.domRestrict φ z

Tactic:
simpa using

Goal state after:
case refine_1
𝕜 : Type u_1
E : Type u_2
s : Set E
φ : E → ℝ
inst✝⁸ : RCLike 𝕜
inst✝⁷ : TopologicalSpace E
inst✝⁶ : AddCommGroup E
inst✝⁵ : Module ℝ E
inst✝⁴ : Module 𝕜 E
inst✝³ : IsScalarTower ℝ 𝕜 E
inst✝² : IsTopologicalAddGroup E
inst✝¹ : ContinuousSMul 𝕜 E
inst✝ : LocallyConvexSpace ℝ E
x : E
a : ℝ
hx : x ∈ s
hax : a < φ x
hsc : IsClosed s
hφc : LowerSemicontinuousOn φ s
hφcv : ConvexOn ℝ s φ
A : Set (E × 𝕜) := {p | p.1 ∈ s ∧ φ p.1 ≤ re p.2}
L : StrongDual 𝕜 (E × 𝕜)
b : ℝ
hLb : re (L (x, ↑a)) < b ∧ ∀ b_1 ∈ {p | p.1 ∈ s ∧ φ p.1 ≤ re p.2}, b < re (L b_1)
u : E →L[𝕜] 𝕜 := L ∘SL inl 𝕜 E 𝕜
c : ℝ := (re (L (0, 1)))⁻¹
z : ↑s
hv : ∀ (v : 𝕜), v * L (0, 1) = L (0, v)
hine : ∀ {w : E}, w ∈ s → re (L (x, 0)) + re (L (0, 1)) * a < re (L (w, 0)) + re (L (0, 1)) * φ w
hc : 0 < c
⊢ (re (L (0, 1)))⁻¹ * re (L (x, 0)) + a ≤ (re (L (0, 1)))⁻¹ * re (L (↑z, 0)) + φ ↑z
Module: Mathlib.Analysis.Convex.Approximation

Goal state before:
pp: s : Set ℝ
f : ℝ → ℝ
x a : ℝ
hx : x ∈ s
hax : a < f x
hsc : IsClosed s
hfc : LowerSemicontinuousOn f s
hf : ConvexOn ℝ s f
l : ℝ →L[ℝ] ℝ
c' : ℝ
hlc'_le : s.domRestrict (⇑re ∘ ⇑l) + const (↑s) c' ≤ s.domRestrict f
hlc'_eq : re (l x) + c' = a
h1 : ∀ (y : ℝ), l 1 * y = l y
y : ℝ
hy : y ∈ s
⊢ l 1 * y + c' ≤ f y

Tactic:
simpa using

Goal state after:
s : Set ℝ
f : ℝ → ℝ
x a : ℝ
hx : x ∈ s
hax : a < f x
hsc : IsClosed s
hfc : LowerSemicontinuousOn f s
hf : ConvexOn ℝ s f
l : ℝ →L[ℝ] ℝ
c' : ℝ
hlc'_le : s.domRestrict (⇑re ∘ ⇑l) + const (↑s) c' ≤ s.domRestrict f
hlc'_eq : re (l x) + c' = a
h1 : ∀ (y : ℝ), l 1 * y = l y
y : ℝ
hy : y ∈ s
⊢ l y + c' ≤ f y
Module: Mathlib.Analysis.Convex.Approximation

Goal state before:
pp: s : Set ℝ
f : ℝ → ℝ
x a : ℝ
hx : x ∈ s
hax : a < f x
hsc : IsClosed s
hfc : LowerSemicontinuousOn f s
hf : ConvexOn ℝ s f
l : ℝ →L[ℝ] ℝ
c' : ℝ
hlc'_le : s.domRestrict (⇑re ∘ ⇑l) + const (↑s) c' ≤ s.domRestrict f
hlc'_eq : re (l x) + c' = a
h1 : ∀ (y : ℝ), l 1 * y = l y
⊢ l 1 * x + c' = a

Tactic:
simpa using

Goal state after:
s : Set ℝ
f : ℝ → ℝ
x a : ℝ
hx : x ∈ s
hax : a < f x
hsc : IsClosed s
hfc : LowerSemicontinuousOn f s
hf : ConvexOn ℝ s f
l : ℝ →L[ℝ] ℝ
c' : ℝ
hlc'_le : s.domRestrict (⇑re ∘ ⇑l) + const (↑s) c' ≤ s.domRestrict f
hlc'_eq : re (l x) + c' = a
h1 : ∀ (y : ℝ), l 1 * y = l y
⊢ l x + c' = a
Module: Mathlib.Analysis.Convex.Approximation

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
s : Set E
φ : E → ℝ
inst✝⁸ : RCLike 𝕜
inst✝⁷ : TopologicalSpace E
inst✝⁶ : AddCommGroup E
inst✝⁵ : Module ℝ E
inst✝⁴ : Module 𝕜 E
inst✝³ : IsScalarTower ℝ 𝕜 E
inst✝² : IsTopologicalAddGroup E
inst✝¹ : ContinuousSMul 𝕜 E
inst✝ : LocallyConvexSpace ℝ E
hsc : IsClosed s
hφc : LowerSemicontinuousOn φ s
hφcv : ConvexOn ℝ s φ
⊢ sSup {f | f ≤ s.domRestrict φ ∧ ∃ l c, f = s.domRestrict (⇑re ∘ ⇑l) + const (↑s) c} = s.domRestrict φ

Tactic:
let A := { p : E × 𝕜 | p.1 ∈ s ∧ φ p.1 ≤ re p.2 }

Goal state after:
𝕜 : Type u_1
E : Type u_2
s : Set E
φ : E → ℝ
inst✝⁸ : RCLike 𝕜
inst✝⁷ : TopologicalSpace E
inst✝⁶ : AddCommGroup E
inst✝⁵ : Module ℝ E
inst✝⁴ : Module 𝕜 E
inst✝³ : IsScalarTower ℝ 𝕜 E
inst✝² : IsTopologicalAddGroup E
inst✝¹ : ContinuousSMul 𝕜 E
inst✝ : LocallyConvexSpace ℝ E
hsc : IsClosed s
hφc : LowerSemicontinuousOn φ s
hφcv : ConvexOn ℝ s φ
A : Set (E × 𝕜) := {p | p.1 ∈ s ∧ φ p.1 ≤ re p.2}
⊢ sSup {f | f ≤ s.domRestrict φ ∧ ∃ l c, f = s.domRestrict (⇑re ∘ ⇑l) + const (↑s) c} = s.domRestrict φ
Module: Mathlib.Analysis.Convex.Between

Goal state before:
pp: case refine_1
R : Type u_1
V : Type u_2
P : Type u_4
inst✝⁵ : Ring R
inst✝⁴ : PartialOrder R
inst✝³ : AddCommGroup V
inst✝² : Module R V
inst✝¹ : AddTorsor V P
inst✝ : IsOrderedRing R
x y : P
h : Wbtw R x y x
⊢ y = x

Tactic:
simpa using

Goal state after:
case refine_1
R : Type u_1
V : Type u_2
P : Type u_4
inst✝⁵ : Ring R
inst✝⁴ : PartialOrder R
inst✝³ : AddCommGroup V
inst✝² : Module R V
inst✝¹ : AddTorsor V P
inst✝ : IsOrderedRing R
x y : P
h : Wbtw R x y x
⊢ y = x
Module: Mathlib.Combinatorics.Hall.Basic

Goal state before:
pp: ι : Type u
α : Type v
t : ι → Finset α
ι' ι'' : Finset ι
h : ι' ⊆ ι''
f : ↑(hallMatchingsOn t ι'')
hinj : Injective ↑f
hc : ∀ (x : ↥ι''), ↑f x ∈ t ↑x
i : ι
hi : i ∈ ι'
j : ι
hj : j ∈ ι'
hh : (fun i ↦ ↑f ⟨↑i, ⋯⟩) ⟨i, hi⟩ = (fun i ↦ ↑f ⟨↑i, ⋯⟩) ⟨j, hj⟩
⊢ ⟨i, hi⟩ = ⟨j, hj⟩

Tactic:
simpa using

Goal state after:
ι : Type u
α : Type v
t : ι → Finset α
ι' ι'' : Finset ι
h : ι' ⊆ ι''
f : ↑(hallMatchingsOn t ι'')
hinj : Injective ↑f
hc : ∀ (x : ↥ι''), ↑f x ∈ t ↑x
i : ι
hi : i ∈ ι'
j : ι
hj : j ∈ ι'
hh : (fun i ↦ ↑f ⟨↑i, ⋯⟩) ⟨i, hi⟩ = (fun i ↦ ↑f ⟨↑i, ⋯⟩) ⟨j, hj⟩
⊢ i = j
Module: Mathlib.Combinatorics.Hall.Basic

Goal state before:
pp: ι : Type u
α : Type v
t : ι → Finset α
ι' : Finset ι
g : ↑(hallMatchingsOn t ι') → ↥ι' → ↥(ι'.biUnion t) := fun f i ↦ ⟨↑f i, ⋯⟩
f f' : ↑(hallMatchingsOn t ι')
h : ∀ (x : ↥ι'), g f x = g f' x
a : ↥ι'
⊢ ↑f a = ↑f' a

Tactic:
simpa using

Goal state after:
ι : Type u
α : Type v
t : ι → Finset α
ι' : Finset ι
g : ↑(hallMatchingsOn t ι') → ↥ι' → ↥(ι'.biUnion t) := fun f i ↦ ⟨↑f i, ⋯⟩
f f' : ↑(hallMatchingsOn t ι')
h : ∀ (x : ↥ι'), g f x = g f' x
a : ↥ι'
⊢ ↑f a = ↑f' a
Module: Mathlib.Analysis.Convex.Approximation

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
φ : E → ℝ
inst✝⁹ : RCLike 𝕜
inst✝⁸ : TopologicalSpace E
inst✝⁷ : AddCommGroup E
inst✝⁶ : Module ℝ E
inst✝⁵ : Module 𝕜 E
inst✝⁴ : IsScalarTower ℝ 𝕜 E
inst✝³ : IsTopologicalAddGroup E
inst✝² : ContinuousSMul 𝕜 E
inst✝¹ : LocallyConvexSpace ℝ E
inst✝ : HereditarilyLindelofSpace E
hφc : LowerSemicontinuous φ
hφcv : ConvexOn ℝ univ φ
l : ℕ → E →L[𝕜] 𝕜
c : ℕ → ℝ
hle : ∀ (i : ℕ), univ.domRestrict (⇑re ∘ ⇑(l i)) + const (↑univ) (c i) ≤ univ.domRestrict φ
hsup : ⨆ i, univ.domRestrict (⇑re ∘ ⇑(l i)) + const (↑univ) (c i) = univ.domRestrict φ
x : E
⊢ (⨆ i, ⇑re ∘ ⇑(l i) + const E (c i)) x = φ x

Tactic:
simpa using

Goal state after:
𝕜 : Type u_1
E : Type u_2
φ : E → ℝ
inst✝⁹ : RCLike 𝕜
inst✝⁸ : TopologicalSpace E
inst✝⁷ : AddCommGroup E
inst✝⁶ : Module ℝ E
inst✝⁵ : Module 𝕜 E
inst✝⁴ : IsScalarTower ℝ 𝕜 E
inst✝³ : IsTopologicalAddGroup E
inst✝² : ContinuousSMul 𝕜 E
inst✝¹ : LocallyConvexSpace ℝ E
inst✝ : HereditarilyLindelofSpace E
hφc : LowerSemicontinuous φ
hφcv : ConvexOn ℝ univ φ
l : ℕ → E →L[𝕜] 𝕜
c : ℕ → ℝ
hle : ∀ (i : ℕ), univ.domRestrict (⇑re ∘ ⇑(l i)) + const (↑univ) (c i) ≤ univ.domRestrict φ
hsup : ⨆ i, univ.domRestrict (⇑re ∘ ⇑(l i)) + const (↑univ) (c i) = univ.domRestrict φ
x : E
⊢ ⨆ i, re ((l i) x) + c i = φ x
Module: Mathlib.CategoryTheory.CofilteredSystem

Goal state before:
pp: J : Type u
inst✝¹ : Category.{v_1, u} J
F : J ⥤ Type v
inst✝ : IsCofilteredOrEmpty J
Fsur : ∀ ⦃i j : J⦄ (f : i ⟶ j), Function.Surjective ⇑(ConcreteCategory.hom (F.map f))
i j : J
f g : i ⟶ j
k : J
φ : k ⟶ i
hφ : φ ≫ f = φ ≫ g
this :
  (fun x ↦ (ConcreteCategory.hom (F.map f)) ((ConcreteCategory.hom (F.map φ)) x)) = fun x ↦
    (ConcreteCategory.hom (F.map g)) ((ConcreteCategory.hom (F.map φ)) x)
⊢ ConcreteCategory.hom (F.map f) = ConcreteCategory.hom (F.map g)

Tactic:
simpa using

Goal state after:
J : Type u
inst✝¹ : Category.{v_1, u} J
F : J ⥤ Type v
inst✝ : IsCofilteredOrEmpty J
Fsur : ∀ ⦃i j : J⦄ (f : i ⟶ j), Function.Surjective ⇑(ConcreteCategory.hom (F.map f))
i j : J
f g : i ⟶ j
k : J
φ : k ⟶ i
hφ : φ ≫ f = φ ≫ g
this :
  (fun x ↦ (ConcreteCategory.hom (F.map f)) ((ConcreteCategory.hom (F.map φ)) x)) = fun x ↦
    (ConcreteCategory.hom (F.map g)) ((ConcreteCategory.hom (F.map φ)) x)
⊢ ConcreteCategory.hom (F.map f) = ConcreteCategory.hom (F.map g)
Module: Mathlib.LinearAlgebra.Matrix.Permutation

Goal state before:
pp: n : Type u_1
inst✝³ : DecidableEq n
σ : Perm n
inst✝² : Fintype n
𝕜 : Type u_3
inst✝¹ : RCLike 𝕜
inst✝ : Nonempty n
inhabited_h : Inhabited n
⊢ 1 ≤ ‖Perm.permMatrix 𝕜 σ‖

Tactic:
simpa using

Goal state after:
n : Type u_1
inst✝³ : DecidableEq n
σ : Perm n
inst✝² : Fintype n
𝕜 : Type u_3
inst✝¹ : RCLike 𝕜
inst✝ : Nonempty n
inhabited_h : Inhabited n
⊢ 1 ≤ ‖Perm.permMatrix 𝕜 σ‖