Documentation

Mathlib.Algebra.Function.Support

Support of a function #

In this file we define Function.support f = {x | f x ≠ 0} and prove its basic properties. We also define Function.mulSupport f = {x | f x ≠ 1}.

def Function.support {α : Type u_1} {A : Type u_3} [Zero A] (f : αA) :
Set α

support of a function is the set of points x such that f x ≠ 0.

Equations
Instances For
    def Function.mulSupport {α : Type u_1} {M : Type u_5} [One M] (f : αM) :
    Set α

    mulSupport of a function is the set of points x such that f x ≠ 1.

    Equations
    Instances For
      theorem Function.support_eq_preimage {α : Type u_1} {M : Type u_5} [Zero M] (f : αM) :
      theorem Function.mulSupport_eq_preimage {α : Type u_1} {M : Type u_5} [One M] (f : αM) :
      theorem Function.nmem_support {α : Type u_1} {M : Type u_5} [Zero M] {f : αM} {x : α} :
      xFunction.support f f x = 0
      theorem Function.nmem_mulSupport {α : Type u_1} {M : Type u_5} [One M] {f : αM} {x : α} :
      xFunction.mulSupport f f x = 1
      theorem Function.compl_support {α : Type u_1} {M : Type u_5} [Zero M] {f : αM} :
      (Function.support f) = {x : α | f x = 0}
      theorem Function.compl_mulSupport {α : Type u_1} {M : Type u_5} [One M] {f : αM} :
      (Function.mulSupport f) = {x : α | f x = 1}
      @[simp]
      theorem Function.mem_support {α : Type u_1} {M : Type u_5} [Zero M] {f : αM} {x : α} :
      @[simp]
      theorem Function.mem_mulSupport {α : Type u_1} {M : Type u_5} [One M] {f : αM} {x : α} :
      @[simp]
      theorem Function.support_subset_iff {α : Type u_1} {M : Type u_5} [Zero M] {f : αM} {s : Set α} :
      Function.support f s ∀ (x : α), f x 0x s
      @[simp]
      theorem Function.mulSupport_subset_iff {α : Type u_1} {M : Type u_5} [One M] {f : αM} {s : Set α} :
      Function.mulSupport f s ∀ (x : α), f x 1x s
      theorem Function.support_subset_iff' {α : Type u_1} {M : Type u_5} [Zero M] {f : αM} {s : Set α} :
      Function.support f s xs, f x = 0
      theorem Function.mulSupport_subset_iff' {α : Type u_1} {M : Type u_5} [One M] {f : αM} {s : Set α} :
      Function.mulSupport f s xs, f x = 1
      theorem Function.support_eq_iff {α : Type u_1} {M : Type u_5} [Zero M] {f : αM} {s : Set α} :
      Function.support f = s (xs, f x 0) xs, f x = 0
      theorem Function.mulSupport_eq_iff {α : Type u_1} {M : Type u_5} [One M] {f : αM} {s : Set α} :
      Function.mulSupport f = s (xs, f x 1) xs, f x = 1
      theorem Function.support_extend_zero_subset {α : Type u_1} {M : Type u_5} {N : Type u_6} [Zero N] {f : αM} {g : αN} :
      theorem Function.mulSupport_extend_one_subset {α : Type u_1} {M : Type u_5} {N : Type u_6} [One N] {f : αM} {g : αN} :
      theorem Function.support_extend_zero {α : Type u_1} {M : Type u_5} {N : Type u_6} [Zero N] {f : αM} {g : αN} (hf : Function.Injective f) :
      theorem Function.mulSupport_extend_one {α : Type u_1} {M : Type u_5} {N : Type u_6} [One N] {f : αM} {g : αN} (hf : Function.Injective f) :
      theorem Function.support_disjoint_iff {α : Type u_1} {M : Type u_5} [Zero M] {f : αM} {s : Set α} :
      theorem Function.mulSupport_disjoint_iff {α : Type u_1} {M : Type u_5} [One M] {f : αM} {s : Set α} :
      theorem Function.disjoint_support_iff {α : Type u_1} {M : Type u_5} [Zero M] {f : αM} {s : Set α} :
      theorem Function.disjoint_mulSupport_iff {α : Type u_1} {M : Type u_5} [One M] {f : αM} {s : Set α} :
      @[simp]
      theorem Function.support_eq_empty_iff {α : Type u_1} {M : Type u_5} [Zero M] {f : αM} :
      @[simp]
      theorem Function.mulSupport_eq_empty_iff {α : Type u_1} {M : Type u_5} [One M] {f : αM} :
      @[simp]
      theorem Function.support_nonempty_iff {α : Type u_1} {M : Type u_5} [Zero M] {f : αM} :
      @[simp]
      theorem Function.mulSupport_nonempty_iff {α : Type u_1} {M : Type u_5} [One M] {f : αM} :
      theorem Function.range_subset_insert_image_support {α : Type u_1} {M : Type u_5} [Zero M] (f : αM) :
      theorem Function.range_eq_image_or_of_support_subset {α : Type u_1} {M : Type u_5} [Zero M] {f : αM} {k : Set α} (h : Function.support f k) :
      Set.range f = f '' k Set.range f = insert 0 (f '' k)
      theorem Function.range_eq_image_or_of_mulSupport_subset {α : Type u_1} {M : Type u_5} [One M] {f : αM} {k : Set α} (h : Function.mulSupport f k) :
      Set.range f = f '' k Set.range f = insert 1 (f '' k)
      @[simp]
      theorem Function.support_zero' {α : Type u_1} {M : Type u_5} [Zero M] :
      @[simp]
      theorem Function.mulSupport_one' {α : Type u_1} {M : Type u_5} [One M] :
      @[simp]
      theorem Function.support_zero {α : Type u_1} {M : Type u_5} [Zero M] :
      (Function.support fun (x : α) => 0) =
      @[simp]
      theorem Function.mulSupport_one {α : Type u_1} {M : Type u_5} [One M] :
      (Function.mulSupport fun (x : α) => 1) =
      theorem Function.support_const {α : Type u_1} {M : Type u_5} [Zero M] {c : M} (hc : c 0) :
      (Function.support fun (x : α) => c) = Set.univ
      theorem Function.mulSupport_const {α : Type u_1} {M : Type u_5} [One M] {c : M} (hc : c 1) :
      (Function.mulSupport fun (x : α) => c) = Set.univ
      theorem Function.support_binop_subset {α : Type u_1} {M : Type u_5} {N : Type u_6} {P : Type u_7} [Zero M] [Zero N] [Zero P] (op : MNP) (op1 : op 0 0 = 0) (f : αM) (g : αN) :
      (Function.support fun (x : α) => op (f x) (g x)) Function.support f Function.support g
      theorem Function.mulSupport_binop_subset {α : Type u_1} {M : Type u_5} {N : Type u_6} {P : Type u_7} [One M] [One N] [One P] (op : MNP) (op1 : op 1 1 = 1) (f : αM) (g : αN) :
      theorem Function.support_comp_subset {α : Type u_1} {M : Type u_5} {N : Type u_6} [Zero M] [Zero N] {g : MN} (hg : g 0 = 0) (f : αM) :
      theorem Function.mulSupport_comp_subset {α : Type u_1} {M : Type u_5} {N : Type u_6} [One M] [One N] {g : MN} (hg : g 1 = 1) (f : αM) :
      theorem Function.support_subset_comp {α : Type u_1} {M : Type u_5} {N : Type u_6} [Zero M] [Zero N] {g : MN} (hg : ∀ {x : M}, g x = 0x = 0) (f : αM) :
      theorem Function.mulSupport_subset_comp {α : Type u_1} {M : Type u_5} {N : Type u_6} [One M] [One N] {g : MN} (hg : ∀ {x : M}, g x = 1x = 1) (f : αM) :
      theorem Function.support_comp_eq {α : Type u_1} {M : Type u_5} {N : Type u_6} [Zero M] [Zero N] (g : MN) (hg : ∀ {x : M}, g x = 0 x = 0) (f : αM) :
      theorem Function.mulSupport_comp_eq {α : Type u_1} {M : Type u_5} {N : Type u_6} [One M] [One N] (g : MN) (hg : ∀ {x : M}, g x = 1 x = 1) (f : αM) :
      theorem Function.support_comp_eq_of_range_subset {α : Type u_1} {M : Type u_5} {N : Type u_6} [Zero M] [Zero N] {g : MN} {f : αM} (hg : ∀ {x : M}, x Set.range f(g x = 0 x = 0)) :
      theorem Function.mulSupport_comp_eq_of_range_subset {α : Type u_1} {M : Type u_5} {N : Type u_6} [One M] [One N] {g : MN} {f : αM} (hg : ∀ {x : M}, x Set.range f(g x = 1 x = 1)) :
      theorem Function.support_comp_eq_preimage {α : Type u_1} {β : Type u_2} {M : Type u_5} [Zero M] (g : βM) (f : αβ) :
      theorem Function.mulSupport_comp_eq_preimage {α : Type u_1} {β : Type u_2} {M : Type u_5} [One M] (g : βM) (f : αβ) :
      theorem Function.support_prod_mk {α : Type u_1} {M : Type u_5} {N : Type u_6} [Zero M] [Zero N] (f : αM) (g : αN) :
      (Function.support fun (x : α) => (f x, g x)) = Function.support f Function.support g
      theorem Function.mulSupport_prod_mk {α : Type u_1} {M : Type u_5} {N : Type u_6} [One M] [One N] (f : αM) (g : αN) :
      theorem Function.support_prod_mk' {α : Type u_1} {M : Type u_5} {N : Type u_6} [Zero M] [Zero N] (f : αM × N) :
      Function.support f = (Function.support fun (x : α) => (f x).1) Function.support fun (x : α) => (f x).2
      theorem Function.mulSupport_prod_mk' {α : Type u_1} {M : Type u_5} {N : Type u_6} [One M] [One N] (f : αM × N) :
      Function.mulSupport f = (Function.mulSupport fun (x : α) => (f x).1) Function.mulSupport fun (x : α) => (f x).2
      theorem Function.support_along_fiber_subset {α : Type u_1} {β : Type u_2} {M : Type u_5} [Zero M] (f : α × βM) (a : α) :
      (Function.support fun (b : β) => f (a, b)) Prod.snd '' Function.support f
      theorem Function.mulSupport_along_fiber_subset {α : Type u_1} {β : Type u_2} {M : Type u_5} [One M] (f : α × βM) (a : α) :
      (Function.mulSupport fun (b : β) => f (a, b)) Prod.snd '' Function.mulSupport f
      theorem Function.support_add {α : Type u_1} {M : Type u_5} [AddZeroClass M] (f : αM) (g : αM) :
      theorem Function.mulSupport_mul {α : Type u_1} {M : Type u_5} [MulOneClass M] (f : αM) (g : αM) :
      theorem Function.support_nsmul {α : Type u_1} {M : Type u_5} [AddMonoid M] (f : αM) (n : ) :
      (Function.support fun (x : α) => n f x) Function.support f
      theorem Function.mulSupport_pow {α : Type u_1} {M : Type u_5} [Monoid M] (f : αM) (n : ) :
      (Function.mulSupport fun (x : α) => f x ^ n) Function.mulSupport f
      @[simp]
      theorem Function.support_neg {α : Type u_1} {G : Type u_10} [SubtractionMonoid G] (f : αG) :
      (Function.support fun (x : α) => -f x) = Function.support f
      @[simp]
      theorem Function.mulSupport_inv {α : Type u_1} {G : Type u_10} [DivisionMonoid G] (f : αG) :
      @[simp]
      theorem Function.support_neg' {α : Type u_1} {G : Type u_10} [SubtractionMonoid G] (f : αG) :
      @[simp]
      theorem Function.mulSupport_inv' {α : Type u_1} {G : Type u_10} [DivisionMonoid G] (f : αG) :
      theorem Function.support_add_neg {α : Type u_1} {G : Type u_10} [SubtractionMonoid G] (f : αG) (g : αG) :
      theorem Function.mulSupport_mul_inv {α : Type u_1} {G : Type u_10} [DivisionMonoid G] (f : αG) (g : αG) :
      theorem Function.support_sub {α : Type u_1} {G : Type u_10} [SubtractionMonoid G] (f : αG) (g : αG) :
      theorem Function.mulSupport_div {α : Type u_1} {G : Type u_10} [DivisionMonoid G] (f : αG) (g : αG) :
      @[simp]
      theorem Function.support_one {α : Type u_1} (R : Type u_8) [Zero R] [One R] [NeZero 1] :
      Function.support 1 = Set.univ
      @[simp]
      theorem Function.mulSupport_zero {α : Type u_1} (R : Type u_8) [Zero R] [One R] [NeZero 1] :
      theorem Function.support_mul_subset_left {α : Type u_1} {M : Type u_5} [MulZeroClass M] (f : αM) (g : αM) :
      (Function.support fun (x : α) => f x * g x) Function.support f
      theorem Function.support_mul_subset_right {α : Type u_1} {M : Type u_5} [MulZeroClass M] (f : αM) (g : αM) :
      (Function.support fun (x : α) => f x * g x) Function.support g
      @[simp]
      theorem Function.support_mul {α : Type u_1} {M : Type u_5} [MulZeroClass M] [NoZeroDivisors M] (f : αM) (g : αM) :
      @[simp]
      theorem Function.support_mul' {α : Type u_1} {M : Type u_5} [MulZeroClass M] [NoZeroDivisors M] (f : αM) (g : αM) :
      @[simp]
      theorem Function.support_pow {α : Type u_1} {M : Type u_5} [MonoidWithZero M] [NoZeroDivisors M] {n : } (f : αM) (hn : n 0) :
      (Function.support fun (a : α) => f a ^ n) = Function.support f
      @[simp]
      theorem Function.support_pow' {α : Type u_1} {M : Type u_5} [MonoidWithZero M] [NoZeroDivisors M] {n : } (f : αM) (hn : n 0) :
      @[simp]
      theorem Function.support_inv {α : Type u_1} {G₀ : Type u_12} [GroupWithZero G₀] (f : αG₀) :
      (Function.support fun (a : α) => (f a)⁻¹) = Function.support f
      @[simp]
      theorem Function.support_inv' {α : Type u_1} {G₀ : Type u_12} [GroupWithZero G₀] (f : αG₀) :
      @[simp]
      theorem Function.support_div {α : Type u_1} {G₀ : Type u_12} [GroupWithZero G₀] (f : αG₀) (g : αG₀) :
      @[simp]
      theorem Function.support_div' {α : Type u_1} {G₀ : Type u_12} [GroupWithZero G₀] (f : αG₀) (g : αG₀) :
      theorem Function.mulSupport_one_add {α : Type u_1} {R : Type u_8} [One R] [AddLeftCancelMonoid R] (f : αR) :
      (Function.mulSupport fun (x : α) => 1 + f x) = Function.support f
      theorem Function.mulSupport_one_add' {α : Type u_1} {R : Type u_8} [One R] [AddLeftCancelMonoid R] (f : αR) :
      theorem Function.mulSupport_add_one {α : Type u_1} {R : Type u_8} [One R] [AddRightCancelMonoid R] (f : αR) :
      (Function.mulSupport fun (x : α) => f x + 1) = Function.support f
      theorem Function.mulSupport_add_one' {α : Type u_1} {R : Type u_8} [One R] [AddRightCancelMonoid R] (f : αR) :
      theorem Function.mulSupport_one_sub' {α : Type u_1} {R : Type u_8} [One R] [AddGroup R] (f : αR) :
      theorem Function.mulSupport_one_sub {α : Type u_1} {R : Type u_8} [One R] [AddGroup R] (f : αR) :
      (Function.mulSupport fun (x : α) => 1 - f x) = Function.support f
      theorem Set.image_inter_support_eq {α : Type u_1} {β : Type u_2} {M : Type u_3} [Zero M] {f : αM} {s : Set β} {g : βα} :
      theorem Set.image_inter_mulSupport_eq {α : Type u_1} {β : Type u_2} {M : Type u_3} [One M] {f : αM} {s : Set β} {g : βα} :
      theorem Pi.support_single_subset {A : Type u_1} {B : Type u_2} [DecidableEq A] [Zero B] {a : A} {b : B} :
      theorem Pi.mulSupport_mulSingle_subset {A : Type u_1} {B : Type u_2} [DecidableEq A] [One B] {a : A} {b : B} :
      theorem Pi.support_single_zero {A : Type u_1} {B : Type u_2} [DecidableEq A] [Zero B] {a : A} :
      @[simp]
      theorem Pi.support_single_of_ne {A : Type u_1} {B : Type u_2} [DecidableEq A] [Zero B] {a : A} {b : B} (h : b 0) :
      @[simp]
      theorem Pi.mulSupport_mulSingle_of_ne {A : Type u_1} {B : Type u_2} [DecidableEq A] [One B] {a : A} {b : B} (h : b 1) :
      theorem Pi.support_single {A : Type u_1} {B : Type u_2} [DecidableEq A] [Zero B] {a : A} {b : B} [DecidableEq B] :
      Function.support (Pi.single a b) = if b = 0 then else {a}
      theorem Pi.mulSupport_mulSingle {A : Type u_1} {B : Type u_2} [DecidableEq A] [One B] {a : A} {b : B} [DecidableEq B] :
      Function.mulSupport (Pi.mulSingle a b) = if b = 1 then else {a}
      theorem Pi.support_single_disjoint {A : Type u_1} {B : Type u_2} [DecidableEq A] [Zero B] {b : B} {b' : B} (hb : b 0) (hb' : b' 0) {i : A} {j : A} :
      theorem Pi.mulSupport_mulSingle_disjoint {A : Type u_1} {B : Type u_2} [DecidableEq A] [One B] {b : B} {b' : B} (hb : b 1) (hb' : b' 1) {i : A} {j : A} :