Locus of unequal values of finitely supported dependent functions

Let N : α → Type* be a type family, assume that N a has a 0 for all a : α and let f g : Π₀ a, N a be finitely supported dependent functions.

Main definition

• DFinsupp.neLocus f g : Finset α, the finite subset of α where f and g differ. In the case in which N a is an additive group for all a, DFinsupp.neLocus f g coincides with DFinsupp.support (f - g).

def DFinsupp.neLocus {α : Type u_1} {N : α → Type u_2} [DecidableEq α] [(a : α) → DecidableEq (N a)] [(a : α) → Zero (N a)] (f g : Π₀ (a : α), N a) : Finset α

Given two finitely supported functions f g : α →₀ N, Finsupp.neLocus f g is the Finset where f and g differ. This generalizes (f - g).support to situations without subtraction.

Equations

• f.neLocus g = {x ∈ f.support ∪ g.support | f x ≠ g x}

@[simp]

theorem DFinsupp.mem_neLocus {α : Type u_1} {N : α → Type u_2} [DecidableEq α] [(a : α) → DecidableEq (N a)] [(a : α) → Zero (N a)] {f g : Π₀ (a : α), N a} {a : α} : a ∈ f.neLocus g ↔ f a ≠ g a

theorem DFinsupp.not_mem_neLocus {α : Type u_1} {N : α → Type u_2} [DecidableEq α] [(a : α) → DecidableEq (N a)] [(a : α) → Zero (N a)] {f g : Π₀ (a : α), N a} {a : α} : a ∉ f.neLocus g ↔ f a = g a

@[simp]

theorem DFinsupp.coe_neLocus {α : Type u_1} {N : α → Type u_2} [DecidableEq α] [(a : α) → DecidableEq (N a)] [(a : α) → Zero (N a)] (f g : Π₀ (a : α), N a) : ↑(f.neLocus g) = {x : α | f x ≠ g x}

@[simp]

theorem DFinsupp.neLocus_eq_empty {α : Type u_1} {N : α → Type u_2} [DecidableEq α] [(a : α) → DecidableEq (N a)] [(a : α) → Zero (N a)] {f g : Π₀ (a : α), N a} : f.neLocus g = ∅ ↔ f = g

@[simp]

theorem DFinsupp.nonempty_neLocus_iff {α : Type u_1} {N : α → Type u_2} [DecidableEq α] [(a : α) → DecidableEq (N a)] [(a : α) → Zero (N a)] {f g : Π₀ (a : α), N a} : (f.neLocus g).Nonempty ↔ f ≠ g

theorem DFinsupp.neLocus_comm {α : Type u_1} {N : α → Type u_2} [DecidableEq α] [(a : α) → DecidableEq (N a)] [(a : α) → Zero (N a)] (f g : Π₀ (a : α), N a) : f.neLocus g = g.neLocus f

@[simp]

theorem DFinsupp.neLocus_zero_right {α : Type u_1} {N : α → Type u_2} [DecidableEq α] [(a : α) → DecidableEq (N a)] [(a : α) → Zero (N a)] (f : Π₀ (a : α), N a) : f.neLocus 0 = f.support

@[simp]

theorem DFinsupp.neLocus_zero_left {α : Type u_1} {N : α → Type u_2} [DecidableEq α] [(a : α) → DecidableEq (N a)] [(a : α) → Zero (N a)] (f : Π₀ (a : α), N a) : neLocus 0 f = f.support

theorem DFinsupp.subset_mapRange_neLocus {α : Type u_1} {N : α → Type u_2} [DecidableEq α] {M : α → Type u_3} [(a : α) → Zero (N a)] [(a : α) → Zero (M a)] [(a : α) → DecidableEq (N a)] [(a : α) → DecidableEq (M a)] (f g : Π₀ (a : α), N a) {F : (a : α) → N a → M a} (F0 : ∀ (a : α), F a 0 = 0) : (mapRange F F0 f).neLocus (mapRange F F0 g) ⊆ f.neLocus g

theorem DFinsupp.zipWith_neLocus_eq_left {α : Type u_1} {N : α → Type u_2} [DecidableEq α] {M : α → Type u_3} {P : α → Type u_4} [(a : α) → Zero (N a)] [(a : α) → Zero (M a)] [(a : α) → Zero (P a)] [(a : α) → DecidableEq (N a)] [(a : α) → DecidableEq (P a)] {F : (a : α) → M a → N a → P a} (F0 : ∀ (a : α), F a 0 0 = 0) (f : Π₀ (a : α), M a) (g₁ g₂ : Π₀ (a : α), N a) (hF : ∀ (a : α) (f : M a), Function.Injective fun (g : N a) => F a f g) : (zipWith F F0 f g₁).neLocus (zipWith F F0 f g₂) = g₁.neLocus g₂

theorem DFinsupp.zipWith_neLocus_eq_right {α : Type u_1} {N : α → Type u_2} [DecidableEq α] {M : α → Type u_3} {P : α → Type u_4} [(a : α) → Zero (N a)] [(a : α) → Zero (M a)] [(a : α) → Zero (P a)] [(a : α) → DecidableEq (M a)] [(a : α) → DecidableEq (P a)] {F : (a : α) → M a → N a → P a} (F0 : ∀ (a : α), F a 0 0 = 0) (f₁ f₂ : Π₀ (a : α), M a) (g : Π₀ (a : α), N a) (hF : ∀ (a : α) (g : N a), Function.Injective fun (f : M a) => F a f g) : (zipWith F F0 f₁ g).neLocus (zipWith F F0 f₂ g) = f₁.neLocus f₂

theorem DFinsupp.mapRange_neLocus_eq {α : Type u_1} {N : α → Type u_2} [DecidableEq α] {M : α → Type u_3} [(a : α) → Zero (N a)] [(a : α) → Zero (M a)] [(a : α) → DecidableEq (N a)] [(a : α) → DecidableEq (M a)] (f g : Π₀ (a : α), N a) {F : (a : α) → N a → M a} (F0 : ∀ (a : α), F a 0 = 0) (hF : ∀ (a : α), Function.Injective (F a)) : (mapRange F F0 f).neLocus (mapRange F F0 g) = f.neLocus g

@[simp]

theorem DFinsupp.neLocus_add_left {α : Type u_1} {N : α → Type u_2} [DecidableEq α] [(a : α) → DecidableEq (N a)] [(a : α) → AddLeftCancelMonoid (N a)] (f g h : Π₀ (a : α), N a) : (f + g).neLocus (f + h) = g.neLocus h

@[simp]

theorem DFinsupp.neLocus_add_right {α : Type u_1} {N : α → Type u_2} [DecidableEq α] [(a : α) → DecidableEq (N a)] [(a : α) → AddRightCancelMonoid (N a)] (f g h : Π₀ (a : α), N a) : (f + h).neLocus (g + h) = f.neLocus g

@[simp]

theorem DFinsupp.neLocus_neg_neg {α : Type u_1} {N : α → Type u_2} [DecidableEq α] [(a : α) → DecidableEq (N a)] [(a : α) → AddGroup (N a)] (f g : Π₀ (a : α), N a) : (-f).neLocus (-g) = f.neLocus g

theorem DFinsupp.neLocus_neg {α : Type u_1} {N : α → Type u_2} [DecidableEq α] [(a : α) → DecidableEq (N a)] [(a : α) → AddGroup (N a)] (f g : Π₀ (a : α), N a) : (-f).neLocus g = f.neLocus (-g)

theorem DFinsupp.neLocus_eq_support_sub {α : Type u_1} {N : α → Type u_2} [DecidableEq α] [(a : α) → DecidableEq (N a)] [(a : α) → AddGroup (N a)] (f g : Π₀ (a : α), N a) : f.neLocus g = (f - g).support

@[simp]

theorem DFinsupp.neLocus_sub_left {α : Type u_1} {N : α → Type u_2} [DecidableEq α] [(a : α) → DecidableEq (N a)] [(a : α) → AddGroup (N a)] (f g₁ g₂ : Π₀ (a : α), N a) : (f - g₁).neLocus (f - g₂) = g₁.neLocus g₂

@[simp]

theorem DFinsupp.neLocus_sub_right {α : Type u_1} {N : α → Type u_2} [DecidableEq α] [(a : α) → DecidableEq (N a)] [(a : α) → AddGroup (N a)] (f₁ f₂ g : Π₀ (a : α), N a) : (f₁ - g).neLocus (f₂ - g) = f₁.neLocus f₂

@[simp]

theorem DFinsupp.neLocus_self_add_right {α : Type u_1} {N : α → Type u_2} [DecidableEq α] [(a : α) → DecidableEq (N a)] [(a : α) → AddGroup (N a)] (f g : Π₀ (a : α), N a) : f.neLocus (f + g) = g.support

@[simp]

theorem DFinsupp.neLocus_self_add_left {α : Type u_1} {N : α → Type u_2} [DecidableEq α] [(a : α) → DecidableEq (N a)] [(a : α) → AddGroup (N a)] (f g : Π₀ (a : α), N a) : (f + g).neLocus f = g.support

@[simp]

theorem DFinsupp.neLocus_self_sub_right {α : Type u_1} {N : α → Type u_2} [DecidableEq α] [(a : α) → DecidableEq (N a)] [(a : α) → AddGroup (N a)] (f g : Π₀ (a : α), N a) : f.neLocus (f - g) = g.support

@[simp]

theorem DFinsupp.neLocus_self_sub_left {α : Type u_1} {N : α → Type u_2} [DecidableEq α] [(a : α) → DecidableEq (N a)] [(a : α) → AddGroup (N a)] (f g : Π₀ (a : α), N a) : (f - g).neLocus f = g.support