-- Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean
theorem MemLp.induction [_i : Fact (1 ≤ p)] (hp_ne_top : p ≠ ∞) (P : (α → E) → Prop)
    (h_ind : ∀ (c : E) ⦃s⦄, MeasurableSet s → μ s < ∞ → P (s.indicator fun _ => c))
    (h_add : ∀ ⦃f g : α → E⦄, Disjoint (support f) (support g) → MemLp f p μ → MemLp g p μ →
      P f → P g → P (f + g))
    (h_closed : IsClosed { f : Lp E p μ | P f })
    (h_ae : ∀ ⦃f g⦄, f =ᵐ[μ] g → MemLp f p μ → P f → P g) :
    ∀ ⦃f : α → E⦄, MemLp f p μ → P f

Proof state:
case h.e'_1.h
α : Type u_1
E : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
μ : Measure α
_i : Fact (1 ≤ p)
hp_ne_top : p ≠ ⊤
P : (α → E) → Prop
h_ind : ∀ (c : E) ⦃s : Set α⦄, MeasurableSet s → μ s < ⊤ → P (s.indicator fun x => c)
h_add : ∀ ⦃f g : α → E⦄, Disjoint (support f) (support g) → MemLp f p μ → MemLp g p μ → P f → P g → P (f + g)
h_closed : IsClosed {f | P ↑↑f}
h_ae : ∀ ⦃f g : α → E⦄, f =ᶠ[ae μ] g → MemLp f p μ → P f → P g
s : Set α
hs : MeasurableSet s
h : MemLp (⇑(SimpleFunc.piecewise s hs (SimpleFunc.const α 0) (SimpleFunc.const α 0))) p μ
x✝ : α
⊢ (SimpleFunc.piecewise s hs (SimpleFunc.const α 0) (SimpleFunc.const α 0)) x✝ = ∅.indicator (fun x => 0) x✝

Tactic:
simp [const]

Result:
no goals

Proof state:
α : Type u_1
E : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
μ : Measure α
_i : Fact (1 ≤ p)
hp_ne_top : p ≠ ⊤
P : (α → E) → Prop
h_ind : ∀ (c : E) ⦃s : Set α⦄, MeasurableSet s → μ s < ⊤ → P (s.indicator fun x => c)
h_add : ∀ ⦃f g : α → E⦄, Disjoint (support f) (support g) → MemLp f p μ → MemLp g p μ → P f → P g → P (f + g)
h_closed : IsClosed {f | P ↑↑f}
h_ae : ∀ ⦃f g : α → E⦄, f =ᶠ[ae μ] g → MemLp f p μ → P f → P g
⊢ ∀ (f : α →ₛ E), MemLp (⇑f) p μ → P ⇑f

Tactic:
apply SimpleFunc.induction

Result:
case h_ind
α : Type u_1
E : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
μ : Measure α
_i : Fact (1 ≤ p)
hp_ne_top : p ≠ ⊤
P : (α → E) → Prop
h_ind : ∀ (c : E) ⦃s : Set α⦄, MeasurableSet s → μ s < ⊤ → P (s.indicator fun x => c)
h_add : ∀ ⦃f g : α → E⦄, Disjoint (support f) (support g) → MemLp f p μ → MemLp g p μ → P f → P g → P (f + g)
h_closed : IsClosed {f | P ↑↑f}
h_ae : ∀ ⦃f g : α → E⦄, f =ᶠ[ae μ] g → MemLp f p μ → P f → P g
⊢ ∀ (c : E) {s : Set α} (hs : MeasurableSet s),
    MemLp (⇑(SimpleFunc.piecewise s hs (SimpleFunc.const α c) (SimpleFunc.const α 0))) p μ →
      P ⇑(SimpleFunc.piecewise s hs (SimpleFunc.const α c) (SimpleFunc.const α 0))

case h_add
α : Type u_1
E : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
μ : Measure α
_i : Fact (1 ≤ p)
hp_ne_top : p ≠ ⊤
P : (α → E) → Prop
h_ind : ∀ (c : E) ⦃s : Set α⦄, MeasurableSet s → μ s < ⊤ → P (s.indicator fun x => c)
h_add : ∀ ⦃f g : α → E⦄, Disjoint (support f) (support g) → MemLp f p μ → MemLp g p μ → P f → P g → P (f + g)
h_closed : IsClosed {f | P ↑↑f}
h_ae : ∀ ⦃f g : α → E⦄, f =ᶠ[ae μ] g → MemLp f p μ → P f → P g
⊢ ∀ ⦃f g : α →ₛ E⦄,
    Disjoint (support ⇑f) (support ⇑g) →
      (MemLp (⇑f) p μ → P ⇑f) → (MemLp (⇑g) p μ → P ⇑g) → MemLp (⇑(f + g)) p μ → P ⇑(f + g)

Proof state:
case h.e'_1
α : Type u_1
E : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
μ : Measure α
_i : Fact (1 ≤ p)
hp_ne_top : p ≠ ⊤
P : (α → E) → Prop
h_ind : ∀ (c : E) ⦃s : Set α⦄, MeasurableSet s → μ s < ⊤ → P (s.indicator fun x => c)
h_add : ∀ ⦃f g : α → E⦄, Disjoint (support f) (support g) → MemLp f p μ → MemLp g p μ → P f → P g → P (f + g)
h_closed : IsClosed {f | P ↑↑f}
h_ae : ∀ ⦃f g : α → E⦄, f =ᶠ[ae μ] g → MemLp f p μ → P f → P g
s : Set α
hs : MeasurableSet s
h : MemLp (⇑(SimpleFunc.piecewise s hs (SimpleFunc.const α 0) (SimpleFunc.const α 0))) p μ
⊢ ⇑(SimpleFunc.piecewise s hs (SimpleFunc.const α 0) (SimpleFunc.const α 0)) = ∅.indicator fun x => 0

Tactic:
ext

Result:
case h.e'_1.h
α : Type u_1
E : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
μ : Measure α
_i : Fact (1 ≤ p)
hp_ne_top : p ≠ ⊤
P : (α → E) → Prop
h_ind : ∀ (c : E) ⦃s : Set α⦄, MeasurableSet s → μ s < ⊤ → P (s.indicator fun x => c)
h_add : ∀ ⦃f g : α → E⦄, Disjoint (support f) (support g) → MemLp f p μ → MemLp g p μ → P f → P g → P (f + g)
h_closed : IsClosed {f | P ↑↑f}
h_ae : ∀ ⦃f g : α → E⦄, f =ᶠ[ae μ] g → MemLp f p μ → P f → P g
s : Set α
hs : MeasurableSet s
h : MemLp (⇑(SimpleFunc.piecewise s hs (SimpleFunc.const α 0) (SimpleFunc.const α 0))) p μ
x✝ : α
⊢ (SimpleFunc.piecewise s hs (SimpleFunc.const α 0) (SimpleFunc.const α 0)) x✝ = ∅.indicator (fun x => 0) x✝

Proof state:
α : Type u_1
E : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
μ : Measure α
_i : Fact (1 ≤ p)
hp_ne_top : p ≠ ⊤
P : (α → E) → Prop
h_ind : ∀ (c : E) ⦃s : Set α⦄, MeasurableSet s → μ s < ⊤ → P (s.indicator fun x => c)
h_add : ∀ ⦃f g : α → E⦄, Disjoint (support f) (support g) → MemLp f p μ → MemLp g p μ → P f → P g → P (f + g)
h_closed : IsClosed {f | P ↑↑f}
h_ae : ∀ ⦃f g : α → E⦄, f =ᶠ[ae μ] g → MemLp f p μ → P f → P g
this : ∀ (f : α →ₛ E), MemLp (⇑f) p μ → P ⇑f
f : ↥(Lp.simpleFunc E p μ)
⊢ P ↑↑↑f

Tactic:
exact
  h_ae (Lp.simpleFunc.toSimpleFunc_eq_toFun f) (Lp.simpleFunc.memLp f)
    (this (Lp.simpleFunc.toSimpleFunc f) (Lp.simpleFunc.memLp f))

Result:
no goals

Proof state:
case neg
α : Type u_1
E : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
μ : Measure α
_i : Fact (1 ≤ p)
hp_ne_top : p ≠ ⊤
P : (α → E) → Prop
h_ind : ∀ (c : E) ⦃s : Set α⦄, MeasurableSet s → μ s < ⊤ → P (s.indicator fun x => c)
h_add : ∀ ⦃f g : α → E⦄, Disjoint (support f) (support g) → MemLp f p μ → MemLp g p μ → P f → P g → P (f + g)
h_closed : IsClosed {f | P ↑↑f}
h_ae : ∀ ⦃f g : α → E⦄, f =ᶠ[ae μ] g → MemLp f p μ → P f → P g
c : E
s : Set α
hs : MeasurableSet s
h : MemLp (⇑(SimpleFunc.piecewise s hs (SimpleFunc.const α c) (SimpleFunc.const α 0))) p μ
hc : ¬c = 0
⊢ P ⇑(SimpleFunc.piecewise s hs (SimpleFunc.const α c) (SimpleFunc.const α 0))

Tactic:
have hp_pos : p ≠ 0 := (lt_of_lt_of_le zero_lt_one _i.elim).ne'

Result:
case neg
α : Type u_1
E : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
μ : Measure α
_i : Fact (1 ≤ p)
hp_ne_top : p ≠ ⊤
P : (α → E) → Prop
h_ind : ∀ (c : E) ⦃s : Set α⦄, MeasurableSet s → μ s < ⊤ → P (s.indicator fun x => c)
h_add : ∀ ⦃f g : α → E⦄, Disjoint (support f) (support g) → MemLp f p μ → MemLp g p μ → P f → P g → P (f + g)
h_closed : IsClosed {f | P ↑↑f}
h_ae : ∀ ⦃f g : α → E⦄, f =ᶠ[ae μ] g → MemLp f p μ → P f → P g
c : E
s : Set α
hs : MeasurableSet s
h : MemLp (⇑(SimpleFunc.piecewise s hs (SimpleFunc.const α c) (SimpleFunc.const α 0))) p μ
hc : ¬c = 0
hp_pos : p ≠ 0
⊢ P ⇑(SimpleFunc.piecewise s hs (SimpleFunc.const α c) (SimpleFunc.const α 0))