mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.TrapezoidalRule

Goal state before:
pp: case hb
f : ℝ → ℝ
ζ a b : ℝ
a_lt_b : a < b
h_df : DifferentiableOn ℝ f (Set.Icc a b)
h_ddf : DifferentiableOn ℝ (_root_.derivWithin f (Set.Icc a b)) (Set.Icc a b)
fpp_bound : ∀ (x : ℝ), |iteratedDerivWithin 2 f (Set.Icc a b) x| ≤ ζ
g : ℝ → ℝ := ⋯
dg : ℝ → ℝ := ⋯
ddg : ℝ → ℝ := ⋯
y : ℝ
hy : y ∈ Set.Icc a b
this : Fact (y ∈ Set.Icc a b)
⊢ ContinuousWithinAt f (Set.Icc a b) y

Tactic:
·

Goal state after:
case hb
f : ℝ → ℝ
ζ a b : ℝ
a_lt_b : a < b
h_df : DifferentiableOn ℝ f (Set.Icc a b)
h_ddf : DifferentiableOn ℝ (_root_.derivWithin f (Set.Icc a b)) (Set.Icc a b)
fpp_bound : ∀ (x : ℝ), |iteratedDerivWithin 2 f (Set.Icc a b) x| ≤ ζ
g : ℝ → ℝ := ⋯
dg : ℝ → ℝ := ⋯
ddg : ℝ → ℝ := ⋯
y : ℝ
hy : y ∈ Set.Icc a b
this : Fact (y ∈ Set.Icc a b)
⊢ ContinuousWithinAt f (Set.Icc a b) y
Module: Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Basic

Goal state before:
pp: case «0»
X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : T2Space X
inst✝ : LocallyCompactSpace X
s₀ s₁ t : Set X
s₀_compact : IsCompact s₀
s₁_compact : IsCompact s₁
t_compact : IsCompact t
disj : Disjoint s₀ s₁
hst : s₀ ∪ s₁ ⊆ t
so : Fin 2 → Set X := fun j ↦ if j = 0 then s₀ᶜ else s₁ᶜ
hso : so = fun j ↦ if j = 0 then s₀ᶜ else s₁ᶜ
⊢ IsOpen (so ((fun i ↦ i) ⟨0, ⋯⟩))

pp: case «1»
X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : T2Space X
inst✝ : LocallyCompactSpace X
s₀ s₁ t : Set X
s₀_compact : IsCompact s₀
s₁_compact : IsCompact s₁
t_compact : IsCompact t
disj : Disjoint s₀ s₁
hst : s₀ ∪ s₁ ⊆ t
so : Fin 2 → Set X := fun j ↦ if j = 0 then s₀ᶜ else s₁ᶜ
hso : so = fun j ↦ if j = 0 then s₀ᶜ else s₁ᶜ
⊢ IsOpen (so ((fun i ↦ i) ⟨1, ⋯⟩))

Tactic:
·

Goal state after:
case «0»
X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : T2Space X
inst✝ : LocallyCompactSpace X
s₀ s₁ t : Set X
s₀_compact : IsCompact s₀
s₁_compact : IsCompact s₁
t_compact : IsCompact t
disj : Disjoint s₀ s₁
hst : s₀ ∪ s₁ ⊆ t
so : Fin 2 → Set X := fun j ↦ if j = 0 then s₀ᶜ else s₁ᶜ
hso : so = fun j ↦ if j = 0 then s₀ᶜ else s₁ᶜ
⊢ IsOpen (so ((fun i ↦ i) ⟨0, ⋯⟩))
Module: Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Basic

Goal state before:
pp: case «1»
X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : T2Space X
inst✝ : LocallyCompactSpace X
s₀ s₁ t : Set X
s₀_compact : IsCompact s₀
s₁_compact : IsCompact s₁
t_compact : IsCompact t
disj : Disjoint s₀ s₁
hst : s₀ ∪ s₁ ⊆ t
so : Fin 2 → Set X := fun j ↦ if j = 0 then s₀ᶜ else s₁ᶜ
hso : so = fun j ↦ if j = 0 then s₀ᶜ else s₁ᶜ
⊢ IsOpen (so ((fun i ↦ i) ⟨1, ⋯⟩))

Tactic:
·

Goal state after:
case «1»
X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : T2Space X
inst✝ : LocallyCompactSpace X
s₀ s₁ t : Set X
s₀_compact : IsCompact s₀
s₁_compact : IsCompact s₁
t_compact : IsCompact t
disj : Disjoint s₀ s₁
hst : s₀ ∪ s₁ ⊆ t
so : Fin 2 → Set X := fun j ↦ if j = 0 then s₀ᶜ else s₁ᶜ
hso : so = fun j ↦ if j = 0 then s₀ᶜ else s₁ᶜ
⊢ IsOpen (so ((fun i ↦ i) ⟨1, ⋯⟩))
Module: Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Basic

Goal state before:
pp: case left
X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : T2Space X
inst✝ : LocallyCompactSpace X
s₀ s₁ t : Set X
s₀_compact : IsCompact s₀
s₁_compact : IsCompact s₁
t_compact : IsCompact t
disj : s₀ᶜ ∪ s₁ᶜ = univ
hst : s₀ ∪ s₁ ⊆ t
so : Fin 2 → Set X := ⋯
hso : so = fun j ↦ if j = 0 then s₀ᶜ else s₁ᶜ
soopen : ∀ (j : Fin 2), IsOpen (so j)
x : X
hx : x ∈ t
h : x ∈ s₀ᶜ ∨ x ∈ s₁ᶜ
⊢ x ∈ s₀ᶜ → ∃ i, x ∈ if i = 0 then s₀ᶜ else s₁ᶜ

pp: case right
X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : T2Space X
inst✝ : LocallyCompactSpace X
s₀ s₁ t : Set X
s₀_compact : IsCompact s₀
s₁_compact : IsCompact s₁
t_compact : IsCompact t
disj : s₀ᶜ ∪ s₁ᶜ = univ
hst : s₀ ∪ s₁ ⊆ t
so : Fin 2 → Set X := ⋯
hso : so = fun j ↦ if j = 0 then s₀ᶜ else s₁ᶜ
soopen : ∀ (j : Fin 2), IsOpen (so j)
x : X
hx : x ∈ t
h : x ∈ s₀ᶜ ∨ x ∈ s₁ᶜ
⊢ x ∈ s₁ᶜ → ∃ i, x ∈ if i = 0 then s₀ᶜ else s₁ᶜ

Tactic:
·

Goal state after:
case left
X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : T2Space X
inst✝ : LocallyCompactSpace X
s₀ s₁ t : Set X
s₀_compact : IsCompact s₀
s₁_compact : IsCompact s₁
t_compact : IsCompact t
disj : s₀ᶜ ∪ s₁ᶜ = univ
hst : s₀ ∪ s₁ ⊆ t
so : Fin 2 → Set X := ⋯
hso : so = fun j ↦ if j = 0 then s₀ᶜ else s₁ᶜ
soopen : ∀ (j : Fin 2), IsOpen (so j)
x : X
hx : x ∈ t
h : x ∈ s₀ᶜ ∨ x ∈ s₁ᶜ
⊢ x ∈ s₀ᶜ → ∃ i, x ∈ if i = 0 then s₀ᶜ else s₁ᶜ
Module: Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Basic

Goal state before:
pp: case right
X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : T2Space X
inst✝ : LocallyCompactSpace X
s₀ s₁ t : Set X
s₀_compact : IsCompact s₀
s₁_compact : IsCompact s₁
t_compact : IsCompact t
disj : s₀ᶜ ∪ s₁ᶜ = univ
hst : s₀ ∪ s₁ ⊆ t
so : Fin 2 → Set X := ⋯
hso : so = fun j ↦ if j = 0 then s₀ᶜ else s₁ᶜ
soopen : ∀ (j : Fin 2), IsOpen (so j)
x : X
hx : x ∈ t
h : x ∈ s₀ᶜ ∨ x ∈ s₁ᶜ
⊢ x ∈ s₁ᶜ → ∃ i, x ∈ if i = 0 then s₀ᶜ else s₁ᶜ

Tactic:
·

Goal state after:
case right
X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : T2Space X
inst✝ : LocallyCompactSpace X
s₀ s₁ t : Set X
s₀_compact : IsCompact s₀
s₁_compact : IsCompact s₁
t_compact : IsCompact t
disj : s₀ᶜ ∪ s₁ᶜ = univ
hst : s₀ ∪ s₁ ⊆ t
so : Fin 2 → Set X := ⋯
hso : so = fun j ↦ if j = 0 then s₀ᶜ else s₁ᶜ
soopen : ∀ (j : Fin 2), IsOpen (so j)
x : X
hx : x ∈ t
h : x ∈ s₀ᶜ ∨ x ∈ s₁ᶜ
⊢ x ∈ s₁ᶜ → ∃ i, x ∈ if i = 0 then s₀ᶜ else s₁ᶜ
Module: Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Basic

Goal state before:
pp: case h.refine_1
X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : T2Space X
inst✝ : LocallyCompactSpace X
s₀ s₁ t : Set X
s₀_compact : IsCompact s₀
s₁_compact : IsCompact s₁
t_compact : IsCompact t
disj : Disjoint s₀ s₁
hst : s₀ ∪ s₁ ⊆ t
so : Fin 2 → Set X := fun j ↦ if j = 0 then s₀ᶜ else s₁ᶜ
hso : so = fun j ↦ if j = 0 then s₀ᶜ else s₁ᶜ
soopen : ∀ (j : Fin 2), IsOpen (so j)
hsot : t ⊆ ⋃ j, so j
f : Fin 2 → C(X, ℝ)
f_supp_in_so : ∀ (i : Fin 2), tsupport ⇑(f i) ⊆ so i
sum_f_one_on_t : EqOn (∑ i, ⇑(f i)) 1 t
f_in_icc : ∀ (i : Fin 2) (x : X), (f i) x ∈ Icc 0 1
f_hcs : ∀ (i : Fin 2), HasCompactSupport ⇑(f i)
sum_one_x : ∀ x ∈ t, (f 0) x + (f 1) x = 1
⊢ EqOn (⇑{ toContinuousMap := f 1, hasCompactSupport' := ⋯ }.nnrealPart) 1 s₀

pp: case h.refine_2
X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : T2Space X
inst✝ : LocallyCompactSpace X
s₀ s₁ t : Set X
s₀_compact : IsCompact s₀
s₁_compact : IsCompact s₁
t_compact : IsCompact t
disj : Disjoint s₀ s₁
hst : s₀ ∪ s₁ ⊆ t
so : Fin 2 → Set X := fun j ↦ if j = 0 then s₀ᶜ else s₁ᶜ
hso : so = fun j ↦ if j = 0 then s₀ᶜ else s₁ᶜ
soopen : ∀ (j : Fin 2), IsOpen (so j)
hsot : t ⊆ ⋃ j, so j
f : Fin 2 → C(X, ℝ)
f_supp_in_so : ∀ (i : Fin 2), tsupport ⇑(f i) ⊆ so i
sum_f_one_on_t : EqOn (∑ i, ⇑(f i)) 1 t
f_in_icc : ∀ (i : Fin 2) (x : X), (f i) x ∈ Icc 0 1
f_hcs : ∀ (i : Fin 2), HasCompactSupport ⇑(f i)
sum_one_x : ∀ x ∈ t, (f 0) x + (f 1) x = 1
⊢ EqOn (⇑{ toContinuousMap := f 0, hasCompactSupport' := ⋯ }.nnrealPart) 1 s₁

pp: case h.refine_3
X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : T2Space X
inst✝ : LocallyCompactSpace X
s₀ s₁ t : Set X
s₀_compact : IsCompact s₀
s₁_compact : IsCompact s₁
t_compact : IsCompact t
disj : Disjoint s₀ s₁
hst : s₀ ∪ s₁ ⊆ t
so : Fin 2 → Set X := fun j ↦ if j = 0 then s₀ᶜ else s₁ᶜ
hso : so = fun j ↦ if j = 0 then s₀ᶜ else s₁ᶜ
soopen : ∀ (j : Fin 2), IsOpen (so j)
hsot : t ⊆ ⋃ j, so j
f : Fin 2 → C(X, ℝ)
f_supp_in_so : ∀ (i : Fin 2), tsupport ⇑(f i) ⊆ so i
sum_f_one_on_t : EqOn (∑ i, ⇑(f i)) 1 t
f_in_icc : ∀ (i : Fin 2) (x : X), (f i) x ∈ Icc 0 1
f_hcs : ∀ (i : Fin 2), HasCompactSupport ⇑(f i)
sum_one_x : ∀ x ∈ t, (f 0) x + (f 1) x = 1
⊢ EqOn
    (⇑{ toContinuousMap := f 1, hasCompactSupport' := ⋯ }.nnrealPart +
      ⇑{ toContinuousMap := f 0, hasCompactSupport' := ⋯ }.nnrealPart)
    1 t

Tactic:
·

Goal state after:
case h.refine_1
X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : T2Space X
inst✝ : LocallyCompactSpace X
s₀ s₁ t : Set X
s₀_compact : IsCompact s₀
s₁_compact : IsCompact s₁
t_compact : IsCompact t
disj : Disjoint s₀ s₁
hst : s₀ ∪ s₁ ⊆ t
so : Fin 2 → Set X := fun j ↦ if j = 0 then s₀ᶜ else s₁ᶜ
hso : so = fun j ↦ if j = 0 then s₀ᶜ else s₁ᶜ
soopen : ∀ (j : Fin 2), IsOpen (so j)
hsot : t ⊆ ⋃ j, so j
f : Fin 2 → C(X, ℝ)
f_supp_in_so : ∀ (i : Fin 2), tsupport ⇑(f i) ⊆ so i
sum_f_one_on_t : EqOn (∑ i, ⇑(f i)) 1 t
f_in_icc : ∀ (i : Fin 2) (x : X), (f i) x ∈ Icc 0 1
f_hcs : ∀ (i : Fin 2), HasCompactSupport ⇑(f i)
sum_one_x : ∀ x ∈ t, (f 0) x + (f 1) x = 1
⊢ EqOn (⇑{ toContinuousMap := f 1, hasCompactSupport' := ⋯ }.nnrealPart) 1 s₀
Module: Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Basic

Goal state before:
pp: case h.refine_2
X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : T2Space X
inst✝ : LocallyCompactSpace X
s₀ s₁ t : Set X
s₀_compact : IsCompact s₀
s₁_compact : IsCompact s₁
t_compact : IsCompact t
disj : Disjoint s₀ s₁
hst : s₀ ∪ s₁ ⊆ t
so : Fin 2 → Set X := fun j ↦ if j = 0 then s₀ᶜ else s₁ᶜ
hso : so = fun j ↦ if j = 0 then s₀ᶜ else s₁ᶜ
soopen : ∀ (j : Fin 2), IsOpen (so j)
hsot : t ⊆ ⋃ j, so j
f : Fin 2 → C(X, ℝ)
f_supp_in_so : ∀ (i : Fin 2), tsupport ⇑(f i) ⊆ so i
sum_f_one_on_t : EqOn (∑ i, ⇑(f i)) 1 t
f_in_icc : ∀ (i : Fin 2) (x : X), (f i) x ∈ Icc 0 1
f_hcs : ∀ (i : Fin 2), HasCompactSupport ⇑(f i)
sum_one_x : ∀ x ∈ t, (f 0) x + (f 1) x = 1
⊢ EqOn (⇑{ toContinuousMap := f 0, hasCompactSupport' := ⋯ }.nnrealPart) 1 s₁

pp: case h.refine_3
X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : T2Space X
inst✝ : LocallyCompactSpace X
s₀ s₁ t : Set X
s₀_compact : IsCompact s₀
s₁_compact : IsCompact s₁
t_compact : IsCompact t
disj : Disjoint s₀ s₁
hst : s₀ ∪ s₁ ⊆ t
so : Fin 2 → Set X := fun j ↦ if j = 0 then s₀ᶜ else s₁ᶜ
hso : so = fun j ↦ if j = 0 then s₀ᶜ else s₁ᶜ
soopen : ∀ (j : Fin 2), IsOpen (so j)
hsot : t ⊆ ⋃ j, so j
f : Fin 2 → C(X, ℝ)
f_supp_in_so : ∀ (i : Fin 2), tsupport ⇑(f i) ⊆ so i
sum_f_one_on_t : EqOn (∑ i, ⇑(f i)) 1 t
f_in_icc : ∀ (i : Fin 2) (x : X), (f i) x ∈ Icc 0 1
f_hcs : ∀ (i : Fin 2), HasCompactSupport ⇑(f i)
sum_one_x : ∀ x ∈ t, (f 0) x + (f 1) x = 1
⊢ EqOn
    (⇑{ toContinuousMap := f 1, hasCompactSupport' := ⋯ }.nnrealPart +
      ⇑{ toContinuousMap := f 0, hasCompactSupport' := ⋯ }.nnrealPart)
    1 t

Tactic:
·

Goal state after:
case h.refine_2
X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : T2Space X
inst✝ : LocallyCompactSpace X
s₀ s₁ t : Set X
s₀_compact : IsCompact s₀
s₁_compact : IsCompact s₁
t_compact : IsCompact t
disj : Disjoint s₀ s₁
hst : s₀ ∪ s₁ ⊆ t
so : Fin 2 → Set X := fun j ↦ if j = 0 then s₀ᶜ else s₁ᶜ
hso : so = fun j ↦ if j = 0 then s₀ᶜ else s₁ᶜ
soopen : ∀ (j : Fin 2), IsOpen (so j)
hsot : t ⊆ ⋃ j, so j
f : Fin 2 → C(X, ℝ)
f_supp_in_so : ∀ (i : Fin 2), tsupport ⇑(f i) ⊆ so i
sum_f_one_on_t : EqOn (∑ i, ⇑(f i)) 1 t
f_in_icc : ∀ (i : Fin 2) (x : X), (f i) x ∈ Icc 0 1
f_hcs : ∀ (i : Fin 2), HasCompactSupport ⇑(f i)
sum_one_x : ∀ x ∈ t, (f 0) x + (f 1) x = 1
⊢ EqOn (⇑{ toContinuousMap := f 0, hasCompactSupport' := ⋯ }.nnrealPart) 1 s₁
Module: Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Basic

Goal state before:
pp: case h.refine_3
X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : T2Space X
inst✝ : LocallyCompactSpace X
s₀ s₁ t : Set X
s₀_compact : IsCompact s₀
s₁_compact : IsCompact s₁
t_compact : IsCompact t
disj : Disjoint s₀ s₁
hst : s₀ ∪ s₁ ⊆ t
so : Fin 2 → Set X := fun j ↦ if j = 0 then s₀ᶜ else s₁ᶜ
hso : so = fun j ↦ if j = 0 then s₀ᶜ else s₁ᶜ
soopen : ∀ (j : Fin 2), IsOpen (so j)
hsot : t ⊆ ⋃ j, so j
f : Fin 2 → C(X, ℝ)
f_supp_in_so : ∀ (i : Fin 2), tsupport ⇑(f i) ⊆ so i
sum_f_one_on_t : EqOn (∑ i, ⇑(f i)) 1 t
f_in_icc : ∀ (i : Fin 2) (x : X), (f i) x ∈ Icc 0 1
f_hcs : ∀ (i : Fin 2), HasCompactSupport ⇑(f i)
sum_one_x : ∀ x ∈ t, (f 0) x + (f 1) x = 1
⊢ EqOn
    (⇑{ toContinuousMap := f 1, hasCompactSupport' := ⋯ }.nnrealPart +
      ⇑{ toContinuousMap := f 0, hasCompactSupport' := ⋯ }.nnrealPart)
    1 t

Tactic:
·

Goal state after:
case h.refine_3
X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : T2Space X
inst✝ : LocallyCompactSpace X
s₀ s₁ t : Set X
s₀_compact : IsCompact s₀
s₁_compact : IsCompact s₁
t_compact : IsCompact t
disj : Disjoint s₀ s₁
hst : s₀ ∪ s₁ ⊆ t
so : Fin 2 → Set X := fun j ↦ if j = 0 then s₀ᶜ else s₁ᶜ
hso : so = fun j ↦ if j = 0 then s₀ᶜ else s₁ᶜ
soopen : ∀ (j : Fin 2), IsOpen (so j)
hsot : t ⊆ ⋃ j, so j
f : Fin 2 → C(X, ℝ)
f_supp_in_so : ∀ (i : Fin 2), tsupport ⇑(f i) ⊆ so i
sum_f_one_on_t : EqOn (∑ i, ⇑(f i)) 1 t
f_in_icc : ∀ (i : Fin 2) (x : X), (f i) x ∈ Icc 0 1
f_hcs : ∀ (i : Fin 2), HasCompactSupport ⇑(f i)
sum_one_x : ∀ x ∈ t, (f 0) x + (f 1) x = 1
⊢ EqOn
    (⇑{ toContinuousMap := f 1, hasCompactSupport' := ⋯ }.nnrealPart +
      ⇑{ toContinuousMap := f 0, hasCompactSupport' := ⋯ }.nnrealPart)
    1 t
Module: Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Basic

Goal state before:
pp: case pos
X : Type u_1
inst✝² : TopologicalSpace X
Λ : (X →C_c ℝ≥0) →ₗ[ℝ≥0] ℝ≥0
inst✝¹ : T2Space X
inst✝ : LocallyCompactSpace X
K₁ K₂ : Compacts X
disj : Disjoint ↑K₁ ↑K₂
b : ℝ≥0
f : X →C_c ℝ≥0
hf : f ∈ {f | ∀ x ∈ K₁ ⊔ K₂, 1 ≤ f x}
Λf_eq_b : Λ f = b
hsuppf : ∀ x ∈ K₁ ⊔ K₂, x ∈ support ⇑f
hsubsuppf : ↑K₁ ∪ ↑K₂ ⊆ tsupport ⇑f
g₁ g₂ : X →C_c ℝ≥0
hg₁ : EqOn (⇑g₁) 1 K₁.carrier
hg₂ : EqOn (⇑g₂) 1 K₂.carrier
sum_g : EqOn (⇑(g₁ + g₂)) 1 (tsupport f.toFun)
x : X
h : f x = 0
⊢ ↑(f x) = ↑((⇑g₁ + ⇑g₂) x) * ↑(f x)

pp: case neg
X : Type u_1
inst✝² : TopologicalSpace X
Λ : (X →C_c ℝ≥0) →ₗ[ℝ≥0] ℝ≥0
inst✝¹ : T2Space X
inst✝ : LocallyCompactSpace X
K₁ K₂ : Compacts X
disj : Disjoint ↑K₁ ↑K₂
b : ℝ≥0
f : X →C_c ℝ≥0
hf : f ∈ {f | ∀ x ∈ K₁ ⊔ K₂, 1 ≤ f x}
Λf_eq_b : Λ f = b
hsuppf : ∀ x ∈ K₁ ⊔ K₂, x ∈ support ⇑f
hsubsuppf : ↑K₁ ∪ ↑K₂ ⊆ tsupport ⇑f
g₁ g₂ : X →C_c ℝ≥0
hg₁ : EqOn (⇑g₁) 1 K₁.carrier
hg₂ : EqOn (⇑g₂) 1 K₂.carrier
sum_g : EqOn (⇑(g₁ + g₂)) 1 (tsupport f.toFun)
x : X
h : ¬f x = 0
⊢ ↑(f x) = ↑((⇑g₁ + ⇑g₂) x) * ↑(f x)

Tactic:
·

Goal state after:
case pos
X : Type u_1
inst✝² : TopologicalSpace X
Λ : (X →C_c ℝ≥0) →ₗ[ℝ≥0] ℝ≥0
inst✝¹ : T2Space X
inst✝ : LocallyCompactSpace X
K₁ K₂ : Compacts X
disj : Disjoint ↑K₁ ↑K₂
b : ℝ≥0
f : X →C_c ℝ≥0
hf : f ∈ {f | ∀ x ∈ K₁ ⊔ K₂, 1 ≤ f x}
Λf_eq_b : Λ f = b
hsuppf : ∀ x ∈ K₁ ⊔ K₂, x ∈ support ⇑f
hsubsuppf : ↑K₁ ∪ ↑K₂ ⊆ tsupport ⇑f
g₁ g₂ : X →C_c ℝ≥0
hg₁ : EqOn (⇑g₁) 1 K₁.carrier
hg₂ : EqOn (⇑g₂) 1 K₂.carrier
sum_g : EqOn (⇑(g₁ + g₂)) 1 (tsupport f.toFun)
x : X
h : f x = 0
⊢ ↑(f x) = ↑((⇑g₁ + ⇑g₂) x) * ↑(f x)
Module: Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Basic

Goal state before:
pp: case neg
X : Type u_1
inst✝² : TopologicalSpace X
Λ : (X →C_c ℝ≥0) →ₗ[ℝ≥0] ℝ≥0
inst✝¹ : T2Space X
inst✝ : LocallyCompactSpace X
K₁ K₂ : Compacts X
disj : Disjoint ↑K₁ ↑K₂
b : ℝ≥0
f : X →C_c ℝ≥0
hf : f ∈ {f | ∀ x ∈ K₁ ⊔ K₂, 1 ≤ f x}
Λf_eq_b : Λ f = b
hsuppf : ∀ x ∈ K₁ ⊔ K₂, x ∈ support ⇑f
hsubsuppf : ↑K₁ ∪ ↑K₂ ⊆ tsupport ⇑f
g₁ g₂ : X →C_c ℝ≥0
hg₁ : EqOn (⇑g₁) 1 K₁.carrier
hg₂ : EqOn (⇑g₂) 1 K₂.carrier
sum_g : EqOn (⇑(g₁ + g₂)) 1 (tsupport f.toFun)
x : X
h : ¬f x = 0
⊢ ↑(f x) = ↑((⇑g₁ + ⇑g₂) x) * ↑(f x)

Tactic:
·

Goal state after:
case neg
X : Type u_1
inst✝² : TopologicalSpace X
Λ : (X →C_c ℝ≥0) →ₗ[ℝ≥0] ℝ≥0
inst✝¹ : T2Space X
inst✝ : LocallyCompactSpace X
K₁ K₂ : Compacts X
disj : Disjoint ↑K₁ ↑K₂
b : ℝ≥0
f : X →C_c ℝ≥0
hf : f ∈ {f | ∀ x ∈ K₁ ⊔ K₂, 1 ≤ f x}
Λf_eq_b : Λ f = b
hsuppf : ∀ x ∈ K₁ ⊔ K₂, x ∈ support ⇑f
hsubsuppf : ↑K₁ ∪ ↑K₂ ⊆ tsupport ⇑f
g₁ g₂ : X →C_c ℝ≥0
hg₁ : EqOn (⇑g₁) 1 K₁.carrier
hg₂ : EqOn (⇑g₂) 1 K₂.carrier
sum_g : EqOn (⇑(g₁ + g₂)) 1 (tsupport f.toFun)
x : X
h : ¬f x = 0
⊢ ↑(f x) = ↑((⇑g₁ + ⇑g₂) x) * ↑(f x)
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.TrapezoidalRule

Goal state before:
pp: case pos
f : ℝ → ℝ
ζ a b : ℝ
a_lt_b : a < b
h_df : DifferentiableOn ℝ f (Set.Icc a b)
h_ddf : DifferentiableOn ℝ (_root_.derivWithin f (Set.Icc a b)) (Set.Icc a b)
fpp_bound : ∀ (x : ℝ), |iteratedDerivWithin 2 f (Set.Icc a b) x| ≤ ζ
N : ℕ
N_nonzero : 0 < N
h : ℝ := (b - a) / ↑N
ak : ℕ → ℝ := fun k ↦ a + ↑k * h
h0 : ∀ (k : ℕ), ak (k + 1) - ak k = h
hab : 0 < b - a
hpos : 0 < h
hb : b = a + ↑N * h
k : ℕ
hk : k < N
h1 : a ≤ ak k
h2 : ak (k + 1) ≤ b
h3 : Set.Icc (ak k) (ak (k + 1)) ⊆ Set.Icc a b
h4 : ak k < ak (k + 1)
h5 :
  EqOn (_root_.derivWithin f (Set.Icc a b)) (_root_.derivWithin f (Set.Icc (ak k) (ak (k + 1))))
    (Set.Icc (ak k) (ak (k + 1)))
h6 :
  EqOn (iteratedDerivWithin 2 f (Set.Icc a b)) (iteratedDerivWithin 2 f (Set.Icc (ak k) (ak (k + 1))))
    (Set.Icc (ak k) (ak (k + 1)))
x : ℝ
hx : x ∈ Set.Icc (ak k) (ak (k + 1))
⊢ |iteratedDerivWithin 2 f (Set.Icc (ak k) (ak (k + 1))) x| ≤ ζ

pp: case neg
f : ℝ → ℝ
ζ a b : ℝ
a_lt_b : a < b
h_df : DifferentiableOn ℝ f (Set.Icc a b)
h_ddf : DifferentiableOn ℝ (_root_.derivWithin f (Set.Icc a b)) (Set.Icc a b)
fpp_bound : ∀ (x : ℝ), |iteratedDerivWithin 2 f (Set.Icc a b) x| ≤ ζ
N : ℕ
N_nonzero : 0 < N
h : ℝ := (b - a) / ↑N
ak : ℕ → ℝ := fun k ↦ a + ↑k * h
h0 : ∀ (k : ℕ), ak (k + 1) - ak k = h
hab : 0 < b - a
hpos : 0 < h
hb : b = a + ↑N * h
k : ℕ
hk : k < N
h1 : a ≤ ak k
h2 : ak (k + 1) ≤ b
h3 : Set.Icc (ak k) (ak (k + 1)) ⊆ Set.Icc a b
h4 : ak k < ak (k + 1)
h5 :
  EqOn (_root_.derivWithin f (Set.Icc a b)) (_root_.derivWithin f (Set.Icc (ak k) (ak (k + 1))))
    (Set.Icc (ak k) (ak (k + 1)))
h6 :
  EqOn (iteratedDerivWithin 2 f (Set.Icc a b)) (iteratedDerivWithin 2 f (Set.Icc (ak k) (ak (k + 1))))
    (Set.Icc (ak k) (ak (k + 1)))
x : ℝ
hx : x ∉ Set.Icc (ak k) (ak (k + 1))
⊢ |iteratedDerivWithin 2 f (Set.Icc (ak k) (ak (k + 1))) x| ≤ ζ

Tactic:
·

Goal state after:
case pos
f : ℝ → ℝ
ζ a b : ℝ
a_lt_b : a < b
h_df : DifferentiableOn ℝ f (Set.Icc a b)
h_ddf : DifferentiableOn ℝ (_root_.derivWithin f (Set.Icc a b)) (Set.Icc a b)
fpp_bound : ∀ (x : ℝ), |iteratedDerivWithin 2 f (Set.Icc a b) x| ≤ ζ
N : ℕ
N_nonzero : 0 < N
h : ℝ := (b - a) / ↑N
ak : ℕ → ℝ := fun k ↦ a + ↑k * h
h0 : ∀ (k : ℕ), ak (k + 1) - ak k = h
hab : 0 < b - a
hpos : 0 < h
hb : b = a + ↑N * h
k : ℕ
hk : k < N
h1 : a ≤ ak k
h2 : ak (k + 1) ≤ b
h3 : Set.Icc (ak k) (ak (k + 1)) ⊆ Set.Icc a b
h4 : ak k < ak (k + 1)
h5 :
  EqOn (_root_.derivWithin f (Set.Icc a b)) (_root_.derivWithin f (Set.Icc (ak k) (ak (k + 1))))
    (Set.Icc (ak k) (ak (k + 1)))
h6 :
  EqOn (iteratedDerivWithin 2 f (Set.Icc a b)) (iteratedDerivWithin 2 f (Set.Icc (ak k) (ak (k + 1))))
    (Set.Icc (ak k) (ak (k + 1)))
x : ℝ
hx : x ∈ Set.Icc (ak k) (ak (k + 1))
⊢ |iteratedDerivWithin 2 f (Set.Icc (ak k) (ak (k + 1))) x| ≤ ζ
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.TrapezoidalRule

Goal state before:
pp: case neg
f : ℝ → ℝ
ζ a b : ℝ
a_lt_b : a < b
h_df : DifferentiableOn ℝ f (Set.Icc a b)
h_ddf : DifferentiableOn ℝ (_root_.derivWithin f (Set.Icc a b)) (Set.Icc a b)
fpp_bound : ∀ (x : ℝ), |iteratedDerivWithin 2 f (Set.Icc a b) x| ≤ ζ
N : ℕ
N_nonzero : 0 < N
h : ℝ := (b - a) / ↑N
ak : ℕ → ℝ := fun k ↦ a + ↑k * h
h0 : ∀ (k : ℕ), ak (k + 1) - ak k = h
hab : 0 < b - a
hpos : 0 < h
hb : b = a + ↑N * h
k : ℕ
hk : k < N
h1 : a ≤ ak k
h2 : ak (k + 1) ≤ b
h3 : Set.Icc (ak k) (ak (k + 1)) ⊆ Set.Icc a b
h4 : ak k < ak (k + 1)
h5 :
  EqOn (_root_.derivWithin f (Set.Icc a b)) (_root_.derivWithin f (Set.Icc (ak k) (ak (k + 1))))
    (Set.Icc (ak k) (ak (k + 1)))
h6 :
  EqOn (iteratedDerivWithin 2 f (Set.Icc a b)) (iteratedDerivWithin 2 f (Set.Icc (ak k) (ak (k + 1))))
    (Set.Icc (ak k) (ak (k + 1)))
x : ℝ
hx : x ∉ Set.Icc (ak k) (ak (k + 1))
⊢ |iteratedDerivWithin 2 f (Set.Icc (ak k) (ak (k + 1))) x| ≤ ζ

Tactic:
·

Goal state after:
case neg
f : ℝ → ℝ
ζ a b : ℝ
a_lt_b : a < b
h_df : DifferentiableOn ℝ f (Set.Icc a b)
h_ddf : DifferentiableOn ℝ (_root_.derivWithin f (Set.Icc a b)) (Set.Icc a b)
fpp_bound : ∀ (x : ℝ), |iteratedDerivWithin 2 f (Set.Icc a b) x| ≤ ζ
N : ℕ
N_nonzero : 0 < N
h : ℝ := (b - a) / ↑N
ak : ℕ → ℝ := fun k ↦ a + ↑k * h
h0 : ∀ (k : ℕ), ak (k + 1) - ak k = h
hab : 0 < b - a
hpos : 0 < h
hb : b = a + ↑N * h
k : ℕ
hk : k < N
h1 : a ≤ ak k
h2 : ak (k + 1) ≤ b
h3 : Set.Icc (ak k) (ak (k + 1)) ⊆ Set.Icc a b
h4 : ak k < ak (k + 1)
h5 :
  EqOn (_root_.derivWithin f (Set.Icc a b)) (_root_.derivWithin f (Set.Icc (ak k) (ak (k + 1))))
    (Set.Icc (ak k) (ak (k + 1)))
h6 :
  EqOn (iteratedDerivWithin 2 f (Set.Icc a b)) (iteratedDerivWithin 2 f (Set.Icc (ak k) (ak (k + 1))))
    (Set.Icc (ak k) (ak (k + 1)))
x : ℝ
hx : x ∉ Set.Icc (ak k) (ak (k + 1))
⊢ |iteratedDerivWithin 2 f (Set.Icc (ak k) (ak (k + 1))) x| ≤ ζ
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.TrapezoidalRule

Goal state before:
pp: case refine_1
f : ℝ → ℝ
ζ a b : ℝ
a_lt_b : a < b
h_df : DifferentiableOn ℝ f (Set.Icc a b)
h_ddf : DifferentiableOn ℝ (_root_.derivWithin f (Set.Icc a b)) (Set.Icc a b)
fpp_bound : ∀ (x : ℝ), |iteratedDerivWithin 2 f (Set.Icc a b) x| ≤ ζ
N : ℕ
N_nonzero : 0 < N
h : ℝ := (b - a) / ↑N
ak : ℕ → ℝ := fun k ↦ a + ↑k * h
h0 : ∀ (k : ℕ), ak (k + 1) - ak k = h
hab : 0 < b - a
hpos : 0 < h
hb : b = a + ↑N * h
k : ℕ
hk : k < N
h1 : a ≤ ak k
h2 : ak (k + 1) ≤ b
h3 : Set.Icc (ak k) (ak (k + 1)) ⊆ Set.Icc a b
h4 : ak k < ak (k + 1)
h5 :
  EqOn (_root_.derivWithin f (Set.Icc a b)) (_root_.derivWithin f (Set.Icc (ak k) (ak (k + 1))))
    (Set.Icc (ak k) (ak (k + 1)))
h6 :
  EqOn (iteratedDerivWithin 2 f (Set.Icc a b)) (iteratedDerivWithin 2 f (Set.Icc (ak k) (ak (k + 1))))
    (Set.Icc (ak k) (ak (k + 1)))
h7 : ∀ (x : ℝ), |iteratedDerivWithin 2 f (Set.Icc (ak k) (ak (k + 1))) x| ≤ ζ
⊢ DifferentiableOn ℝ (_root_.derivWithin f (Set.Icc (ak k) (ak (k + 1)))) (Set.Icc (ak k) (ak (k + 1)))

pp: case refine_2
f : ℝ → ℝ
ζ a b : ℝ
a_lt_b : a < b
h_df : DifferentiableOn ℝ f (Set.Icc a b)
h_ddf : DifferentiableOn ℝ (_root_.derivWithin f (Set.Icc a b)) (Set.Icc a b)
fpp_bound : ∀ (x : ℝ), |iteratedDerivWithin 2 f (Set.Icc a b) x| ≤ ζ
N : ℕ
N_nonzero : 0 < N
h : ℝ := (b - a) / ↑N
ak : ℕ → ℝ := fun k ↦ a + ↑k * h
h0 : ∀ (k : ℕ), ak (k + 1) - ak k = h
hab : 0 < b - a
hpos : 0 < h
hb : b = a + ↑N * h
k : ℕ
hk : k < N
h1 : a ≤ ak k
h2 : ak (k + 1) ≤ b
h3 : Set.Icc (ak k) (ak (k + 1)) ⊆ Set.Icc a b
h4 : ak k < ak (k + 1)
h5 :
  EqOn (_root_.derivWithin f (Set.Icc a b)) (_root_.derivWithin f (Set.Icc (ak k) (ak (k + 1))))
    (Set.Icc (ak k) (ak (k + 1)))
h6 :
  EqOn (iteratedDerivWithin 2 f (Set.Icc a b)) (iteratedDerivWithin 2 f (Set.Icc (ak k) (ak (k + 1))))
    (Set.Icc (ak k) (ak (k + 1)))
h7 : ∀ (x : ℝ), |iteratedDerivWithin 2 f (Set.Icc (ak k) (ak (k + 1))) x| ≤ ζ
⊢ (ak (k + 1) - ak k) ^ 3 * ζ / 12 = ζ / 12 * h ^ 3

Tactic:
·

Goal state after:
case refine_1
f : ℝ → ℝ
ζ a b : ℝ
a_lt_b : a < b
h_df : DifferentiableOn ℝ f (Set.Icc a b)
h_ddf : DifferentiableOn ℝ (_root_.derivWithin f (Set.Icc a b)) (Set.Icc a b)
fpp_bound : ∀ (x : ℝ), |iteratedDerivWithin 2 f (Set.Icc a b) x| ≤ ζ
N : ℕ
N_nonzero : 0 < N
h : ℝ := (b - a) / ↑N
ak : ℕ → ℝ := fun k ↦ a + ↑k * h
h0 : ∀ (k : ℕ), ak (k + 1) - ak k = h
hab : 0 < b - a
hpos : 0 < h
hb : b = a + ↑N * h
k : ℕ
hk : k < N
h1 : a ≤ ak k
h2 : ak (k + 1) ≤ b
h3 : Set.Icc (ak k) (ak (k + 1)) ⊆ Set.Icc a b
h4 : ak k < ak (k + 1)
h5 :
  EqOn (_root_.derivWithin f (Set.Icc a b)) (_root_.derivWithin f (Set.Icc (ak k) (ak (k + 1))))
    (Set.Icc (ak k) (ak (k + 1)))
h6 :
  EqOn (iteratedDerivWithin 2 f (Set.Icc a b)) (iteratedDerivWithin 2 f (Set.Icc (ak k) (ak (k + 1))))
    (Set.Icc (ak k) (ak (k + 1)))
h7 : ∀ (x : ℝ), |iteratedDerivWithin 2 f (Set.Icc (ak k) (ak (k + 1))) x| ≤ ζ
⊢ DifferentiableOn ℝ (_root_.derivWithin f (Set.Icc (ak k) (ak (k + 1)))) (Set.Icc (ak k) (ak (k + 1)))