Documentation

Mathlib.Algebra.Group.Indicator

Indicator function #

Set.indicator (s : Set α) (f : α → β) (a : α) is f a if a ∈ s and is 0 otherwise.
Set.mulIndicator (s : Set α) (f : α → β) (a : α) is f a if a ∈ s and is 1 otherwise.

Implementation note #

In mathematics, an indicator function or a characteristic function is a function used to indicate membership of an element in a set s, having the value 1 for all elements of s and the value 0 otherwise. But since it is usually used to restrict a function to a certain set s, we let the indicator function take the value f x for some function f, instead of 1. If the usual indicator function is needed, just set f to be the constant function fun _ ↦ 1.

The indicator function is implemented non-computably, to avoid having to pass around Decidable arguments. This is in contrast with the design of Pi.single or Set.piecewise.

Tags #

indicator, characteristic

noncomputable def Set.mulIndicator {α : Type u_1} {M : Type u_3} [One M] (s : Set α) (f : α → M) (x : α) :

M

Set.mulIndicator s f a is f a if a ∈ s, 1 otherwise.

Equations

s.mulIndicator f x = if x ∈ s then f x else 1

noncomputable def Set.indicator {α : Type u_1} {M : Type u_3} [Zero M] (s : Set α) (f : α → M) (x : α) :

M

Set.indicator s f a is f a if a ∈ s, 0 otherwise.

Equations

s.indicator f x = if x ∈ s then f x else 0

@[simp]

theorem Set.piecewise_eq_mulIndicator {α : Type u_1} {M : Type u_3} [One M] {s : Set α} {f : α → M} [DecidablePred fun (x : α) => x ∈ s] :

s.piecewise f 1 = s.mulIndicator f

@[simp]

theorem Set.piecewise_eq_indicator {α : Type u_1} {M : Type u_3} [Zero M] {s : Set α} {f : α → M} [DecidablePred fun (x : α) => x ∈ s] :

s.piecewise f 0 = s.indicator f

theorem Set.mulIndicator_apply {α : Type u_1} {M : Type u_3} [One M] (s : Set α) (f : α → M) (a : α) [Decidable (a ∈ s)] :

s.mulIndicator f a = if a ∈ s then f a else 1

theorem Set.indicator_apply {α : Type u_1} {M : Type u_3} [Zero M] (s : Set α) (f : α → M) (a : α) [Decidable (a ∈ s)] :

s.indicator f a = if a ∈ s then f a else 0

@[simp]

theorem Set.mulIndicator_of_mem {α : Type u_1} {M : Type u_3} [One M] {s : Set α} {a : α} (h : a ∈ s) (f : α → M) :

s.mulIndicator f a = f a

@[simp]

theorem Set.indicator_of_mem {α : Type u_1} {M : Type u_3} [Zero M] {s : Set α} {a : α} (h : a ∈ s) (f : α → M) :

s.indicator f a = f a

@[simp]

theorem Set.mulIndicator_of_not_mem {α : Type u_1} {M : Type u_3} [One M] {s : Set α} {a : α} (h : a ∉ s) (f : α → M) :

s.mulIndicator f a = 1

@[simp]

theorem Set.indicator_of_not_mem {α : Type u_1} {M : Type u_3} [Zero M] {s : Set α} {a : α} (h : a ∉ s) (f : α → M) :

s.indicator f a = 0

theorem Set.mulIndicator_eq_one_or_self {α : Type u_1} {M : Type u_3} [One M] (s : Set α) (f : α → M) (a : α) :

s.mulIndicator f a = 1 ∨ s.mulIndicator f a = f a

theorem Set.indicator_eq_zero_or_self {α : Type u_1} {M : Type u_3} [Zero M] (s : Set α) (f : α → M) (a : α) :

s.indicator f a = 0 ∨ s.indicator f a = f a

@[simp]

theorem Set.mulIndicator_apply_eq_self {α : Type u_1} {M : Type u_3} [One M] {s : Set α} {f : α → M} {a : α} :

s.mulIndicator f a = f a ↔ a ∉ s → f a = 1

@[simp]

theorem Set.indicator_apply_eq_self {α : Type u_1} {M : Type u_3} [Zero M] {s : Set α} {f : α → M} {a : α} :

s.indicator f a = f a ↔ a ∉ s → f a = 0

@[simp]

theorem Set.mulIndicator_eq_self {α : Type u_1} {M : Type u_3} [One M] {s : Set α} {f : α → M} :

s.mulIndicator f = f ↔ Function.mulSupport f ⊆ s

@[simp]

theorem Set.indicator_eq_self {α : Type u_1} {M : Type u_3} [Zero M] {s : Set α} {f : α → M} :

s.indicator f = f ↔ Function.support f ⊆ s

theorem Set.mulIndicator_eq_self_of_superset {α : Type u_1} {M : Type u_3} [One M] {s t : Set α} {f : α → M} (h1 : s.mulIndicator f = f) (h2 : s ⊆ t) :

t.mulIndicator f = f

theorem Set.indicator_eq_self_of_superset {α : Type u_1} {M : Type u_3} [Zero M] {s t : Set α} {f : α → M} (h1 : s.indicator f = f) (h2 : s ⊆ t) :

t.indicator f = f

@[simp]

theorem Set.mulIndicator_apply_eq_one {α : Type u_1} {M : Type u_3} [One M] {s : Set α} {f : α → M} {a : α} :

s.mulIndicator f a = 1 ↔ a ∈ s → f a = 1

@[simp]

theorem Set.indicator_apply_eq_zero {α : Type u_1} {M : Type u_3} [Zero M] {s : Set α} {f : α → M} {a : α} :

s.indicator f a = 0 ↔ a ∈ s → f a = 0

@[simp]

theorem Set.mulIndicator_eq_one {α : Type u_1} {M : Type u_3} [One M] {s : Set α} {f : α → M} :

(s.mulIndicator f = fun (x : α) => 1) ↔ Disjoint (Function.mulSupport f) s

@[simp]

theorem Set.indicator_eq_zero {α : Type u_1} {M : Type u_3} [Zero M] {s : Set α} {f : α → M} :

(s.indicator f = fun (x : α) => 0) ↔ Disjoint (Function.support f) s

@[simp]

theorem Set.mulIndicator_eq_one' {α : Type u_1} {M : Type u_3} [One M] {s : Set α} {f : α → M} :

s.mulIndicator f = 1 ↔ Disjoint (Function.mulSupport f) s

@[simp]

theorem Set.indicator_eq_zero' {α : Type u_1} {M : Type u_3} [Zero M] {s : Set α} {f : α → M} :

s.indicator f = 0 ↔ Disjoint (Function.support f) s

theorem Set.mulIndicator_apply_ne_one {α : Type u_1} {M : Type u_3} [One M] {s : Set α} {f : α → M} {a : α} :

s.mulIndicator f a ≠ 1 ↔ a ∈ s ∩ Function.mulSupport f

theorem Set.indicator_apply_ne_zero {α : Type u_1} {M : Type u_3} [Zero M] {s : Set α} {f : α → M} {a : α} :

s.indicator f a ≠ 0 ↔ a ∈ s ∩ Function.support f

@[simp]

theorem Set.mulSupport_mulIndicator {α : Type u_1} {M : Type u_3} [One M] {s : Set α} {f : α → M} :

Function.mulSupport (s.mulIndicator f) = s ∩ Function.mulSupport f

@[simp]

theorem Set.support_indicator {α : Type u_1} {M : Type u_3} [Zero M] {s : Set α} {f : α → M} :

Function.support (s.indicator f) = s ∩ Function.support f

theorem Set.mem_of_mulIndicator_ne_one {α : Type u_1} {M : Type u_3} [One M] {s : Set α} {f : α → M} {a : α} (h : s.mulIndicator f a ≠ 1) :

a ∈ s

If a multiplicative indicator function is not equal to 1 at a point, then that point is in the set.

theorem Set.mem_of_indicator_ne_zero {α : Type u_1} {M : Type u_3} [Zero M] {s : Set α} {f : α → M} {a : α} (h : s.indicator f a ≠ 0) :

a ∈ s

If an additive indicator function is not equal to 0 at a point, then that point is in the set.

theorem Set.eqOn_mulIndicator {α : Type u_1} {M : Type u_3} [One M] {s : Set α} {f : α → M} :

EqOn (s.mulIndicator f) f s

See Set.eqOn_mulIndicator' for the version with sᶜ.

theorem Set.eqOn_indicator {α : Type u_1} {M : Type u_3} [Zero M] {s : Set α} {f : α → M} :

EqOn (s.indicator f) f s

See Set.eqOn_indicator' for the version with sᶜ

theorem Set.eqOn_mulIndicator' {α : Type u_1} {M : Type u_3} [One M] {s : Set α} {f : α → M} :

EqOn (s.mulIndicator f) 1 sᶜ

See Set.eqOn_mulIndicator for the version with s.

theorem Set.eqOn_indicator' {α : Type u_1} {M : Type u_3} [Zero M] {s : Set α} {f : α → M} :

EqOn (s.indicator f) 0 sᶜ

See Set.eqOn_indicator for the version with s.

theorem Set.mulSupport_mulIndicator_subset {α : Type u_1} {M : Type u_3} [One M] {s : Set α} {f : α → M} :

Function.mulSupport (s.mulIndicator f) ⊆ s

theorem Set.support_indicator_subset {α : Type u_1} {M : Type u_3} [Zero M] {s : Set α} {f : α → M} :

Function.support (s.indicator f) ⊆ s

@[simp]

theorem Set.mulIndicator_mulSupport {α : Type u_1} {M : Type u_3} [One M] {f : α → M} :

(Function.mulSupport f).mulIndicator f = f

@[simp]

theorem Set.indicator_support {α : Type u_1} {M : Type u_3} [Zero M] {f : α → M} :

(Function.support f).indicator f = f

@[simp]

theorem Set.mulIndicator_range_comp {α : Type u_1} {M : Type u_3} [One M] {ι : Sort u_5} (f : ι → α) (g : α → M) :

(range f).mulIndicator g ∘ f = g ∘ f

@[simp]

theorem Set.indicator_range_comp {α : Type u_1} {M : Type u_3} [Zero M] {ι : Sort u_5} (f : ι → α) (g : α → M) :

(range f).indicator g ∘ f = g ∘ f

theorem Set.mulIndicator_congr {α : Type u_1} {M : Type u_3} [One M] {s : Set α} {f g : α → M} (h : EqOn f g s) :

s.mulIndicator f = s.mulIndicator g

theorem Set.indicator_congr {α : Type u_1} {M : Type u_3} [Zero M] {s : Set α} {f g : α → M} (h : EqOn f g s) :

s.indicator f = s.indicator g

theorem Set.mulIndicator_eq_mulIndicator {α : Type u_1} {β : Type u_2} {M : Type u_3} [One M] {s : Set α} {f : α → M} {a : α} {t : Set β} {g : β → M} {b : β} (h1 : a ∈ s ↔ b ∈ t) (h2 : f a = g b) :

s.mulIndicator f a = t.mulIndicator g b

theorem Set.indicator_eq_indicator {α : Type u_1} {β : Type u_2} {M : Type u_3} [Zero M] {s : Set α} {f : α → M} {a : α} {t : Set β} {g : β → M} {b : β} (h1 : a ∈ s ↔ b ∈ t) (h2 : f a = g b) :

s.indicator f a = t.indicator g b

theorem Set.mulIndicator_const_eq_mulIndicator_const {α : Type u_1} {β : Type u_2} {M : Type u_3} [One M] {s : Set α} {a : α} {t : Set β} {b : β} {c : M} (h : a ∈ s ↔ b ∈ t) :

s.mulIndicator (fun (x : α) => c) a = t.mulIndicator (fun (x : β) => c) b

theorem Set.indicator_const_eq_indicator_const {α : Type u_1} {β : Type u_2} {M : Type u_3} [Zero M] {s : Set α} {a : α} {t : Set β} {b : β} {c : M} (h : a ∈ s ↔ b ∈ t) :

s.indicator (fun (x : α) => c) a = t.indicator (fun (x : β) => c) b

@[simp]

theorem Set.mulIndicator_univ {α : Type u_1} {M : Type u_3} [One M] (f : α → M) :

univ.mulIndicator f = f

@[simp]

theorem Set.indicator_univ {α : Type u_1} {M : Type u_3} [Zero M] (f : α → M) :

univ.indicator f = f

@[simp]

theorem Set.mulIndicator_empty {α : Type u_1} {M : Type u_3} [One M] (f : α → M) :

∅.mulIndicator f = fun (x : α) => 1

@[simp]

theorem Set.indicator_empty {α : Type u_1} {M : Type u_3} [Zero M] (f : α → M) :

∅.indicator f = fun (x : α) => 0

theorem Set.mulIndicator_empty' {α : Type u_1} {M : Type u_3} [One M] (f : α → M) :

∅.mulIndicator f = 1

theorem Set.indicator_empty' {α : Type u_1} {M : Type u_3} [Zero M] (f : α → M) :

∅.indicator f = 0

@[simp]

theorem Set.mulIndicator_one {α : Type u_1} (M : Type u_3) [One M] (s : Set α) :

(s.mulIndicator fun (x : α) => 1) = fun (x : α) => 1

@[simp]

theorem Set.indicator_zero {α : Type u_1} (M : Type u_3) [Zero M] (s : Set α) :

(s.indicator fun (x : α) => 0) = fun (x : α) => 0

@[simp]

theorem Set.mulIndicator_one' {α : Type u_1} (M : Type u_3) [One M] {s : Set α} :

s.mulIndicator 1 = 1

@[simp]

theorem Set.indicator_zero' {α : Type u_1} (M : Type u_3) [Zero M] {s : Set α} :

s.indicator 0 = 0

theorem Set.mulIndicator_mulIndicator {α : Type u_1} {M : Type u_3} [One M] (s t : Set α) (f : α → M) :

s.mulIndicator (t.mulIndicator f) = (s ∩ t).mulIndicator f

theorem Set.indicator_indicator {α : Type u_1} {M : Type u_3} [Zero M] (s t : Set α) (f : α → M) :

s.indicator (t.indicator f) = (s ∩ t).indicator f

@[simp]

theorem Set.mulIndicator_inter_mulSupport {α : Type u_1} {M : Type u_3} [One M] (s : Set α) (f : α → M) :

(s ∩ Function.mulSupport f).mulIndicator f = s.mulIndicator f

@[simp]

theorem Set.indicator_inter_support {α : Type u_1} {M : Type u_3} [Zero M] (s : Set α) (f : α → M) :

(s ∩ Function.support f).indicator f = s.indicator f

theorem Set.comp_mulIndicator {α : Type u_1} {β : Type u_2} {M : Type u_3} [One M] (h : M → β) (f : α → M) {s : Set α} {x : α} [DecidablePred fun (x : α) => x ∈ s] :

h (s.mulIndicator f x) = s.piecewise (h ∘ f) (Function.const α (h 1)) x

theorem Set.comp_indicator {α : Type u_1} {β : Type u_2} {M : Type u_3} [Zero M] (h : M → β) (f : α → M) {s : Set α} {x : α} [DecidablePred fun (x : α) => x ∈ s] :

h (s.indicator f x) = s.piecewise (h ∘ f) (Function.const α (h 0)) x

theorem Set.mulIndicator_comp_right {α : Type u_1} {β : Type u_2} {M : Type u_3} [One M] {s : Set α} (f : β → α) {g : α → M} {x : β} :

(f ⁻¹' s).mulIndicator (g ∘ f) x = s.mulIndicator g (f x)

theorem Set.indicator_comp_right {α : Type u_1} {β : Type u_2} {M : Type u_3} [Zero M] {s : Set α} (f : β → α) {g : α → M} {x : β} :

(f ⁻¹' s).indicator (g ∘ f) x = s.indicator g (f x)

theorem Set.mulIndicator_image {α : Type u_1} {β : Type u_2} {M : Type u_3} [One M] {s : Set α} {f : β → M} {g : α → β} (hg : Function.Injective g) {x : α} :

(g '' s).mulIndicator f (g x) = s.mulIndicator (f ∘ g) x

theorem Set.indicator_image {α : Type u_1} {β : Type u_2} {M : Type u_3} [Zero M] {s : Set α} {f : β → M} {g : α → β} (hg : Function.Injective g) {x : α} :

(g '' s).indicator f (g x) = s.indicator (f ∘ g) x

theorem Set.mulIndicator_comp_of_one {α : Type u_1} {M : Type u_3} {N : Type u_4} [One M] [One N] {s : Set α} {f : α → M} {g : M → N} (hg : g 1 = 1) :

s.mulIndicator (g ∘ f) = g ∘ s.mulIndicator f

theorem Set.indicator_comp_of_zero {α : Type u_1} {M : Type u_3} {N : Type u_4} [Zero M] [Zero N] {s : Set α} {f : α → M} {g : M → N} (hg : g 0 = 0) :

s.indicator (g ∘ f) = g ∘ s.indicator f

theorem Set.comp_mulIndicator_const {α : Type u_1} {M : Type u_3} {N : Type u_4} [One M] [One N] {s : Set α} (c : M) (f : M → N) (hf : f 1 = 1) :

(fun (x : α) => f (s.mulIndicator (fun (x : α) => c) x)) = s.mulIndicator fun (x : α) => f c

theorem Set.comp_indicator_const {α : Type u_1} {M : Type u_3} {N : Type u_4} [Zero M] [Zero N] {s : Set α} (c : M) (f : M → N) (hf : f 0 = 0) :

(fun (x : α) => f (s.indicator (fun (x : α) => c) x)) = s.indicator fun (x : α) => f c

theorem Set.mulIndicator_preimage {α : Type u_1} {M : Type u_3} [One M] (s : Set α) (f : α → M) (B : Set M) :

s.mulIndicator f ⁻¹' B = s.ite (f ⁻¹' B) (1 ⁻¹' B)

theorem Set.indicator_preimage {α : Type u_1} {M : Type u_3} [Zero M] (s : Set α) (f : α → M) (B : Set M) :

s.indicator f ⁻¹' B = s.ite (f ⁻¹' B) (0 ⁻¹' B)

theorem Set.mulIndicator_one_preimage {α : Type u_1} {M : Type u_3} [One M] {t : Set α} (s : Set M) :

t.mulIndicator 1 ⁻¹' s ∈ {univ , ∅}

theorem Set.indicator_zero_preimage {α : Type u_1} {M : Type u_3} [Zero M] {t : Set α} (s : Set M) :

t.indicator 0 ⁻¹' s ∈ {univ , ∅}

theorem Set.mulIndicator_const_preimage_eq_union {α : Type u_1} {M : Type u_3} [One M] (U : Set α) (s : Set M) (a : M) [Decidable (a ∈ s)] [Decidable (1 ∈ s)] :

(U.mulIndicator fun (x : α) => a) ⁻¹' s = (if a ∈ s then U else ∅) ∪ if 1 ∈ s then Uᶜ else ∅

theorem Set.indicator_const_preimage_eq_union {α : Type u_1} {M : Type u_3} [Zero M] (U : Set α) (s : Set M) (a : M) [Decidable (a ∈ s)] [Decidable (0 ∈ s)] :

(U.indicator fun (x : α) => a) ⁻¹' s = (if a ∈ s then U else ∅) ∪ if 0 ∈ s then Uᶜ else ∅

theorem Set.mulIndicator_const_preimage {α : Type u_1} {M : Type u_3} [One M] (U : Set α) (s : Set M) (a : M) :

(U.mulIndicator fun (x : α) => a) ⁻¹' s ∈ {univ , U, Uᶜ , ∅}

theorem Set.indicator_const_preimage {α : Type u_1} {M : Type u_3} [Zero M] (U : Set α) (s : Set M) (a : M) :

(U.indicator fun (x : α) => a) ⁻¹' s ∈ {univ , U, Uᶜ , ∅}

theorem Set.indicator_one_preimage {α : Type u_1} {M : Type u_3} [One M] [Zero M] (U : Set α) (s : Set M) :

U.indicator 1 ⁻¹' s ∈ {univ , U, Uᶜ , ∅}

theorem Set.mulIndicator_preimage_of_not_mem {α : Type u_1} {M : Type u_3} [One M] (s : Set α) (f : α → M) {t : Set M} (ht : 1 ∉ t) :

s.mulIndicator f ⁻¹' t = f ⁻¹' t ∩ s

theorem Set.indicator_preimage_of_not_mem {α : Type u_1} {M : Type u_3} [Zero M] (s : Set α) (f : α → M) {t : Set M} (ht : 0 ∉ t) :

s.indicator f ⁻¹' t = f ⁻¹' t ∩ s

theorem Set.mem_range_mulIndicator {α : Type u_1} {M : Type u_3} [One M] {r : M} {s : Set α} {f : α → M} :

r ∈ range (s.mulIndicator f) ↔ r = 1 ∧ s ≠ univ ∨ r ∈ f '' s

theorem Set.mem_range_indicator {α : Type u_1} {M : Type u_3} [Zero M] {r : M} {s : Set α} {f : α → M} :

r ∈ range (s.indicator f) ↔ r = 0 ∧ s ≠ univ ∨ r ∈ f '' s

theorem Set.mulIndicator_rel_mulIndicator {α : Type u_1} {M : Type u_3} [One M] {s : Set α} {f g : α → M} {a : α} {r : M → M → Prop} (h1 : r 1 1) (ha : a ∈ s → r (f a) (g a)) :

r (s.mulIndicator f a) (s.mulIndicator g a)

theorem Set.indicator_rel_indicator {α : Type u_1} {M : Type u_3} [Zero M] {s : Set α} {f g : α → M} {a : α} {r : M → M → Prop} (h1 : r 0 0) (ha : a ∈ s → r (f a) (g a)) :

r (s.indicator f a) (s.indicator g a)

theorem Set.mulIndicator_union_mul_inter_apply {α : Type u_1} {M : Type u_3} [MulOneClass M] (f : α → M) (s t : Set α) (a : α) :

(s ∪ t).mulIndicator f a * (s ∩ t).mulIndicator f a = s.mulIndicator f a * t.mulIndicator f a

theorem Set.indicator_union_add_inter_apply {α : Type u_1} {M : Type u_3} [AddZeroClass M] (f : α → M) (s t : Set α) (a : α) :

(s ∪ t).indicator f a + (s ∩ t).indicator f a = s.indicator f a + t.indicator f a

theorem Set.mulIndicator_union_mul_inter {α : Type u_1} {M : Type u_3} [MulOneClass M] (f : α → M) (s t : Set α) :

(s ∪ t).mulIndicator f * (s ∩ t).mulIndicator f = s.mulIndicator f * t.mulIndicator f

theorem Set.indicator_union_add_inter {α : Type u_1} {M : Type u_3} [AddZeroClass M] (f : α → M) (s t : Set α) :

(s ∪ t).indicator f + (s ∩ t).indicator f = s.indicator f + t.indicator f

theorem Set.mulIndicator_union_of_not_mem_inter {α : Type u_1} {M : Type u_3} [MulOneClass M] {s t : Set α} {a : α} (h : a ∉ s ∩ t) (f : α → M) :

(s ∪ t).mulIndicator f a = s.mulIndicator f a * t.mulIndicator f a

theorem Set.indicator_union_of_not_mem_inter {α : Type u_1} {M : Type u_3} [AddZeroClass M] {s t : Set α} {a : α} (h : a ∉ s ∩ t) (f : α → M) :

(s ∪ t).indicator f a = s.indicator f a + t.indicator f a

theorem Set.mulIndicator_union_of_disjoint {α : Type u_1} {M : Type u_3} [MulOneClass M] {s t : Set α} (h : Disjoint s t) (f : α → M) :

(s ∪ t).mulIndicator f = fun (a : α) => s.mulIndicator f a * t.mulIndicator f a

theorem Set.indicator_union_of_disjoint {α : Type u_1} {M : Type u_3} [AddZeroClass M] {s t : Set α} (h : Disjoint s t) (f : α → M) :

(s ∪ t).indicator f = fun (a : α) => s.indicator f a + t.indicator f a

theorem Set.mulIndicator_symmDiff {α : Type u_1} {M : Type u_3} [MulOneClass M] (s t : Set α) (f : α → M) :

(symmDiff s t).mulIndicator f = (s \ t).mulIndicator f * (t \ s).mulIndicator f

theorem Set.indicator_symmDiff {α : Type u_1} {M : Type u_3} [AddZeroClass M] (s t : Set α) (f : α → M) :

(symmDiff s t).indicator f = (s \ t).indicator f + (t \ s).indicator f

theorem Set.mulIndicator_mul {α : Type u_1} {M : Type u_3} [MulOneClass M] (s : Set α) (f g : α → M) :

(s.mulIndicator fun (a : α) => f a * g a) = fun (a : α) => s.mulIndicator f a * s.mulIndicator g a

theorem Set.indicator_add {α : Type u_1} {M : Type u_3} [AddZeroClass M] (s : Set α) (f g : α → M) :

(s.indicator fun (a : α) => f a + g a) = fun (a : α) => s.indicator f a + s.indicator g a

theorem Set.mulIndicator_mul' {α : Type u_1} {M : Type u_3} [MulOneClass M] (s : Set α) (f g : α → M) :

s.mulIndicator (f * g) = s.mulIndicator f * s.mulIndicator g

theorem Set.indicator_add' {α : Type u_1} {M : Type u_3} [AddZeroClass M] (s : Set α) (f g : α → M) :

s.indicator (f + g) = s.indicator f + s.indicator g

@[simp]

theorem Set.mulIndicator_compl_mul_self_apply {α : Type u_1} {M : Type u_3} [MulOneClass M] (s : Set α) (f : α → M) (a : α) :

sᶜ.mulIndicator f a * s.mulIndicator f a = f a

@[simp]

theorem Set.indicator_compl_add_self_apply {α : Type u_1} {M : Type u_3} [AddZeroClass M] (s : Set α) (f : α → M) (a : α) :

sᶜ.indicator f a + s.indicator f a = f a

@[simp]

theorem Set.mulIndicator_compl_mul_self {α : Type u_1} {M : Type u_3} [MulOneClass M] (s : Set α) (f : α → M) :

sᶜ.mulIndicator f * s.mulIndicator f = f

@[simp]

theorem Set.indicator_compl_add_self {α : Type u_1} {M : Type u_3} [AddZeroClass M] (s : Set α) (f : α → M) :

sᶜ.indicator f + s.indicator f = f

@[simp]

theorem Set.mulIndicator_self_mul_compl_apply {α : Type u_1} {M : Type u_3} [MulOneClass M] (s : Set α) (f : α → M) (a : α) :

s.mulIndicator f a * sᶜ.mulIndicator f a = f a

@[simp]

theorem Set.indicator_self_add_compl_apply {α : Type u_1} {M : Type u_3} [AddZeroClass M] (s : Set α) (f : α → M) (a : α) :

s.indicator f a + sᶜ.indicator f a = f a

@[simp]

theorem Set.mulIndicator_self_mul_compl {α : Type u_1} {M : Type u_3} [MulOneClass M] (s : Set α) (f : α → M) :

s.mulIndicator f * sᶜ.mulIndicator f = f

@[simp]

theorem Set.indicator_self_add_compl {α : Type u_1} {M : Type u_3} [AddZeroClass M] (s : Set α) (f : α → M) :

s.indicator f + sᶜ.indicator f = f

theorem Set.mulIndicator_mul_eq_left {α : Type u_1} {M : Type u_3} [MulOneClass M] {f g : α → M} (h : Disjoint (Function.mulSupport f) (Function.mulSupport g)) :

(Function.mulSupport f).mulIndicator (f * g) = f

theorem Set.indicator_add_eq_left {α : Type u_1} {M : Type u_3} [AddZeroClass M] {f g : α → M} (h : Disjoint (Function.support f) (Function.support g)) :

(Function.support f).indicator (f + g) = f

theorem Set.mulIndicator_mul_eq_right {α : Type u_1} {M : Type u_3} [MulOneClass M] {f g : α → M} (h : Disjoint (Function.mulSupport f) (Function.mulSupport g)) :

(Function.mulSupport g).mulIndicator (f * g) = g

theorem Set.indicator_add_eq_right {α : Type u_1} {M : Type u_3} [AddZeroClass M] {f g : α → M} (h : Disjoint (Function.support f) (Function.support g)) :

(Function.support g).indicator (f + g) = g

theorem Set.mulIndicator_mul_compl_eq_piecewise {α : Type u_1} {M : Type u_3} [MulOneClass M] {s : Set α} [DecidablePred fun (x : α) => x ∈ s] (f g : α → M) :

s.mulIndicator f * sᶜ.mulIndicator g = s.piecewise f g

theorem Set.indicator_add_compl_eq_piecewise {α : Type u_1} {M : Type u_3} [AddZeroClass M] {s : Set α} [DecidablePred fun (x : α) => x ∈ s] (f g : α → M) :

s.indicator f + sᶜ.indicator g = s.piecewise f g

noncomputable def Set.mulIndicatorHom {α : Type u_5} (M : Type u_6) [MulOneClass M] (s : Set α) :

(α → M) →* α → M

Set.mulIndicator as a monoidHom.

Equations

Set.mulIndicatorHom M s = { toFun := s.mulIndicator, map_one' := ⋯, map_mul' := ⋯ }

noncomputable def Set.indicatorHom {α : Type u_5} (M : Type u_6) [AddZeroClass M] (s : Set α) :

(α → M) →+ α → M

Set.indicator as an addMonoidHom.

Equations

Set.indicatorHom M s = { toFun := s.indicator, map_zero' := ⋯, map_add' := ⋯ }

theorem Set.mulIndicator_inv' {α : Type u_1} {G : Type u_5} [Group G] (s : Set α) (f : α → G) :

s.mulIndicator f⁻¹ = (s.mulIndicator f)⁻¹

theorem Set.indicator_neg' {α : Type u_1} {G : Type u_5} [AddGroup G] (s : Set α) (f : α → G) :

s.indicator (-f) = -s.indicator f

theorem Set.mulIndicator_inv {α : Type u_1} {G : Type u_5} [Group G] (s : Set α) (f : α → G) :

(s.mulIndicator fun (a : α) => (f a)⁻¹) = fun (a : α) => (s.mulIndicator f a)⁻¹

theorem Set.indicator_neg {α : Type u_1} {G : Type u_5} [AddGroup G] (s : Set α) (f : α → G) :

(s.indicator fun (a : α) => -f a) = fun (a : α) => -s.indicator f a

theorem Set.mulIndicator_div {α : Type u_1} {G : Type u_5} [Group G] (s : Set α) (f g : α → G) :

(s.mulIndicator fun (a : α) => f a / g a) = fun (a : α) => s.mulIndicator f a / s.mulIndicator g a

theorem Set.indicator_sub {α : Type u_1} {G : Type u_5} [AddGroup G] (s : Set α) (f g : α → G) :

(s.indicator fun (a : α) => f a - g a) = fun (a : α) => s.indicator f a - s.indicator g a

theorem Set.mulIndicator_div' {α : Type u_1} {G : Type u_5} [Group G] (s : Set α) (f g : α → G) :

s.mulIndicator (f / g) = s.mulIndicator f / s.mulIndicator g

theorem Set.indicator_sub' {α : Type u_1} {G : Type u_5} [AddGroup G] (s : Set α) (f g : α → G) :

s.indicator (f - g) = s.indicator f - s.indicator g

theorem Set.mulIndicator_compl {α : Type u_1} {G : Type u_5} [Group G] (s : Set α) (f : α → G) :

sᶜ.mulIndicator f = f * (s.mulIndicator f)⁻¹

theorem Set.indicator_compl' {α : Type u_1} {G : Type u_5} [AddGroup G] (s : Set α) (f : α → G) :

sᶜ.indicator f = f + -s.indicator f

theorem Set.mulIndicator_compl' {α : Type u_1} {G : Type u_5} [Group G] (s : Set α) (f : α → G) :

sᶜ.mulIndicator f = f / s.mulIndicator f

theorem Set.indicator_compl {α : Type u_1} {G : Type u_5} [AddGroup G] (s : Set α) (f : α → G) :

sᶜ.indicator f = f - s.indicator f

theorem Set.mulIndicator_diff {α : Type u_1} {G : Type u_5} [Group G] {s t : Set α} (h : s ⊆ t) (f : α → G) :

(t \ s).mulIndicator f = t.mulIndicator f * (s.mulIndicator f)⁻¹

theorem Set.indicator_diff' {α : Type u_1} {G : Type u_5} [AddGroup G] {s t : Set α} (h : s ⊆ t) (f : α → G) :

(t \ s).indicator f = t.indicator f + -s.indicator f

theorem Set.mulIndicator_diff' {α : Type u_1} {G : Type u_5} [Group G] {s t : Set α} (h : s ⊆ t) (f : α → G) :

(t \ s).mulIndicator f = t.mulIndicator f / s.mulIndicator f

theorem Set.indicator_diff {α : Type u_1} {G : Type u_5} [AddGroup G] {s t : Set α} (h : s ⊆ t) (f : α → G) :

(t \ s).indicator f = t.indicator f - s.indicator f

theorem Set.apply_mulIndicator_symmDiff {α : Type u_1} {β : Type u_2} {G : Type u_5} [Group G] {g : G → β} (hg : ∀ (x : G), g x⁻¹ = g x) (s t : Set α) (f : α → G) (x : α) :

g ((symmDiff s t).mulIndicator f x) = g (s.mulIndicator f x / t.mulIndicator f x)

theorem Set.apply_indicator_symmDiff {α : Type u_1} {β : Type u_2} {G : Type u_5} [AddGroup G] {g : G → β} (hg : ∀ (x : G), g (-x) = g x) (s t : Set α) (f : α → G) (x : α) :

g ((symmDiff s t).indicator f x) = g (s.indicator f x - t.indicator f x)

theorem MonoidHom.map_mulIndicator {α : Type u_1} {M : Type u_5} {N : Type u_6} [MulOneClass M] [MulOneClass N] (f : M →* N) (s : Set α) (g : α → M) (x : α) :

f (s.mulIndicator g x) = s.mulIndicator (⇑f ∘ g) x

theorem AddMonoidHom.map_indicator {α : Type u_1} {M : Type u_5} {N : Type u_6} [AddZeroClass M] [AddZeroClass N] (f : M →+ N) (s : Set α) (g : α → M) (x : α) :

f (s.indicator g x) = s.indicator (⇑f ∘ g) x