Theorem: MeasureTheory.SimpleFunc.induction
File: Mathlib/MeasureTheory/Function/SimpleFunc.lean

tactic: generalize h : f.range \ {0} = s
state_before: α✝ : Type ?u.922804
β : Type ?u.922807
γ✝ : Type ?u.922810
δ : Type ?u.922813
α : Type u_1
γ : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : AddMonoid γ
P : (α →ₛ γ) → Prop
h_ind : ∀ (c : γ) {s : Set α} (hs : MeasurableSet s), P (piecewise s hs (const α c) (const α 0))
h_add : ∀ ⦃f g : α →ₛ γ⦄, Disjoint (support ↑f) (support ↑g) → P f → P g → P (f + g)
f : α →ₛ γ
⊢ P f

tactic: rw [← Finset.coe_inj, Finset.coe_sdiff, Finset.coe_singleton, SimpleFunc.coe_range] at h
state_before: α✝ : Type ?u.922804
β : Type ?u.922807
γ✝ : Type ?u.922810
δ : Type ?u.922813
α : Type u_1
γ : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : AddMonoid γ
P : (α →ₛ γ) → Prop
h_ind : ∀ (c : γ) {s : Set α} (hs : MeasurableSet s), P (piecewise s hs (const α c) (const α 0))
h_add : ∀ ⦃f g : α →ₛ γ⦄, Disjoint (support ↑f) (support ↑g) → P f → P g → P (f + g)
f : α →ₛ γ
s : Finset γ
h : SimpleFunc.range f \ {0} = s
⊢ P f

tactic: rw [Finset.coe_empty, diff_eq_empty, range_subset_singleton] at h
state_before: case empty
α✝ : Type ?u.922804
β : Type ?u.922807
γ✝ : Type ?u.922810
δ : Type ?u.922813
α : Type u_1
γ : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : AddMonoid γ
P : (α →ₛ γ) → Prop
h_ind : ∀ (c : γ) {s : Set α} (hs : MeasurableSet s), P (piecewise s hs (const α c) (const α 0))
h_add : ∀ ⦃f g : α →ₛ γ⦄, Disjoint (support ↑f) (support ↑g) → P f → P g → P (f + g)
f : α →ₛ γ
h : range ↑f \ {0} = ↑∅
⊢ P f

tactic: convert h_ind 0 MeasurableSet.univ
state_before: case empty
α✝ : Type ?u.922804
β : Type ?u.922807
γ✝ : Type ?u.922810
δ : Type ?u.922813
α : Type u_1
γ : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : AddMonoid γ
P : (α →ₛ γ) → Prop
h_ind : ∀ (c : γ) {s : Set α} (hs : MeasurableSet s), P (piecewise s hs (const α c) (const α 0))
h_add : ∀ ⦃f g : α →ₛ γ⦄, Disjoint (support ↑f) (support ↑g) → P f → P g → P (f + g)
f : α →ₛ γ
h : ↑f = Function.const α 0
⊢ P f

tactic: ext x
state_before: case h.e'_1
α✝ : Type ?u.922804
β : Type ?u.922807
γ✝ : Type ?u.922810
δ : Type ?u.922813
α : Type u_1
γ : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : AddMonoid γ
P : (α →ₛ γ) → Prop
h_ind : ∀ (c : γ) {s : Set α} (hs : MeasurableSet s), P (piecewise s hs (const α c) (const α 0))
h_add : ∀ ⦃f g : α →ₛ γ⦄, Disjoint (support ↑f) (support ↑g) → P f → P g → P (f + g)
f : α →ₛ γ
h : ↑f = Function.const α 0
⊢ f = piecewise univ (_ : MeasurableSet univ) (const α 0) (const α 0)

tactic: simp [h]
state_before: case h.e'_1.H
α✝ : Type ?u.922804
β : Type ?u.922807
γ✝ : Type ?u.922810
δ : Type ?u.922813
α : Type u_1
γ : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : AddMonoid γ
P : (α →ₛ γ) → Prop
h_ind : ∀ (c : γ) {s : Set α} (hs : MeasurableSet s), P (piecewise s hs (const α c) (const α 0))
h_add : ∀ ⦃f g : α →ₛ γ⦄, Disjoint (support ↑f) (support ↑g) → P f → P g → P (f + g)
f : α →ₛ γ
h : ↑f = Function.const α 0
x : α
⊢ ↑f x = ↑(piecewise univ (_ : MeasurableSet univ) (const α 0) (const α 0)) x

tactic: have mx := f.measurableSet_preimage {x}
state_before: case insert
α✝ : Type ?u.922804
β : Type ?u.922807
γ✝ : Type ?u.922810
δ : Type ?u.922813
α : Type u_1
γ : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : AddMonoid γ
P : (α →ₛ γ) → Prop
h_ind : ∀ (c : γ) {s : Set α} (hs : MeasurableSet s), P (piecewise s hs (const α c) (const α 0))
h_add : ∀ ⦃f g : α →ₛ γ⦄, Disjoint (support ↑f) (support ↑g) → P f → P g → P (f + g)
x : γ
s : Finset γ
hxs : ¬x ∈ s
ih : ∀ (f : α →ₛ γ), range ↑f \ {0} = ↑s → P f
f : α →ₛ γ
h : range ↑f \ {0} = ↑(insert x s)
⊢ P f

tactic: let g := SimpleFunc.piecewise (f ⁻¹' {x}) mx 0 f
state_before: case insert
α✝ : Type ?u.922804
β : Type ?u.922807
γ✝ : Type ?u.922810
δ : Type ?u.922813
α : Type u_1
γ : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : AddMonoid γ
P : (α →ₛ γ) → Prop
h_ind : ∀ (c : γ) {s : Set α} (hs : MeasurableSet s), P (piecewise s hs (const α c) (const α 0))
h_add : ∀ ⦃f g : α →ₛ γ⦄, Disjoint (support ↑f) (support ↑g) → P f → P g → P (f + g)
x : γ
s : Finset γ
hxs : ¬x ∈ s
ih : ∀ (f : α →ₛ γ), range ↑f \ {0} = ↑s → P f
f : α →ₛ γ
h : range ↑f \ {0} = ↑(insert x s)
mx : MeasurableSet (↑f ⁻¹' {x})
⊢ P f

tactic: convert h_add _ Pg (h_ind x mx)
state_before: case insert
α✝ : Type ?u.922804
β : Type ?u.922807
γ✝ : Type ?u.922810
δ : Type ?u.922813
α : Type u_1
γ : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : AddMonoid γ
P : (α →ₛ γ) → Prop
h_ind : ∀ (c : γ) {s : Set α} (hs : MeasurableSet s), P (piecewise s hs (const α c) (const α 0))
h_add : ∀ ⦃f g : α →ₛ γ⦄, Disjoint (support ↑f) (support ↑g) → P f → P g → P (f + g)
x : γ
s : Finset γ
hxs : ¬x ∈ s
ih : ∀ (f : α →ₛ γ), range ↑f \ {0} = ↑s → P f
f : α →ₛ γ
h : range ↑f \ {0} = ↑(insert x s)
mx : MeasurableSet (↑f ⁻¹' {x})
g : α →ₛ γ := piecewise (↑f ⁻¹' {x}) mx 0 f
Pg : P g
⊢ P f

tactic: rw [disjoint_iff_inf_le]
state_before: case insert
α✝ : Type ?u.922804
β : Type ?u.922807
γ✝ : Type ?u.922810
δ : Type ?u.922813
α : Type u_1
γ : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : AddMonoid γ
P : (α →ₛ γ) → Prop
h_ind : ∀ (c : γ) {s : Set α} (hs : MeasurableSet s), P (piecewise s hs (const α c) (const α 0))
h_add : ∀ ⦃f g : α →ₛ γ⦄, Disjoint (support ↑f) (support ↑g) → P f → P g → P (f + g)
x : γ
s : Finset γ
hxs : ¬x ∈ s
ih : ∀ (f : α →ₛ γ), range ↑f \ {0} = ↑s → P f
f : α →ₛ γ
h : range ↑f \ {0} = ↑(insert x s)
mx : MeasurableSet (↑f ⁻¹' {x})
g : α →ₛ γ := piecewise (↑f ⁻¹' {x}) mx 0 f
Pg : P g
⊢ Disjoint (support ↑g) (support ↑(piecewise (↑f ⁻¹' {x}) mx (const α x) (const α 0)))

tactic: rintro y
state_before: case insert
α✝ : Type ?u.922804
β : Type ?u.922807
γ✝ : Type ?u.922810
δ : Type ?u.922813
α : Type u_1
γ : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : AddMonoid γ
P : (α →ₛ γ) → Prop
h_ind : ∀ (c : γ) {s : Set α} (hs : MeasurableSet s), P (piecewise s hs (const α c) (const α 0))
h_add : ∀ ⦃f g : α →ₛ γ⦄, Disjoint (support ↑f) (support ↑g) → P f → P g → P (f + g)
x : γ
s : Finset γ
hxs : ¬x ∈ s
ih : ∀ (f : α →ₛ γ), range ↑f \ {0} = ↑s → P f
f : α →ₛ γ
h : range ↑f \ {0} = ↑(insert x s)
mx : MeasurableSet (↑f ⁻¹' {x})
g : α →ₛ γ := piecewise (↑f ⁻¹' {x}) mx 0 f
Pg : P g
⊢ support ↑g ⊓ support ↑(piecewise (↑f ⁻¹' {x}) mx (const α x) (const α 0)) ≤ ⊥

tactic: by_cases hy : y ∈ f ⁻¹' {x}
state_before: case insert
α✝ : Type ?u.922804
β : Type ?u.922807
γ✝ : Type ?u.922810
δ : Type ?u.922813
α : Type u_1
γ : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : AddMonoid γ
P : (α →ₛ γ) → Prop
h_ind : ∀ (c : γ) {s : Set α} (hs : MeasurableSet s), P (piecewise s hs (const α c) (const α 0))
h_add : ∀ ⦃f g : α →ₛ γ⦄, Disjoint (support ↑f) (support ↑g) → P f → P g → P (f + g)
x : γ
s : Finset γ
hxs : ¬x ∈ s
ih : ∀ (f : α →ₛ γ), range ↑f \ {0} = ↑s → P f
f : α →ₛ γ
h : range ↑f \ {0} = ↑(insert x s)
mx : MeasurableSet (↑f ⁻¹' {x})
g : α →ₛ γ := piecewise (↑f ⁻¹' {x}) mx 0 f
Pg : P g
y : α
⊢ y ∈ support ↑g ⊓ support ↑(piecewise (↑f ⁻¹' {x}) mx (const α x) (const α 0)) → y ∈ ⊥

tactic: apply ih
state_before: α✝ : Type ?u.922804
β : Type ?u.922807
γ✝ : Type ?u.922810
δ : Type ?u.922813
α : Type u_1
γ : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : AddMonoid γ
P : (α →ₛ γ) → Prop
h_ind : ∀ (c : γ) {s : Set α} (hs : MeasurableSet s), P (piecewise s hs (const α c) (const α 0))
h_add : ∀ ⦃f g : α →ₛ γ⦄, Disjoint (support ↑f) (support ↑g) → P f → P g → P (f + g)
x : γ
s : Finset γ
hxs : ¬x ∈ s
ih : ∀ (f : α →ₛ γ), range ↑f \ {0} = ↑s → P f
f : α →ₛ γ
h : range ↑f \ {0} = ↑(insert x s)
mx : MeasurableSet (↑f ⁻¹' {x})
g : α →ₛ γ := piecewise (↑f ⁻¹' {x}) mx 0 f
⊢ P g

tactic: simp only [SimpleFunc.coe_piecewise, range_piecewise]
state_before: case h
α✝ : Type ?u.922804
β : Type ?u.922807
γ✝ : Type ?u.922810
δ : Type ?u.922813
α : Type u_1
γ : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : AddMonoid γ
P : (α →ₛ γ) → Prop
h_ind : ∀ (c : γ) {s : Set α} (hs : MeasurableSet s), P (piecewise s hs (const α c) (const α 0))
h_add : ∀ ⦃f g : α →ₛ γ⦄, Disjoint (support ↑f) (support ↑g) → P f → P g → P (f + g)
x : γ
s : Finset γ
hxs : ¬x ∈ s
ih : ∀ (f : α →ₛ γ), range ↑f \ {0} = ↑s → P f
f : α →ₛ γ
h : range ↑f \ {0} = ↑(insert x s)
mx : MeasurableSet (↑f ⁻¹' {x})
g : α →ₛ γ := piecewise (↑f ⁻¹' {x}) mx 0 f
⊢ range ↑g \ {0} = ↑s

tactic: rw [image_compl_preimage, union_diff_distrib, diff_diff_comm, h, Finset.coe_insert,
  insert_diff_self_of_not_mem, diff_eq_empty.mpr, Set.empty_union]
state_before: case h
α✝ : Type ?u.922804
β : Type ?u.922807
γ✝ : Type ?u.922810
δ : Type ?u.922813
α : Type u_1
γ : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : AddMonoid γ
P : (α →ₛ γ) → Prop
h_ind : ∀ (c : γ) {s : Set α} (hs : MeasurableSet s), P (piecewise s hs (const α c) (const α 0))
h_add : ∀ ⦃f g : α →ₛ γ⦄, Disjoint (support ↑f) (support ↑g) → P f → P g → P (f + g)
x : γ
s : Finset γ
hxs : ¬x ∈ s
ih : ∀ (f : α →ₛ γ), range ↑f \ {0} = ↑s → P f
f : α →ₛ γ
h : range ↑f \ {0} = ↑(insert x s)
mx : MeasurableSet (↑f ⁻¹' {x})
g : α →ₛ γ := piecewise (↑f ⁻¹' {x}) mx 0 f
⊢ (↑0 '' (↑f ⁻¹' {x}) ∪ ↑f '' (↑f ⁻¹' {x})ᶜ) \ {0} = ↑s

tactic: rw [Set.image_subset_iff]
state_before: case h
α✝ : Type ?u.922804
β : Type ?u.922807
γ✝ : Type ?u.922810
δ : Type ?u.922813
α : Type u_1
γ : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : AddMonoid γ
P : (α →ₛ γ) → Prop
h_ind : ∀ (c : γ) {s : Set α} (hs : MeasurableSet s), P (piecewise s hs (const α c) (const α 0))
h_add : ∀ ⦃f g : α →ₛ γ⦄, Disjoint (support ↑f) (support ↑g) → P f → P g → P (f + g)
x : γ
s : Finset γ
hxs : ¬x ∈ s
ih : ∀ (f : α →ₛ γ), range ↑f \ {0} = ↑s → P f
f : α →ₛ γ
h : range ↑f \ {0} = ↑(insert x s)
mx : MeasurableSet (↑f ⁻¹' {x})
g : α →ₛ γ := piecewise (↑f ⁻¹' {x}) mx 0 f
⊢ ↑0 '' (↑f ⁻¹' {x}) ⊆ {0}

tactic: convert Set.subset_univ _
state_before: case h
α✝ : Type ?u.922804
β : Type ?u.922807
γ✝ : Type ?u.922810
δ : Type ?u.922813
α : Type u_1
γ : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : AddMonoid γ
P : (α →ₛ γ) → Prop
h_ind : ∀ (c : γ) {s : Set α} (hs : MeasurableSet s), P (piecewise s hs (const α c) (const α 0))
h_add : ∀ ⦃f g : α →ₛ γ⦄, Disjoint (support ↑f) (support ↑g) → P f → P g → P (f + g)
x : γ
s : Finset γ
hxs : ¬x ∈ s
ih : ∀ (f : α →ₛ γ), range ↑f \ {0} = ↑s → P f
f : α →ₛ γ
h : range ↑f \ {0} = ↑(insert x s)
mx : MeasurableSet (↑f ⁻¹' {x})
g : α →ₛ γ := piecewise (↑f ⁻¹' {x}) mx 0 f
⊢ ↑f ⁻¹' {x} ⊆ ↑0 ⁻¹' {0}

tactic: exact preimage_const_of_mem (mem_singleton _)
state_before: case h.e'_4
α✝ : Type ?u.922804
β : Type ?u.922807
γ✝ : Type ?u.922810
δ : Type ?u.922813
α : Type u_1
γ : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : AddMonoid γ
P : (α →ₛ γ) → Prop
h_ind : ∀ (c : γ) {s : Set α} (hs : MeasurableSet s), P (piecewise s hs (const α c) (const α 0))
h_add : ∀ ⦃f g : α →ₛ γ⦄, Disjoint (support ↑f) (support ↑g) → P f → P g → P (f + g)
x : γ
s : Finset γ
hxs : ¬x ∈ s
ih : ∀ (f : α →ₛ γ), range ↑f \ {0} = ↑s → P f
f : α →ₛ γ
h : range ↑f \ {0} = ↑(insert x s)
mx : MeasurableSet (↑f ⁻¹' {x})
g : α →ₛ γ := piecewise (↑f ⁻¹' {x}) mx 0 f
⊢ ↑0 ⁻¹' {0} = univ

tactic: rwa [Finset.mem_coe]
state_before: case h
α✝ : Type ?u.922804
β : Type ?u.922807
γ✝ : Type ?u.922810
δ : Type ?u.922813
α : Type u_1
γ : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : AddMonoid γ
P : (α →ₛ γ) → Prop
h_ind : ∀ (c : γ) {s : Set α} (hs : MeasurableSet s), P (piecewise s hs (const α c) (const α 0))
h_add : ∀ ⦃f g : α →ₛ γ⦄, Disjoint (support ↑f) (support ↑g) → P f → P g → P (f + g)
x : γ
s : Finset γ
hxs : ¬x ∈ s
ih : ∀ (f : α →ₛ γ), range ↑f \ {0} = ↑s → P f
f : α →ₛ γ
h : range ↑f \ {0} = ↑(insert x s)
mx : MeasurableSet (↑f ⁻¹' {x})
g : α →ₛ γ := piecewise (↑f ⁻¹' {x}) mx 0 f
⊢ ¬x ∈ ↑s

tactic: ext1 y
state_before: case h.e'_1
α✝ : Type ?u.922804
β : Type ?u.922807
γ✝ : Type ?u.922810
δ : Type ?u.922813
α : Type u_1
γ : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : AddMonoid γ
P : (α →ₛ γ) → Prop
h_ind : ∀ (c : γ) {s : Set α} (hs : MeasurableSet s), P (piecewise s hs (const α c) (const α 0))
h_add : ∀ ⦃f g : α →ₛ γ⦄, Disjoint (support ↑f) (support ↑g) → P f → P g → P (f + g)
x : γ
s : Finset γ
hxs : ¬x ∈ s
ih : ∀ (f : α →ₛ γ), range ↑f \ {0} = ↑s → P f
f : α →ₛ γ
h : range ↑f \ {0} = ↑(insert x s)
mx : MeasurableSet (↑f ⁻¹' {x})
g : α →ₛ γ := piecewise (↑f ⁻¹' {x}) mx 0 f
Pg : P g
⊢ f = g + piecewise (↑f ⁻¹' {x}) mx (const α x) (const α 0)

tactic: by_cases hy : y ∈ f ⁻¹' {x}
state_before: case h.e'_1.H
α✝ : Type ?u.922804
β : Type ?u.922807
γ✝ : Type ?u.922810
δ : Type ?u.922813
α : Type u_1
γ : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : AddMonoid γ
P : (α →ₛ γ) → Prop
h_ind : ∀ (c : γ) {s : Set α} (hs : MeasurableSet s), P (piecewise s hs (const α c) (const α 0))
h_add : ∀ ⦃f g : α →ₛ γ⦄, Disjoint (support ↑f) (support ↑g) → P f → P g → P (f + g)
x : γ
s : Finset γ
hxs : ¬x ∈ s
ih : ∀ (f : α →ₛ γ), range ↑f \ {0} = ↑s → P f
f : α →ₛ γ
h : range ↑f \ {0} = ↑(insert x s)
mx : MeasurableSet (↑f ⁻¹' {x})
g : α →ₛ γ := piecewise (↑f ⁻¹' {x}) mx 0 f
Pg : P g
y : α
⊢ ↑f y = ↑(g + piecewise (↑f ⁻¹' {x}) mx (const α x) (const α 0)) y

tactic: simpa [piecewise_eq_of_mem _ _ _ hy, -piecewise_eq_indicator]
state_before: case pos
α✝ : Type ?u.922804
β : Type ?u.922807
γ✝ : Type ?u.922810
δ : Type ?u.922813
α : Type u_1
γ : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : AddMonoid γ
P : (α →ₛ γ) → Prop
h_ind : ∀ (c : γ) {s : Set α} (hs : MeasurableSet s), P (piecewise s hs (const α c) (const α 0))
h_add : ∀ ⦃f g : α →ₛ γ⦄, Disjoint (support ↑f) (support ↑g) → P f → P g → P (f + g)
x : γ
s : Finset γ
hxs : ¬x ∈ s
ih : ∀ (f : α →ₛ γ), range ↑f \ {0} = ↑s → P f
f : α →ₛ γ
h : range ↑f \ {0} = ↑(insert x s)
mx : MeasurableSet (↑f ⁻¹' {x})
g : α →ₛ γ := piecewise (↑f ⁻¹' {x}) mx 0 f
Pg : P g
y : α
hy : y ∈ ↑f ⁻¹' {x}
⊢ ↑f y = ↑(g + piecewise (↑f ⁻¹' {x}) mx (const α x) (const α 0)) y

tactic: simp [piecewise_eq_of_not_mem _ _ _ hy, -piecewise_eq_indicator]
state_before: case neg
α✝ : Type ?u.922804
β : Type ?u.922807
γ✝ : Type ?u.922810
δ : Type ?u.922813
α : Type u_1
γ : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : AddMonoid γ
P : (α →ₛ γ) → Prop
h_ind : ∀ (c : γ) {s : Set α} (hs : MeasurableSet s), P (piecewise s hs (const α c) (const α 0))
h_add : ∀ ⦃f g : α →ₛ γ⦄, Disjoint (support ↑f) (support ↑g) → P f → P g → P (f + g)
x : γ
s : Finset γ
hxs : ¬x ∈ s
ih : ∀ (f : α →ₛ γ), range ↑f \ {0} = ↑s → P f
f : α →ₛ γ
h : range ↑f \ {0} = ↑(insert x s)
mx : MeasurableSet (↑f ⁻¹' {x})
g : α →ₛ γ := piecewise (↑f ⁻¹' {x}) mx 0 f
Pg : P g
y : α
hy : ¬y ∈ ↑f ⁻¹' {x}
⊢ ↑f y = ↑(g + piecewise (↑f ⁻¹' {x}) mx (const α x) (const α 0)) y

tactic: simp [piecewise_eq_of_mem _ _ _ hy, -piecewise_eq_indicator]
state_before: case pos
α✝ : Type ?u.922804
β : Type ?u.922807
γ✝ : Type ?u.922810
δ : Type ?u.922813
α : Type u_1
γ : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : AddMonoid γ
P : (α →ₛ γ) → Prop
h_ind : ∀ (c : γ) {s : Set α} (hs : MeasurableSet s), P (piecewise s hs (const α c) (const α 0))
h_add : ∀ ⦃f g : α →ₛ γ⦄, Disjoint (support ↑f) (support ↑g) → P f → P g → P (f + g)
x : γ
s : Finset γ
hxs : ¬x ∈ s
ih : ∀ (f : α →ₛ γ), range ↑f \ {0} = ↑s → P f
f : α →ₛ γ
h : range ↑f \ {0} = ↑(insert x s)
mx : MeasurableSet (↑f ⁻¹' {x})
g : α →ₛ γ := piecewise (↑f ⁻¹' {x}) mx 0 f
Pg : P g
y : α
hy : y ∈ ↑f ⁻¹' {x}
⊢ y ∈ support ↑g ⊓ support ↑(piecewise (↑f ⁻¹' {x}) mx (const α x) (const α 0)) → y ∈ ⊥

tactic: simp [piecewise_eq_of_not_mem _ _ _ hy, -piecewise_eq_indicator]
state_before: case neg
α✝ : Type ?u.922804
β : Type ?u.922807
γ✝ : Type ?u.922810
δ : Type ?u.922813
α : Type u_1
γ : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : AddMonoid γ
P : (α →ₛ γ) → Prop
h_ind : ∀ (c : γ) {s : Set α} (hs : MeasurableSet s), P (piecewise s hs (const α c) (const α 0))
h_add : ∀ ⦃f g : α →ₛ γ⦄, Disjoint (support ↑f) (support ↑g) → P f → P g → P (f + g)
x : γ
s : Finset γ
hxs : ¬x ∈ s
ih : ∀ (f : α →ₛ γ), range ↑f \ {0} = ↑s → P f
f : α →ₛ γ
h : range ↑f \ {0} = ↑(insert x s)
mx : MeasurableSet (↑f ⁻¹' {x})
g : α →ₛ γ := piecewise (↑f ⁻¹' {x}) mx 0 f
Pg : P g
y : α
hy : ¬y ∈ ↑f ⁻¹' {x}
⊢ y ∈ support ↑g ⊓ support ↑(piecewise (↑f ⁻¹' {x}) mx (const α x) (const α 0)) → y ∈ ⊥