Locus of unequal values of finitely supported functions

Let α N be two Types, assume that N has a 0 and let f g : α →₀ N be finitely supported functions.

Main definition

• Finsupp.neLocus f g : Finset α, the finite subset of α where f and g differ.

In the case in which N is an additive group, Finsupp.neLocus f g coincides with Finsupp.support (f - g).

def Finsupp.neLocus {α : Type u_1} {N : Type u_3} [DecidableEq α] [DecidableEq N] [Zero N] (f g : α →₀ N) : 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.

@[simp]

theorem Finsupp.mem_neLocus {α : Type u_1} {N : Type u_3} [DecidableEq α] [DecidableEq N] [Zero N] {f g : α →₀ N} {a : α} : a ∈ f.neLocus g ↔ f a ≠ g a

theorem Finsupp.notMem_neLocus {α : Type u_1} {N : Type u_3} [DecidableEq α] [DecidableEq N] [Zero N] {f g : α →₀ N} {a : α} : a ∉ f.neLocus g ↔ f a = g a

@[simp]

theorem Finsupp.coe_neLocus {α : Type u_1} {N : Type u_3} [DecidableEq α] [DecidableEq N] [Zero N] (f g : α →₀ N) : ↑(f.neLocus g) = {x : α | f x ≠ g x}

@[simp]

theorem Finsupp.neLocus_eq_empty {α : Type u_1} {N : Type u_3} [DecidableEq α] [DecidableEq N] [Zero N] {f g : α →₀ N} : f.neLocus g = ∅ ↔ f = g

@[simp]

theorem Finsupp.nonempty_neLocus_iff {α : Type u_1} {N : Type u_3} [DecidableEq α] [DecidableEq N] [Zero N] {f g : α →₀ N} : (f.neLocus g).Nonempty ↔ f ≠ g

theorem Finsupp.neLocus_comm {α : Type u_1} {N : Type u_3} [DecidableEq α] [DecidableEq N] [Zero N] (f g : α →₀ N) : f.neLocus g = g.neLocus f

@[simp]

theorem Finsupp.neLocus_zero_right {α : Type u_1} {N : Type u_3} [DecidableEq α] [DecidableEq N] [Zero N] (f : α →₀ N) : f.neLocus 0 = f.support

@[simp]

theorem Finsupp.neLocus_zero_left {α : Type u_1} {N : Type u_3} [DecidableEq α] [DecidableEq N] [Zero N] (f : α →₀ N) : neLocus 0 f = f.support

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

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

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

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

@[simp]

theorem Finsupp.neLocus_add_left {α : Type u_1} {N : Type u_3} [DecidableEq α] [DecidableEq N] [AddLeftCancelMonoid N] (f g h : α →₀ N) : (f + g).neLocus (f + h) = g.neLocus h

@[simp]

theorem Finsupp.neLocus_add_right {α : Type u_1} {N : Type u_3} [DecidableEq α] [DecidableEq N] [AddRightCancelMonoid N] (f g h : α →₀ N) : (f + h).neLocus (g + h) = f.neLocus g

@[simp]

theorem Finsupp.neLocus_neg_neg {α : Type u_1} {N : Type u_3} [DecidableEq α] [DecidableEq N] [AddGroup N] (f g : α →₀ N) : (-f).neLocus (-g) = f.neLocus g

theorem Finsupp.neLocus_neg {α : Type u_1} {N : Type u_3} [DecidableEq α] [DecidableEq N] [AddGroup N] (f g : α →₀ N) : (-f).neLocus g = f.neLocus (-g)

theorem Finsupp.neLocus_eq_support_sub {α : Type u_1} {N : Type u_3} [DecidableEq α] [DecidableEq N] [AddGroup N] (f g : α →₀ N) : f.neLocus g = (f - g).support

@[simp]

theorem Finsupp.neLocus_sub_left {α : Type u_1} {N : Type u_3} [DecidableEq α] [DecidableEq N] [AddGroup N] (f g₁ g₂ : α →₀ N) : (f - g₁).neLocus (f - g₂) = g₁.neLocus g₂

@[simp]

theorem Finsupp.neLocus_sub_right {α : Type u_1} {N : Type u_3} [DecidableEq α] [DecidableEq N] [AddGroup N] (f₁ f₂ g : α →₀ N) : (f₁ - g).neLocus (f₂ - g) = f₁.neLocus f₂

@[simp]

theorem Finsupp.neLocus_self_add_right {α : Type u_1} {N : Type u_3} [DecidableEq α] [DecidableEq N] [AddGroup N] (f g : α →₀ N) : f.neLocus (f + g) = g.support

@[simp]

theorem Finsupp.neLocus_self_add_left {α : Type u_1} {N : Type u_3} [DecidableEq α] [DecidableEq N] [AddGroup N] (f g : α →₀ N) : (f + g).neLocus f = g.support

@[simp]

theorem Finsupp.neLocus_self_sub_right {α : Type u_1} {N : Type u_3} [DecidableEq α] [DecidableEq N] [AddGroup N] (f g : α →₀ N) : f.neLocus (f - g) = g.support

@[simp]

theorem Finsupp.neLocus_self_sub_left {α : Type u_1} {N : Type u_3} [DecidableEq α] [DecidableEq N] [AddGroup N] (f g : α →₀ N) : (f - g).neLocus f = g.support