Type of functions with finite support
For any type α and a type M with zero, we define the type finsupp α M (notation: α →₀ M)
of finitely supported functions from α to M, i.e. the functions which are zero everywhere
on α except on a finite set.
Functions with finite support are used (at least) in the following parts of the library:
monoid_algebra R Mandadd_monoid_algebra R Mare defined asM →₀ R;polynomials and multivariate polynomials are defined as
add_monoid_algebras, hence they usefinsuppunder the hood;the linear combination of a family of vectors
v iwith coefficientsf i(as used, e.g., to define linearly independent familylinear_independent) is defined as a mapfinsupp.total : (ι → M) → (ι →₀ R) →ₗ[R] M.
Some other constructions are naturally equivalent to α →₀ M with some α and M but are defined
in a different way in the library:
multiset α ≃+ α →₀ ℕ;free_abelian_group α ≃+ α →₀ ℤ.
Most of the theory assumes that the range is a commutative additive monoid. This gives us the big
sum operator as a powerful way to construct finsupp elements.
Many constructions based on α →₀ M use semireducible type tags to avoid reusing unwanted type
instances. E.g., monoid_algebra, add_monoid_algebra, and types based on these two have
non-pointwise multiplication.
Notations
This file adds α →₀ M as a global notation for finsupp α M. We also use the following convention
for Type* variables in this file
α,β,γ: types with no additional structure that appear as the first argument tofinsuppsomewhere in the statement;ι: an auxiliary index type;M,M',N,P: types withhas_zeroor(add_)(comm_)monoidstructure;Mis also used for a (semi)module over a (semi)ring.G,H: groups (commutative or not, multiplicative or additive);R,S: (semi)rings.
TODO
This file is currently ~2K lines long, so possibly it should be splitted into smaller chunks;
Add the list of definitions and important lemmas to the module docstring.
Implementation notes
This file is a noncomputable theory and uses classical logic throughout.
Notation
This file defines α →₀ β as notation for finsupp α β.
finsupp α M, denoted α →₀ M, is the type of functions f : α → M such that
f x = 0 for all but finitely many x.
Basic declarations about finsupp
@[instance]
Equations
- finsupp.has_coe_to_fun = {F := λ (_x : α →₀ M), α → M, coe := finsupp.to_fun _inst_1}
@[instance]
Equations
- finsupp.has_zero = {zero := {support := ∅, to_fun := λ (_x : α), 0, mem_support_to_fun := _}}
@[simp]
@[instance]
Equations
- finsupp.inhabited = {default := 0}
@[simp]
theorem
finsupp.fun_support_eq
{α : Type u_1}
{M : Type u_5}
[has_zero M]
(f : α →₀ M) :
function.support ⇑f = ↑(f.support)
theorem
finsupp.coe_fn_injective
{α : Type u_1}
{M : Type u_5}
[has_zero M] :
function.injective (λ (f : α →₀ M) (x : α), ⇑f x)
@[instance]
def
finsupp.finsupp.decidable_eq
{α : Type u_1}
{M : Type u_5}
[has_zero M]
[decidable_eq α]
[decidable_eq M] :
decidable_eq (α →₀ M)
Given fintype α, equiv_fun_on_fintype is the equiv between α →₀ β and α → β.
(All functions on a finite type are finitely supported.)
Equations
- finsupp.equiv_fun_on_fintype = {to_fun := λ (f : α →₀ M) (a : α), ⇑f a, inv_fun := λ (f : α → M), {support := finset.filter (λ (a : α), f a ≠ 0) finset.univ, to_fun := f, mem_support_to_fun := _}, left_inv := _, right_inv := _}
Declarations about single
single a b is the finitely supported function which has
value b at a and zero otherwise.
theorem
finsupp.single_apply
{α : Type u_1}
{M : Type u_5}
[has_zero M]
{a a' : α}
{b : M} :
⇑(finsupp.single a b) a' = ite (a = a') b 0
theorem
finsupp.single_eq_indicator
{α : Type u_1}
{M : Type u_5}
[has_zero M]
{a : α}
{b : M} :
⇑(finsupp.single a b) = {a}.indicator (λ (_x : α), b)
@[simp]
theorem
finsupp.single_eq_same
{α : Type u_1}
{M : Type u_5}
[has_zero M]
{a : α}
{b : M} :
⇑(finsupp.single a b) a = b
@[simp]
theorem
finsupp.single_eq_of_ne
{α : Type u_1}
{M : Type u_5}
[has_zero M]
{a a' : α}
{b : M}
(h : a ≠ a') :
⇑(finsupp.single a b) a' = 0
theorem
finsupp.single_eq_update
{α : Type u_1}
{M : Type u_5}
[has_zero M]
{a : α}
{b : M} :
⇑(finsupp.single a b) = function.update 0 a b
@[simp]
theorem
finsupp.single_zero
{α : Type u_1}
{M : Type u_5}
[has_zero M]
{a : α} :
finsupp.single a 0 = 0
theorem
finsupp.single_of_single_apply
{α : Type u_1}
{M : Type u_5}
[has_zero M]
(a a' : α)
(b : M) :
finsupp.single a (⇑(finsupp.single a' b) a) = ⇑(finsupp.single a' (finsupp.single a' b)) a
theorem
finsupp.support_single_ne_zero
{α : Type u_1}
{M : Type u_5}
[has_zero M]
{a : α}
{b : M}
(hb : b ≠ 0) :
(finsupp.single a b).support = {a}
theorem
finsupp.support_single_subset
{α : Type u_1}
{M : Type u_5}
[has_zero M]
{a : α}
{b : M} :
(finsupp.single a b).support ⊆ {a}
theorem
finsupp.single_apply_mem
{α : Type u_1}
{M : Type u_5}
[has_zero M]
{a : α}
{b : M}
(x : α) :
⇑(finsupp.single a b) x ∈ {0, b}
theorem
finsupp.range_single_subset
{α : Type u_1}
{M : Type u_5}
[has_zero M]
{a : α}
{b : M} :
set.range ⇑(finsupp.single a b) ⊆ {0, b}
theorem
finsupp.single_eq_single_iff
{α : Type u_1}
{M : Type u_5}
[has_zero M]
(a₁ a₂ : α)
(b₁ b₂ : M) :
theorem
finsupp.single_left_inj
{α : Type u_1}
{M : Type u_5}
[has_zero M]
{a a' : α}
{b : M}
(h : b ≠ 0) :
finsupp.single a b = finsupp.single a' b ↔ a = a'
@[simp]
theorem
finsupp.single_eq_zero
{α : Type u_1}
{M : Type u_5}
[has_zero M]
{a : α}
{b : M} :
finsupp.single a b = 0 ↔ b = 0
theorem
finsupp.single_swap
{α : Type u_1}
{M : Type u_5}
[has_zero M]
(a₁ a₂ : α)
(b : M) :
⇑(finsupp.single a₁ b) a₂ = ⇑(finsupp.single a₂ b) a₁
@[instance]
def
finsupp.nontrivial
{α : Type u_1}
{M : Type u_5}
[has_zero M]
[nonempty α]
[nontrivial M] :
nontrivial (α →₀ M)
theorem
finsupp.unique_single
{α : Type u_1}
{M : Type u_5}
[has_zero M]
[unique α]
(x : α →₀ M) :
x = finsupp.single (default α) (⇑x (default α))
@[simp]
theorem
finsupp.unique_single_eq_iff
{α : Type u_1}
{M : Type u_5}
[has_zero M]
{a a' : α}
{b : M}
[unique α]
{b' : M} :
finsupp.single a b = finsupp.single a' b' ↔ b = b'
theorem
finsupp.support_eq_singleton'
{α : Type u_1}
{M : Type u_5}
[has_zero M]
{f : α →₀ M}
{a : α} :
Declarations about on_finset
def
finsupp.on_finset
{α : Type u_1}
{M : Type u_5}
[has_zero M]
(s : finset α)
(f : α → M)
(hf : ∀ (a : α), f a ≠ 0 → a ∈ s) :
α →₀ M
on_finset s f hf is the finsupp function representing f restricted to the finset s.
The function needs to be 0 outside of s. Use this when the set needs to be filtered anyways,
otherwise a better set representation is often available.
Equations
- finsupp.on_finset s f hf = {support := finset.filter (λ (a : α), f a ≠ 0) s, to_fun := f, mem_support_to_fun := _}
@[simp]
theorem
finsupp.on_finset_apply
{α : Type u_1}
{M : Type u_5}
[has_zero M]
{s : finset α}
{f : α → M}
{hf : ∀ (a : α), f a ≠ 0 → a ∈ s}
{a : α} :
⇑(finsupp.on_finset s f hf) a = f a
@[simp]
theorem
finsupp.support_on_finset_subset
{α : Type u_1}
{M : Type u_5}
[has_zero M]
{s : finset α}
{f : α → M}
{hf : ∀ (a : α), f a ≠ 0 → a ∈ s} :
(finsupp.on_finset s f hf).support ⊆ s
theorem
finsupp.support_on_finset
{α : Type u_1}
{M : Type u_5}
[has_zero M]
{s : finset α}
{f : α → M}
(hf : ∀ (a : α), f a ≠ 0 → a ∈ s) :
(finsupp.on_finset s f hf).support = finset.filter (λ (a : α), f a ≠ 0) s
Declarations about map_range
def
finsupp.map_range
{α : Type u_1}
{M : Type u_5}
{N : Type u_7}
[has_zero M]
[has_zero N]
(f : M → N)
(hf : f 0 = 0)
(g : α →₀ M) :
α →₀ N
The composition of f : M → N and g : α →₀ M is
map_range f hf g : α →₀ N, well-defined when f 0 = 0.
Equations
- finsupp.map_range f hf g = finsupp.on_finset g.support (f ∘ ⇑g) _
@[simp]
theorem
finsupp.map_range_apply
{α : Type u_1}
{M : Type u_5}
{N : Type u_7}
[has_zero M]
[has_zero N]
{f : M → N}
{hf : f 0 = 0}
{g : α →₀ M}
{a : α} :
⇑(finsupp.map_range f hf g) a = f (⇑g a)
@[simp]
theorem
finsupp.map_range_zero
{α : Type u_1}
{M : Type u_5}
{N : Type u_7}
[has_zero M]
[has_zero N]
{f : M → N}
{hf : f 0 = 0} :
finsupp.map_range f hf 0 = 0
theorem
finsupp.support_map_range
{α : Type u_1}
{M : Type u_5}
{N : Type u_7}
[has_zero M]
[has_zero N]
{f : M → N}
{hf : f 0 = 0}
{g : α →₀ M} :
(finsupp.map_range f hf g).support ⊆ g.support
@[simp]
theorem
finsupp.map_range_single
{α : Type u_1}
{M : Type u_5}
{N : Type u_7}
[has_zero M]
[has_zero N]
{f : M → N}
{hf : f 0 = 0}
{a : α}
{b : M} :
finsupp.map_range f hf (finsupp.single a b) = finsupp.single a (f b)
Declarations about emb_domain
def
finsupp.emb_domain
{α : Type u_1}
{β : Type u_2}
{M : Type u_5}
[has_zero M]
(f : α ↪ β)
(v : α →₀ M) :
β →₀ M
Given f : α ↪ β and v : α →₀ M, emb_domain f v : β →₀ M
is the finitely supported function whose value at f a : β is v a.
For a b : β outside the range of f, it is zero.
Equations
- finsupp.emb_domain f v = {support := finset.map f v.support, to_fun := λ (a₂ : β), dite (a₂ ∈ finset.map f v.support) (λ (h : a₂ ∈ finset.map f v.support), ⇑v (finset.choose (λ (a₁ : α), ⇑f a₁ = a₂) v.support _)) (λ (h : a₂ ∉ finset.map f v.support), 0), mem_support_to_fun := _}
@[simp]
theorem
finsupp.support_emb_domain
{α : Type u_1}
{β : Type u_2}
{M : Type u_5}
[has_zero M]
(f : α ↪ β)
(v : α →₀ M) :
(finsupp.emb_domain f v).support = finset.map f v.support
@[simp]
theorem
finsupp.emb_domain_zero
{α : Type u_1}
{β : Type u_2}
{M : Type u_5}
[has_zero M]
(f : α ↪ β) :
finsupp.emb_domain f 0 = 0
@[simp]
theorem
finsupp.emb_domain_apply
{α : Type u_1}
{β : Type u_2}
{M : Type u_5}
[has_zero M]
(f : α ↪ β)
(v : α →₀ M)
(a : α) :
⇑(finsupp.emb_domain f v) (⇑f a) = ⇑v a
theorem
finsupp.emb_domain_injective
{α : Type u_1}
{β : Type u_2}
{M : Type u_5}
[has_zero M]
(f : α ↪ β) :
@[simp]
theorem
finsupp.emb_domain_inj
{α : Type u_1}
{β : Type u_2}
{M : Type u_5}
[has_zero M]
{f : α ↪ β}
{l₁ l₂ : α →₀ M} :
finsupp.emb_domain f l₁ = finsupp.emb_domain f l₂ ↔ l₁ = l₂
@[simp]
theorem
finsupp.emb_domain_eq_zero
{α : Type u_1}
{β : Type u_2}
{M : Type u_5}
[has_zero M]
{f : α ↪ β}
{l : α →₀ M} :
finsupp.emb_domain f l = 0 ↔ l = 0
theorem
finsupp.emb_domain_map_range
{α : Type u_1}
{β : Type u_2}
{M : Type u_5}
{N : Type u_7}
[has_zero M]
[has_zero N]
(f : α ↪ β)
(g : M → N)
(p : α →₀ M)
(hg : g 0 = 0) :
finsupp.emb_domain f (finsupp.map_range g hg p) = finsupp.map_range g hg (finsupp.emb_domain f p)
theorem
finsupp.single_of_emb_domain_single
{α : Type u_1}
{β : Type u_2}
{M : Type u_5}
[has_zero M]
(l : α →₀ M)
(f : α ↪ β)
(a : β)
(b : M)
(hb : b ≠ 0)
(h : finsupp.emb_domain f l = finsupp.single a b) :
∃ (x : α), l = finsupp.single x b ∧ ⇑f x = a
Declarations about zip_with
def
finsupp.zip_with
{α : Type u_1}
{M : Type u_5}
{N : Type u_7}
{P : Type u_8}
[has_zero M]
[has_zero N]
[has_zero P]
(f : M → N → P)
(hf : f 0 0 = 0)
(g₁ : α →₀ M)
(g₂ : α →₀ N) :
α →₀ P
zip_with f hf g₁ g₂ is the finitely supported function satisfying
zip_with f hf g₁ g₂ a = f (g₁ a) (g₂ a), and it is well-defined when f 0 0 = 0.
Equations
- finsupp.zip_with f hf g₁ g₂ = finsupp.on_finset (g₁.support ∪ g₂.support) (λ (a : α), f (⇑g₁ a) (⇑g₂ a)) _
Declarations about erase
erase a f is the finitely supported function equal to f except at a where it is equal to
0.
@[simp]
theorem
finsupp.support_erase
{α : Type u_1}
{M : Type u_5}
[has_zero M]
{a : α}
{f : α →₀ M} :
(finsupp.erase a f).support = f.support.erase a
@[simp]
theorem
finsupp.erase_same
{α : Type u_1}
{M : Type u_5}
[has_zero M]
{a : α}
{f : α →₀ M} :
⇑(finsupp.erase a f) a = 0
@[simp]
theorem
finsupp.erase_ne
{α : Type u_1}
{M : Type u_5}
[has_zero M]
{a a' : α}
{f : α →₀ M}
(h : a' ≠ a) :
⇑(finsupp.erase a f) a' = ⇑f a'
@[simp]
theorem
finsupp.erase_single
{α : Type u_1}
{M : Type u_5}
[has_zero M]
{a : α}
{b : M} :
finsupp.erase a (finsupp.single a b) = 0
theorem
finsupp.erase_single_ne
{α : Type u_1}
{M : Type u_5}
[has_zero M]
{a a' : α}
{b : M}
(h : a ≠ a') :
finsupp.erase a (finsupp.single a' b) = finsupp.single a' b
@[simp]
theorem
finsupp.erase_zero
{α : Type u_1}
{M : Type u_5}
[has_zero M]
(a : α) :
finsupp.erase a 0 = 0
Declarations about sum and prod
In most of this section, the domain β is assumed to be an add_monoid.
def
finsupp.sum
{α : Type u_1}
{M : Type u_5}
{N : Type u_7}
[has_zero M]
[add_comm_monoid N]
(f : α →₀ M)
(g : α → M → N) :
N
sum f g is the sum of g a (f a) over the support of f.
def
finsupp.prod
{α : Type u_1}
{M : Type u_5}
{N : Type u_7}
[has_zero M]
[comm_monoid N]
(f : α →₀ M)
(g : α → M → N) :
N
prod f g is the product of g a (f a) over the support of f.
theorem
finsupp.prod_fintype
{α : Type u_1}
{M : Type u_5}
{N : Type u_7}
[has_zero M]
[comm_monoid N]
[fintype α]
(f : α →₀ M)
(g : α → M → N)
(h : ∀ (i : α), g i 0 = 1) :
theorem
finsupp.sum_fintype
{α : Type u_1}
{M : Type u_5}
{N : Type u_7}
[has_zero M]
[add_comm_monoid N]
[fintype α]
(f : α →₀ M)
(g : α → M → N)
(h : ∀ (i : α), g i 0 = 0) :
@[simp]
theorem
finsupp.prod_single_index
{α : Type u_1}
{M : Type u_5}
{N : Type u_7}
[has_zero M]
[comm_monoid N]
{a : α}
{b : M}
{h : α → M → N}
(h_zero : h a 0 = 1) :
(finsupp.single a b).prod h = h a b
@[simp]
theorem
finsupp.sum_single_index
{α : Type u_1}
{M : Type u_5}
{N : Type u_7}
[has_zero M]
[add_comm_monoid N]
{a : α}
{b : M}
{h : α → M → N}
(h_zero : h a 0 = 0) :
(finsupp.single a b).sum h = h a b
theorem
finsupp.sum_map_range_index
{α : Type u_1}
{M : Type u_5}
{M' : Type u_6}
{N : Type u_7}
[has_zero M]
[has_zero M']
[add_comm_monoid N]
{f : M → M'}
{hf : f 0 = 0}
{g : α →₀ M}
{h : α → M' → N}
(h0 : ∀ (a : α), h a 0 = 0) :
(finsupp.map_range f hf g).sum h = g.sum (λ (a : α) (b : M), h a (f b))
theorem
finsupp.prod_map_range_index
{α : Type u_1}
{M : Type u_5}
{M' : Type u_6}
{N : Type u_7}
[has_zero M]
[has_zero M']
[comm_monoid N]
{f : M → M'}
{hf : f 0 = 0}
{g : α →₀ M}
{h : α → M' → N}
(h0 : ∀ (a : α), h a 0 = 1) :
(finsupp.map_range f hf g).prod h = g.prod (λ (a : α) (b : M), h a (f b))
@[simp]
theorem
finsupp.prod_zero_index
{α : Type u_1}
{M : Type u_5}
{N : Type u_7}
[has_zero M]
[comm_monoid N]
{h : α → M → N} :
@[simp]
theorem
finsupp.sum_zero_index
{α : Type u_1}
{M : Type u_5}
{N : Type u_7}
[has_zero M]
[add_comm_monoid N]
{h : α → M → N} :
theorem
finsupp.prod_comm
{α : Type u_1}
{β : Type u_2}
{M : Type u_5}
{M' : Type u_6}
{N : Type u_7}
[has_zero M]
[has_zero M']
[comm_monoid N]
(f : α →₀ M)
(g : β →₀ M')
(h : α → M → β → M' → N) :
theorem
finsupp.sum_comm
{α : Type u_1}
{β : Type u_2}
{M : Type u_5}
{M' : Type u_6}
{N : Type u_7}
[has_zero M]
[has_zero M']
[add_comm_monoid N]
(f : α →₀ M)
(g : β →₀ M')
(h : α → M → β → M' → N) :
@[simp]
theorem
finsupp.prod_ite_eq'
{α : Type u_1}
{M : Type u_5}
{N : Type u_7}
[has_zero M]
[comm_monoid N]
(f : α →₀ M)
(a : α)
(b : α → M → N) :
A restatement of prod_ite_eq with the equality test reversed.
@[simp]
theorem
finsupp.sum_ite_eq'
{α : Type u_1}
{M : Type u_5}
{N : Type u_7}
[has_zero M]
[add_comm_monoid N]
(f : α →₀ M)
(a : α)
(b : α → M → N) :
A restatement of sum_ite_eq with the equality test reversed.
theorem
finsupp.on_finset_prod
{α : Type u_1}
{M : Type u_5}
{N : Type u_7}
[has_zero M]
[comm_monoid N]
{s : finset α}
{f : α → M}
{g : α → M → N}
(hf : ∀ (a : α), f a ≠ 0 → a ∈ s)
(hg : ∀ (a : α), g a 0 = 1) :
(finsupp.on_finset s f hf).prod g = ∏ (a : α) in s, g a (f a)
If g maps a second argument of 0 to 1, then multiplying it over the
result of on_finset is the same as multiplying it over the original
finset.
theorem
finsupp.on_finset_sum
{α : Type u_1}
{M : Type u_5}
{N : Type u_7}
[has_zero M]
[add_comm_monoid N]
{s : finset α}
{f : α → M}
{g : α → M → N}
(hf : ∀ (a : α), f a ≠ 0 → a ∈ s)
(hg : ∀ (a : α), g a 0 = 0) :
(finsupp.on_finset s f hf).sum g = ∑ (a : α) in s, g a (f a)
If g maps a second argument of 0 to 0, summing it over the
result of on_finset is the same as summing it over the original
finset.
Additive monoid structure on α →₀ M
@[instance]
Equations
- finsupp.has_add = {add := finsupp.zip_with has_add.add finsupp.has_add._proof_1}
@[simp]
@[simp]
theorem
finsupp.single_add
{α : Type u_1}
{M : Type u_5}
[add_monoid M]
{a : α}
{b₁ b₂ : M} :
finsupp.single a (b₁ + b₂) = finsupp.single a b₁ + finsupp.single a b₂
@[instance]
Equations
- finsupp.add_monoid = {add := has_add.add finsupp.has_add, add_assoc := _, zero := 0, zero_add := _, add_zero := _}
finsupp.single as an add_monoid_hom.
See finsupp.lsingle for the stronger version as a linear map.
Equations
- finsupp.single_add_hom a = {to_fun := finsupp.single a, map_zero' := _, map_add' := _}
@[simp]
theorem
finsupp.single_add_hom_apply
{α : Type u_1}
{M : Type u_5}
[add_monoid M]
(a : α)
(b : M) :
⇑(finsupp.single_add_hom a) b = finsupp.single a b
Evaluation of a function f : α →₀ M at a point as an additive monoid homomorphism.
See finsupp.lapply for the stronger version as a linear map.
@[simp]
theorem
finsupp.apply_add_hom_apply
{α : Type u_1}
{M : Type u_5}
[add_monoid M]
(a : α)
(g : α →₀ M) :
⇑(finsupp.apply_add_hom a) g = ⇑g a
theorem
finsupp.single_add_erase
{α : Type u_1}
{M : Type u_5}
[add_monoid M]
(a : α)
(f : α →₀ M) :
finsupp.single a (⇑f a) + finsupp.erase a f = f
theorem
finsupp.erase_add_single
{α : Type u_1}
{M : Type u_5}
[add_monoid M]
(a : α)
(f : α →₀ M) :
finsupp.erase a f + finsupp.single a (⇑f a) = f
@[simp]
theorem
finsupp.erase_add
{α : Type u_1}
{M : Type u_5}
[add_monoid M]
(a : α)
(f f' : α →₀ M) :
finsupp.erase a (f + f') = finsupp.erase a f + finsupp.erase a f'
theorem
finsupp.induction
{α : Type u_1}
{M : Type u_5}
[add_monoid M]
{p : (α →₀ M) → Prop}
(f : α →₀ M)
(h0 : p 0)
(ha : ∀ (a : α) (b : M) (f : α →₀ M), a ∉ f.support → b ≠ 0 → p f → p (finsupp.single a b + f)) :
p f
theorem
finsupp.induction₂
{α : Type u_1}
{M : Type u_5}
[add_monoid M]
{p : (α →₀ M) → Prop}
(f : α →₀ M)
(h0 : p 0)
(ha : ∀ (a : α) (b : M) (f : α →₀ M), a ∉ f.support → b ≠ 0 → p f → p (f + finsupp.single a b)) :
p f
@[simp]
theorem
finsupp.add_closure_Union_range_single
{α : Type u_1}
{M : Type u_5}
[add_monoid M] :
add_submonoid.closure (⋃ (a : α), set.range (finsupp.single a)) = ⊤
theorem
finsupp.add_hom_ext
{α : Type u_1}
{M : Type u_5}
{N : Type u_7}
[add_monoid M]
[add_monoid N]
⦃f g : (α →₀ M) →+ N⦄
(H : ∀ (x : α) (y : M), ⇑f (finsupp.single x y) = ⇑g (finsupp.single x y)) :
f = g
If two additive homomorphisms from α →₀ M are equal on each single a b, then
they are equal.
@[ext]
theorem
finsupp.add_hom_ext'
{α : Type u_1}
{M : Type u_5}
{N : Type u_7}
[add_monoid M]
[add_monoid N]
⦃f g : (α →₀ M) →+ N⦄
(H : ∀ (x : α), f.comp (finsupp.single_add_hom x) = g.comp (finsupp.single_add_hom x)) :
f = g
If two additive homomorphisms from α →₀ M are equal on each single a b, then
they are equal.
We formulate this using equality of add_monoid_homs so that ext tactic can apply a type-specific
extensionality lemma after this one. E.g., if the fiber M is ℕ or ℤ, then it suffices to
verify f (single a 1) = g (single a 1).
theorem
finsupp.mul_hom_ext
{α : Type u_1}
{M : Type u_5}
{N : Type u_7}
[add_monoid M]
[monoid N]
⦃f g : multiplicative (α →₀ M) →* N⦄
(H : ∀ (x : α) (y : M), ⇑f (⇑multiplicative.of_add (finsupp.single x y)) = ⇑g (⇑multiplicative.of_add (finsupp.single x y))) :
f = g
@[ext]
theorem
finsupp.mul_hom_ext'
{α : Type u_1}
{M : Type u_5}
{N : Type u_7}
[add_monoid M]
[monoid N]
{f g : multiplicative (α →₀ M) →* N}
(H : ∀ (x : α), f.comp (⇑add_monoid_hom.to_multiplicative (finsupp.single_add_hom x)) = g.comp (⇑add_monoid_hom.to_multiplicative (finsupp.single_add_hom x))) :
f = g
theorem
finsupp.map_range_add
{α : Type u_1}
{M : Type u_5}
{N : Type u_7}
[add_monoid M]
[add_monoid N]
{f : M → N}
{hf : f 0 = 0}
(hf' : ∀ (x y : M), f (x + y) = f x + f y)
(v₁ v₂ : α →₀ M) :
finsupp.map_range f hf (v₁ + v₂) = finsupp.map_range f hf v₁ + finsupp.map_range f hf v₂
theorem
mul_equiv.map_finsupp_prod
{α : Type u_1}
{M : Type u_5}
{N : Type u_7}
{P : Type u_8}
[has_zero M]
[comm_monoid N]
[comm_monoid P]
(h : N ≃* P)
(f : α →₀ M)
(g : α → M → N) :
theorem
add_equiv.map_finsupp_sum
{α : Type u_1}
{M : Type u_5}
{N : Type u_7}
{P : Type u_8}
[has_zero M]
[add_comm_monoid N]
[add_comm_monoid P]
(h : N ≃+ P)
(f : α →₀ M)
(g : α → M → N) :
theorem
add_monoid_hom.map_finsupp_sum
{α : Type u_1}
{M : Type u_5}
{N : Type u_7}
{P : Type u_8}
[has_zero M]
[add_comm_monoid N]
[add_comm_monoid P]
(h : N →+ P)
(f : α →₀ M)
(g : α → M → N) :
theorem
monoid_hom.map_finsupp_prod
{α : Type u_1}
{M : Type u_5}
{N : Type u_7}
{P : Type u_8}
[has_zero M]
[comm_monoid N]
[comm_monoid P]
(h : N →* P)
(f : α →₀ M)
(g : α → M → N) :
theorem
ring_hom.map_finsupp_prod
{α : Type u_1}
{M : Type u_5}
{R : Type u_11}
{S : Type u_12}
[has_zero M]
[comm_semiring R]
[comm_semiring S]
(h : R →+* S)
(f : α →₀ M)
(g : α → M → R) :
@[instance]
Equations
- finsupp.nat_sub = {sub := finsupp.zip_with (λ (m n : ℕ), m - n) finsupp.nat_sub._proof_1}
@[simp]
theorem
finsupp.single_sub
{α : Type u_1}
{a : α}
{n₁ n₂ : ℕ} :
finsupp.single a (n₁ - n₂) = finsupp.single a n₁ - finsupp.single a n₂
theorem
finsupp.sub_single_one_add
{α : Type u_1}
{a : α}
{u u' : α →₀ ℕ}
(h : ⇑u a ≠ 0) :
u - finsupp.single a 1 + u' = u + u' - finsupp.single a 1
theorem
finsupp.add_sub_single_one
{α : Type u_1}
{a : α}
{u u' : α →₀ ℕ}
(h : ⇑u' a ≠ 0) :
u + (u' - finsupp.single a 1) = u + u' - finsupp.single a 1
@[instance]
def
finsupp.add_comm_monoid
{α : Type u_1}
{M : Type u_5}
[add_comm_monoid M] :
add_comm_monoid (α →₀ M)
Equations
- finsupp.add_comm_monoid = {add := add_monoid.add finsupp.add_monoid, add_assoc := _, zero := add_monoid.zero finsupp.add_monoid, zero_add := _, add_zero := _, add_comm := _}
@[instance]
Equations
- finsupp.has_sub = {sub := finsupp.zip_with has_sub.sub finsupp.has_sub._proof_1}
@[instance]
Equations
- finsupp.add_group = {add := add_monoid.add finsupp.add_monoid, add_assoc := _, zero := add_monoid.zero finsupp.add_monoid, zero_add := _, add_zero := _, neg := finsupp.map_range has_neg.neg neg_zero, sub := has_sub.sub finsupp.has_sub, sub_eq_add_neg := _, add_left_neg := _}
@[instance]
def
finsupp.add_comm_group
{α : Type u_1}
{G : Type u_9}
[add_comm_group G] :
add_comm_group (α →₀ G)
Equations
- finsupp.add_comm_group = {add := add_group.add finsupp.add_group, add_assoc := _, zero := add_group.zero finsupp.add_group, zero_add := _, add_zero := _, neg := add_group.neg finsupp.add_group, sub := add_group.sub finsupp.add_group, sub_eq_add_neg := _, add_left_neg := _, add_comm := _}
theorem
finsupp.single_multiset_sum
{α : Type u_1}
{M : Type u_5}
[add_comm_monoid M]
(s : multiset M)
(a : α) :
finsupp.single a s.sum = (multiset.map (finsupp.single a) s).sum
theorem
finsupp.single_finset_sum
{α : Type u_1}
{ι : Type u_4}
{M : Type u_5}
[add_comm_monoid M]
(s : finset ι)
(f : ι → M)
(a : α) :
finsupp.single a (∑ (b : ι) in s, f b) = ∑ (b : ι) in s, finsupp.single a (f b)
theorem
finsupp.single_sum
{α : Type u_1}
{ι : Type u_4}
{M : Type u_5}
{N : Type u_7}
[has_zero M]
[add_comm_monoid N]
(s : ι →₀ M)
(f : ι → M → N)
(a : α) :
finsupp.single a (s.sum f) = s.sum (λ (d : ι) (c : M), finsupp.single a (f d c))
@[simp]
theorem
finsupp.sum_zero
{α : Type u_1}
{M : Type u_5}
{N : Type u_7}
[has_zero M]
[add_comm_monoid N]
{f : α →₀ M} :
@[simp]
theorem
finsupp.prod_inv
{α : Type u_1}
{M : Type u_5}
{G : Type u_9}
[has_zero M]
[comm_group G]
{f : α →₀ M}
{h : α → M → G} :
@[simp]
theorem
finsupp.sum_neg
{α : Type u_1}
{M : Type u_5}
{G : Type u_9}
[has_zero M]
[add_comm_group G]
{f : α →₀ M}
{h : α → M → G} :
theorem
finsupp.sum_add_index
{α : Type u_1}
{M : Type u_5}
{N : Type u_7}
[add_comm_monoid M]
[add_comm_monoid N]
{f g : α →₀ M}
{h : α → M → N}
(h_zero : ∀ (a : α), h a 0 = 0)
(h_add : ∀ (a : α) (b₁ b₂ : M), h a (b₁ + b₂) = h a b₁ + h a b₂) :
theorem
finsupp.prod_add_index
{α : Type u_1}
{M : Type u_5}
{N : Type u_7}
[add_comm_monoid M]
[comm_monoid N]
{f g : α →₀ M}
{h : α → M → N}
(h_zero : ∀ (a : α), h a 0 = 1)
(h_add : ∀ (a : α) (b₁ b₂ : M), h a (b₁ + b₂) = (h a b₁) * h a b₂) :
@[simp]
theorem
finsupp.sum_add_index'
{α : Type u_1}
{M : Type u_5}
{N : Type u_7}
[add_comm_monoid M]
[add_comm_monoid N]
{f g : α →₀ M}
(h : α → M →+ N) :
@[simp]
theorem
finsupp.prod_add_index'
{α : Type u_1}
{M : Type u_5}
{N : Type u_7}
[add_comm_monoid M]
[comm_monoid N]
{f g : α →₀ M}
(h : α → multiplicative M →* N) :
(f + g).prod (λ (a : α) (b : M), ⇑(h a) (⇑multiplicative.of_add b)) = (f.prod (λ (a : α) (b : M), ⇑(h a) (⇑multiplicative.of_add b))) * g.prod (λ (a : α) (b : M), ⇑(h a) (⇑multiplicative.of_add b))
def
finsupp.lift_add_hom
{α : Type u_1}
{M : Type u_5}
{N : Type u_7}
[add_comm_monoid M]
[add_comm_monoid N] :
The canonical isomorphism between families of additive monoid homomorphisms α → (M →+ N)
and monoid homomorphisms (α →₀ M) →+ N.
@[simp]
theorem
finsupp.lift_add_hom_apply
{α : Type u_1}
{M : Type u_5}
{N : Type u_7}
[add_comm_monoid M]
[add_comm_monoid N]
(F : α → M →+ N)
(f : α →₀ M) :
@[simp]
theorem
finsupp.lift_add_hom_symm_apply
{α : Type u_1}
{M : Type u_5}
{N : Type u_7}
[add_comm_monoid M]
[add_comm_monoid N]
(F : (α →₀ M) →+ N)
(x : α) :
⇑(finsupp.lift_add_hom.symm) F x = F.comp (finsupp.single_add_hom x)
theorem
finsupp.lift_add_hom_symm_apply_apply
{α : Type u_1}
{M : Type u_5}
{N : Type u_7}
[add_comm_monoid M]
[add_comm_monoid N]
(F : (α →₀ M) →+ N)
(x : α)
(y : M) :
⇑(⇑(finsupp.lift_add_hom.symm) F x) y = ⇑F (finsupp.single x y)
@[simp]
@[simp]
theorem
finsupp.sum_single
{α : Type u_1}
{M : Type u_5}
[add_comm_monoid M]
(f : α →₀ M) :
f.sum finsupp.single = f
@[simp]
theorem
finsupp.lift_add_hom_apply_single
{α : Type u_1}
{M : Type u_5}
{N : Type u_7}
[add_comm_monoid M]
[add_comm_monoid N]
(f : α → M →+ N)
(a : α)
(b : M) :
⇑(⇑finsupp.lift_add_hom f) (finsupp.single a b) = ⇑(f a) b
@[simp]
theorem
finsupp.lift_add_hom_comp_single
{α : Type u_1}
{M : Type u_5}
{N : Type u_7}
[add_comm_monoid M]
[add_comm_monoid N]
(f : α → M →+ N)
(a : α) :
(⇑finsupp.lift_add_hom f).comp (finsupp.single_add_hom a) = f a
theorem
finsupp.comp_lift_add_hom
{α : Type u_1}
{M : Type u_5}
{N : Type u_7}
{P : Type u_8}
[add_comm_monoid M]
[add_comm_monoid N]
[add_comm_monoid P]
(g : N →+ P)
(f : α → M →+ N) :
g.comp (⇑finsupp.lift_add_hom f) = ⇑finsupp.lift_add_hom (λ (a : α), g.comp (f a))
theorem
finsupp.sum_sub_index
{α : Type u_1}
{β : Type u_2}
{γ : Type u_3}
[add_comm_group β]
[add_comm_group γ]
{f g : α →₀ β}
{h : α → β → γ}
(h_sub : ∀ (a : α) (b₁ b₂ : β), h a (b₁ - b₂) = h a b₁ - h a b₂) :
theorem
finsupp.sum_emb_domain
{α : Type u_1}
{β : Type u_2}
{M : Type u_5}
{N : Type u_7}
[has_zero M]
[add_comm_monoid N]
{v : α →₀ M}
{f : α ↪ β}
{g : β → M → N} :
(finsupp.emb_domain f v).sum g = v.sum (λ (a : α) (b : M), g (⇑f a) b)
theorem
finsupp.prod_emb_domain
{α : Type u_1}
{β : Type u_2}
{M : Type u_5}
{N : Type u_7}
[has_zero M]
[comm_monoid N]
{v : α →₀ M}
{f : α ↪ β}
{g : β → M → N} :
(finsupp.emb_domain f v).prod g = v.prod (λ (a : α) (b : M), g (⇑f a) b)
theorem
finsupp.prod_finset_sum_index
{α : Type u_1}
{ι : Type u_4}
{M : Type u_5}
{N : Type u_7}
[add_comm_monoid M]
[comm_monoid N]
{s : finset ι}
{g : ι → α →₀ M}
{h : α → M → N}
(h_zero : ∀ (a : α), h a 0 = 1)
(h_add : ∀ (a : α) (b₁ b₂ : M), h a (b₁ + b₂) = (h a b₁) * h a b₂) :
theorem
finsupp.sum_finset_sum_index
{α : Type u_1}
{ι : Type u_4}
{M : Type u_5}
{N : Type u_7}
[add_comm_monoid M]
[add_comm_monoid N]
{s : finset ι}
{g : ι → α →₀ M}
{h : α → M → N}
(h_zero : ∀ (a : α), h a 0 = 0)
(h_add : ∀ (a : α) (b₁ b₂ : M), h a (b₁ + b₂) = h a b₁ + h a b₂) :
theorem
finsupp.sum_sum_index
{α : Type u_1}
{β : Type u_2}
{M : Type u_5}
{N : Type u_7}
{P : Type u_8}
[add_comm_monoid M]
[add_comm_monoid N]
[add_comm_monoid P]
{f : α →₀ M}
{g : α → M → β →₀ N}
{h : β → N → P}
(h_zero : ∀ (a : β), h a 0 = 0)
(h_add : ∀ (a : β) (b₁ b₂ : N), h a (b₁ + b₂) = h a b₁ + h a b₂) :
theorem
finsupp.prod_sum_index
{α : Type u_1}
{β : Type u_2}
{M : Type u_5}
{N : Type u_7}
{P : Type u_8}
[add_comm_monoid M]
[add_comm_monoid N]
[comm_monoid P]
{f : α →₀ M}
{g : α → M → β →₀ N}
{h : β → N → P}
(h_zero : ∀ (a : β), h a 0 = 1)
(h_add : ∀ (a : β) (b₁ b₂ : N), h a (b₁ + b₂) = (h a b₁) * h a b₂) :
theorem
finsupp.multiset_sum_sum_index
{α : Type u_1}
{M : Type u_5}
{N : Type u_7}
[add_comm_monoid M]
[add_comm_monoid N]
(f : multiset (α →₀ M))
(h : α → M → N)
(h₀ : ∀ (a : α), h a 0 = 0)
(h₁ : ∀ (a : α) (b₁ b₂ : M), h a (b₁ + b₂) = h a b₁ + h a b₂) :
theorem
finsupp.multiset_map_sum
{α : Type u_1}
{β : Type u_2}
{γ : Type u_3}
{M : Type u_5}
[has_zero M]
{f : α →₀ M}
{m : β → γ}
{h : α → M → multiset β} :
multiset.map m (f.sum h) = f.sum (λ (a : α) (b : M), multiset.map m (h a b))
def
finsupp.map_range.add_monoid_hom
{α : Type u_1}
{M : Type u_5}
{N : Type u_7}
[add_comm_monoid M]
[add_comm_monoid N]
(f : M →+ N) :
Composition with a fixed additive homomorphism is itself an additive homomorphism on functions.
Equations
- finsupp.map_range.add_monoid_hom f = {to_fun := finsupp.map_range ⇑f _, map_zero' := _, map_add' := _}
theorem
finsupp.map_range_multiset_sum
{α : Type u_1}
{M : Type u_5}
{N : Type u_7}
[add_comm_monoid M]
[add_comm_monoid N]
(f : M →+ N)
(m : multiset (α →₀ M)) :
finsupp.map_range ⇑f _ m.sum = (multiset.map (λ (x : α →₀ M), finsupp.map_range ⇑f _ x) m).sum
theorem
finsupp.map_range_finset_sum
{α : Type u_1}
{ι : Type u_4}
{M : Type u_5}
{N : Type u_7}
[add_comm_monoid M]
[add_comm_monoid N]
(f : M →+ N)
(s : finset ι)
(g : ι → α →₀ M) :
finsupp.map_range ⇑f _ (∑ (x : ι) in s, g x) = ∑ (x : ι) in s, finsupp.map_range ⇑f _ (g x)
Declarations about map_domain
def
finsupp.map_domain
{α : Type u_1}
{β : Type u_2}
{M : Type u_5}
[add_comm_monoid M]
(f : α → β)
(v : α →₀ M) :
β →₀ M
Given f : α → β and v : α →₀ M, map_domain f v : β →₀ M
is the finitely supported function whose value at a : β is the sum
of v x over all x such that f x = a.
Equations
- finsupp.map_domain f v = v.sum (λ (a : α), finsupp.single (f a))
theorem
finsupp.map_domain_apply
{α : Type u_1}
{β : Type u_2}
{M : Type u_5}
[add_comm_monoid M]
{f : α → β}
(hf : function.injective f)
(x : α →₀ M)
(a : α) :
⇑(finsupp.map_domain f x) (f a) = ⇑x a
theorem
finsupp.map_domain_notin_range
{α : Type u_1}
{β : Type u_2}
{M : Type u_5}
[add_comm_monoid M]
{f : α → β}
(x : α →₀ M)
(a : β)
(h : a ∉ set.range f) :
⇑(finsupp.map_domain f x) a = 0
theorem
finsupp.map_domain_id
{α : Type u_1}
{M : Type u_5}
[add_comm_monoid M]
{v : α →₀ M} :
finsupp.map_domain id v = v
theorem
finsupp.map_domain_comp
{α : Type u_1}
{β : Type u_2}
{γ : Type u_3}
{M : Type u_5}
[add_comm_monoid M]
{v : α →₀ M}
{f : α → β}
{g : β → γ} :
finsupp.map_domain (g ∘ f) v = finsupp.map_domain g (finsupp.map_domain f v)
theorem
finsupp.map_domain_single
{α : Type u_1}
{β : Type u_2}
{M : Type u_5}
[add_comm_monoid M]
{f : α → β}
{a : α}
{b : M} :
finsupp.map_domain f (finsupp.single a b) = finsupp.single (f a) b
@[simp]
theorem
finsupp.map_domain_zero
{α : Type u_1}
{β : Type u_2}
{M : Type u_5}
[add_comm_monoid M]
{f : α → β} :
finsupp.map_domain f 0 = 0
theorem
finsupp.map_domain_congr
{α : Type u_1}
{β : Type u_2}
{M : Type u_5}
[add_comm_monoid M]
{v : α →₀ M}
{f g : α → β}
(h : ∀ (x : α), x ∈ v.support → f x = g x) :
finsupp.map_domain f v = finsupp.map_domain g v
theorem
finsupp.map_domain_add
{α : Type u_1}
{β : Type u_2}
{M : Type u_5}
[add_comm_monoid M]
{v₁ v₂ : α →₀ M}
{f : α → β} :
finsupp.map_domain f (v₁ + v₂) = finsupp.map_domain f v₁ + finsupp.map_domain f v₂
theorem
finsupp.map_domain_finset_sum
{α : Type u_1}
{β : Type u_2}
{ι : Type u_4}
{M : Type u_5}
[add_comm_monoid M]
{f : α → β}
{s : finset ι}
{v : ι → α →₀ M} :
finsupp.map_domain f (∑ (i : ι) in s, v i) = ∑ (i : ι) in s, finsupp.map_domain f (v i)
theorem
finsupp.map_domain_sum
{α : Type u_1}
{β : Type u_2}
{M : Type u_5}
{N : Type u_7}
[add_comm_monoid M]
[has_zero N]
{f : α → β}
{s : α →₀ N}
{v : α → N → α →₀ M} :
finsupp.map_domain f (s.sum v) = s.sum (λ (a : α) (b : N), finsupp.map_domain f (v a b))
theorem
finsupp.map_domain_support
{α : Type u_1}
{β : Type u_2}
{M : Type u_5}
[add_comm_monoid M]
{f : α → β}
{s : α →₀ M} :
(finsupp.map_domain f s).support ⊆ finset.image f s.support
theorem
finsupp.sum_map_domain_index
{α : Type u_1}
{β : Type u_2}
{M : Type u_5}
{N : Type u_7}
[add_comm_monoid M]
[add_comm_monoid N]
{f : α → β}
{s : α →₀ M}
{h : β → M → N}
(h_zero : ∀ (a : β), h a 0 = 0)
(h_add : ∀ (a : β) (b₁ b₂ : M), h a (b₁ + b₂) = h a b₁ + h a b₂) :
(finsupp.map_domain f s).sum h = s.sum (λ (a : α) (b : M), h (f a) b)
theorem
finsupp.prod_map_domain_index
{α : Type u_1}
{β : Type u_2}
{M : Type u_5}
{N : Type u_7}
[add_comm_monoid M]
[comm_monoid N]
{f : α → β}
{s : α →₀ M}
{h : β → M → N}
(h_zero : ∀ (a : β), h a 0 = 1)
(h_add : ∀ (a : β) (b₁ b₂ : M), h a (b₁ + b₂) = (h a b₁) * h a b₂) :
(finsupp.map_domain f s).prod h = s.prod (λ (a : α) (b : M), h (f a) b)
theorem
finsupp.emb_domain_eq_map_domain
{α : Type u_1}
{β : Type u_2}
{M : Type u_5}
[add_comm_monoid M]
(f : α ↪ β)
(v : α →₀ M) :
finsupp.emb_domain f v = finsupp.map_domain ⇑f v
theorem
finsupp.prod_map_domain_index_inj
{α : Type u_1}
{β : Type u_2}
{M : Type u_5}
{N : Type u_7}
[add_comm_monoid M]
[comm_monoid N]
{f : α → β}
{s : α →₀ M}
{h : β → M → N}
(hf : function.injective f) :
(finsupp.map_domain f s).prod h = s.prod (λ (a : α) (b : M), h (f a) b)
theorem
finsupp.sum_map_domain_index_inj
{α : Type u_1}
{β : Type u_2}
{M : Type u_5}
{N : Type u_7}
[add_comm_monoid M]
[add_comm_monoid N]
{f : α → β}
{s : α →₀ M}
{h : β → M → N}
(hf : function.injective f) :
(finsupp.map_domain f s).sum h = s.sum (λ (a : α) (b : M), h (f a) b)
theorem
finsupp.map_domain_injective
{α : Type u_1}
{β : Type u_2}
{M : Type u_5}
[add_comm_monoid M]
{f : α → β}
(hf : function.injective f) :
Declarations about comap_domain
def
finsupp.comap_domain
{α : Type u_1}
{β : Type u_2}
{M : Type u_5}
[has_zero M]
(f : α → β)
(l : β →₀ M)
(hf : set.inj_on f (f ⁻¹' ↑(l.support))) :
α →₀ M
Given f : α → β, l : β →₀ M and a proof hf that f is injective on
the preimage of l.support, comap_domain f l hf is the finitely supported function
from α to M given by composing l with f.
Equations
- finsupp.comap_domain f l hf = {support := l.support.preimage f hf, to_fun := λ (a : α), ⇑l (f a), mem_support_to_fun := _}
@[simp]
theorem
finsupp.comap_domain_apply
{α : Type u_1}
{β : Type u_2}
{M : Type u_5}
[has_zero M]
(f : α → β)
(l : β →₀ M)
(hf : set.inj_on f (f ⁻¹' ↑(l.support)))
(a : α) :
⇑(finsupp.comap_domain f l hf) a = ⇑l (f a)
theorem
finsupp.sum_comap_domain
{α : Type u_1}
{β : Type u_2}
{M : Type u_5}
{N : Type u_7}
[has_zero M]
[add_comm_monoid N]
(f : α → β)
(l : β →₀ M)
(g : β → M → N)
(hf : set.bij_on f (f ⁻¹' ↑(l.support)) ↑(l.support)) :
(finsupp.comap_domain f l _).sum (g ∘ f) = l.sum g
theorem
finsupp.eq_zero_of_comap_domain_eq_zero
{α : Type u_1}
{β : Type u_2}
{M : Type u_5}
[add_comm_monoid M]
(f : α → β)
(l : β →₀ M)
(hf : set.bij_on f (f ⁻¹' ↑(l.support)) ↑(l.support)) :
finsupp.comap_domain f l _ = 0 → l = 0
theorem
finsupp.map_domain_comap_domain
{α : Type u_1}
{β : Type u_2}
{M : Type u_5}
[add_comm_monoid M]
(f : α → β)
(l : β →₀ M)
(hf : function.injective f)
(hl : ↑(l.support) ⊆ set.range f) :
finsupp.map_domain f (finsupp.comap_domain f l _) = l
Declarations about filter
filter p f is the function which is f a if p a is true and 0 otherwise.
Equations
- finsupp.filter p f = {support := finset.filter (λ (a : α), p a) f.support, to_fun := λ (a : α), ite (p a) (⇑f a) 0, mem_support_to_fun := _}
theorem
finsupp.filter_apply
{α : Type u_1}
{M : Type u_5}
[has_zero M]
(p : α → Prop)
(f : α →₀ M)
(a : α) :
⇑(finsupp.filter p f) a = ite (p a) (⇑f a) 0
theorem
finsupp.filter_eq_indicator
{α : Type u_1}
{M : Type u_5}
[has_zero M]
(p : α → Prop)
(f : α →₀ M) :
⇑(finsupp.filter p f) = {x : α | p x}.indicator ⇑f
@[simp]
theorem
finsupp.filter_apply_pos
{α : Type u_1}
{M : Type u_5}
[has_zero M]
(p : α → Prop)
(f : α →₀ M)
{a : α}
(h : p a) :
⇑(finsupp.filter p f) a = ⇑f a
@[simp]
theorem
finsupp.filter_apply_neg
{α : Type u_1}
{M : Type u_5}
[has_zero M]
(p : α → Prop)
(f : α →₀ M)
{a : α}
(h : ¬p a) :
⇑(finsupp.filter p f) a = 0
@[simp]
theorem
finsupp.support_filter
{α : Type u_1}
{M : Type u_5}
[has_zero M]
(p : α → Prop)
(f : α →₀ M) :
(finsupp.filter p f).support = finset.filter p f.support
theorem
finsupp.filter_zero
{α : Type u_1}
{M : Type u_5}
[has_zero M]
(p : α → Prop) :
finsupp.filter p 0 = 0
@[simp]
theorem
finsupp.filter_single_of_pos
{α : Type u_1}
{M : Type u_5}
[has_zero M]
(p : α → Prop)
{a : α}
{b : M}
(h : p a) :
finsupp.filter p (finsupp.single a b) = finsupp.single a b
@[simp]
theorem
finsupp.filter_single_of_neg
{α : Type u_1}
{M : Type u_5}
[has_zero M]
(p : α → Prop)
{a : α}
{b : M}
(h : ¬p a) :
finsupp.filter p (finsupp.single a b) = 0
theorem
finsupp.filter_pos_add_filter_neg
{α : Type u_1}
{M : Type u_5}
[add_monoid M]
(f : α →₀ M)
(p : α → Prop) :
finsupp.filter p f + finsupp.filter (λ (a : α), ¬p a) f = f
Declarations about frange
frange f is the image of f on the support of f.
Equations
- f.frange = finset.image ⇑f f.support
theorem
finsupp.frange_single
{α : Type u_1}
{M : Type u_5}
[has_zero M]
{x : α}
{y : M} :
(finsupp.single x y).frange ⊆ {y}
Declarations about subtype_domain
subtype_domain p f is the restriction of the finitely supported function
f to the subtype p.
Equations
- finsupp.subtype_domain p f = {support := finset.subtype p f.support, to_fun := ⇑f ∘ coe, mem_support_to_fun := _}
@[simp]
theorem
finsupp.support_subtype_domain
{α : Type u_1}
{M : Type u_5}
[has_zero M]
{p : α → Prop}
{f : α →₀ M} :
(finsupp.subtype_domain p f).support = finset.subtype p f.support
@[simp]
theorem
finsupp.subtype_domain_apply
{α : Type u_1}
{M : Type u_5}
[has_zero M]
{p : α → Prop}
{a : subtype p}
{v : α →₀ M} :
⇑(finsupp.subtype_domain p v) a = ⇑v a.val
@[simp]
theorem
finsupp.subtype_domain_zero
{α : Type u_1}
{M : Type u_5}
[has_zero M]
{p : α → Prop} :
finsupp.subtype_domain p 0 = 0
theorem
finsupp.subtype_domain_eq_zero_iff'
{α : Type u_1}
{M : Type u_5}
[has_zero M]
{p : α → Prop}
{f : α →₀ M} :
finsupp.subtype_domain p f = 0 ↔ ∀ (x : α), p x → ⇑f x = 0
theorem
finsupp.subtype_domain_eq_zero_iff
{α : Type u_1}
{M : Type u_5}
[has_zero M]
{p : α → Prop}
{f : α →₀ M}
(hf : ∀ (x : α), x ∈ f.support → p x) :
finsupp.subtype_domain p f = 0 ↔ f = 0
theorem
finsupp.sum_subtype_domain_index
{α : Type u_1}
{M : Type u_5}
{N : Type u_7}
[has_zero M]
{p : α → Prop}
[add_comm_monoid N]
{v : α →₀ M}
{h : α → M → N}
(hp : ∀ (x : α), x ∈ v.support → p x) :
theorem
finsupp.prod_subtype_domain_index
{α : Type u_1}
{M : Type u_5}
{N : Type u_7}
[has_zero M]
{p : α → Prop}
[comm_monoid N]
{v : α →₀ M}
{h : α → M → N}
(hp : ∀ (x : α), x ∈ v.support → p x) :
@[simp]
theorem
finsupp.subtype_domain_add
{α : Type u_1}
{M : Type u_5}
[add_monoid M]
{p : α → Prop}
{v v' : α →₀ M} :
finsupp.subtype_domain p (v + v') = finsupp.subtype_domain p v + finsupp.subtype_domain p v'
@[instance]
def
finsupp.subtype_domain.is_add_monoid_hom
{α : Type u_1}
{M : Type u_5}
[add_monoid M]
{p : α → Prop} :
finsupp.filter as an add_monoid_hom.
Equations
- finsupp.filter_add_hom p = {to_fun := finsupp.filter p, map_zero' := _, map_add' := _}
@[simp]
theorem
finsupp.filter_add
{α : Type u_1}
{M : Type u_5}
[add_monoid M]
{p : α → Prop}
{v v' : α →₀ M} :
finsupp.filter p (v + v') = finsupp.filter p v + finsupp.filter p v'
theorem
finsupp.subtype_domain_sum
{α : Type u_1}
{ι : Type u_4}
{M : Type u_5}
[add_comm_monoid M]
{p : α → Prop}
{s : finset ι}
{h : ι → α →₀ M} :
finsupp.subtype_domain p (∑ (c : ι) in s, h c) = ∑ (c : ι) in s, finsupp.subtype_domain p (h c)
theorem
finsupp.subtype_domain_finsupp_sum
{α : Type u_1}
{β : Type u_2}
{M : Type u_5}
{N : Type u_7}
[add_comm_monoid M]
{p : α → Prop}
[has_zero N]
{s : β →₀ N}
{h : β → N → α →₀ M} :
finsupp.subtype_domain p (s.sum h) = s.sum (λ (c : β) (d : N), finsupp.subtype_domain p (h c d))
theorem
finsupp.filter_sum
{α : Type u_1}
{ι : Type u_4}
{M : Type u_5}
[add_comm_monoid M]
{p : α → Prop}
(s : finset ι)
(f : ι → α →₀ M) :
finsupp.filter p (∑ (a : ι) in s, f a) = ∑ (a : ι) in s, finsupp.filter p (f a)
theorem
finsupp.filter_eq_sum
{α : Type u_1}
{M : Type u_5}
[add_comm_monoid M]
(p : α → Prop)
(f : α →₀ M) :
finsupp.filter p f = ∑ (i : α) in finset.filter p f.support, finsupp.single i (⇑f i)
@[simp]
theorem
finsupp.subtype_domain_neg
{α : Type u_1}
{G : Type u_9}
[add_group G]
{p : α → Prop}
{v : α →₀ G} :
finsupp.subtype_domain p (-v) = -finsupp.subtype_domain p v
@[simp]
theorem
finsupp.subtype_domain_sub
{α : Type u_1}
{G : Type u_9}
[add_group G]
{p : α → Prop}
{v v' : α →₀ G} :
finsupp.subtype_domain p (v - v') = finsupp.subtype_domain p v - finsupp.subtype_domain p v'
Given f : α →₀ ℕ, f.to_multiset is the multiset with multiplicities given by the values of
f on the elements of α. We define this function as an add_equiv.
@[simp]
theorem
finsupp.to_multiset_single
{α : Type u_1}
(a : α)
(n : ℕ) :
⇑finsupp.to_multiset (finsupp.single a n) = n •ℕ {a}
theorem
finsupp.card_to_multiset
{α : Type u_1}
(f : α →₀ ℕ) :
⇑multiset.card (⇑finsupp.to_multiset f) = f.sum (λ (a : α), id)
@[simp]
@[simp]
@[simp]
theorem
finsupp.count_to_multiset
{α : Type u_1}
(f : α →₀ ℕ)
(a : α) :
multiset.count a (⇑finsupp.to_multiset f) = ⇑f a
theorem
finsupp.mem_support_finset_sum
{α : Type u_1}
{ι : Type u_4}
{M : Type u_5}
[add_comm_monoid M]
{s : finset ι}
{h : ι → α →₀ M}
(a : α)
(ha : a ∈ (∑ (c : ι) in s, h c).support) :
@[simp]
Declarations about curry and uncurry
def
finsupp.curry
{α : Type u_1}
{β : Type u_2}
{M : Type u_5}
[add_comm_monoid M]
(f : α × β →₀ M) :
Given a finitely supported function f from a product type α × β to γ,
curry f is the "curried" finitely supported function from α to the type of
finitely supported functions from β to γ.
Equations
- f.curry = f.sum (λ (p : α × β) (c : M), finsupp.single p.fst (finsupp.single p.snd c))
theorem
finsupp.sum_curry_index
{α : Type u_1}
{β : Type u_2}
{M : Type u_5}
{N : Type u_7}
[add_comm_monoid M]
[add_comm_monoid N]
(f : α × β →₀ M)
(g : α → β → M → N)
(hg₀ : ∀ (a : α) (b : β), g a b 0 = 0)
(hg₁ : ∀ (a : α) (b : β) (c₀ c₁ : M), g a b (c₀ + c₁) = g a b c₀ + g a b c₁) :
def
finsupp.uncurry
{α : Type u_1}
{β : Type u_2}
{M : Type u_5}
[add_comm_monoid M]
(f : α →₀ β →₀ M) :
Given a finitely supported function f from α to the type of
finitely supported functions from β to M,
uncurry f is the "uncurried" finitely supported function from α × β to M.
finsupp_prod_equiv defines the equiv between ((α × β) →₀ M) and (α →₀ (β →₀ M)) given by
currying and uncurrying.
Equations
- finsupp.finsupp_prod_equiv = {to_fun := finsupp.curry _inst_1, inv_fun := finsupp.uncurry _inst_1, left_inv := _, right_inv := _}
theorem
finsupp.filter_curry
{α : Type u_1}
{β : Type u_2}
{M : Type u_5}
[add_comm_monoid M]
(f : α × β →₀ M)
(p : α → Prop) :
(finsupp.filter (λ (a : α × β), p a.fst) f).curry = finsupp.filter p f.curry
theorem
finsupp.support_curry
{α : Type u_1}
{β : Type u_2}
{M : Type u_5}
[add_comm_monoid M]
(f : α × β →₀ M) :
def
finsupp.comap_has_scalar
{α : Type u_1}
{M : Type u_5}
{G : Type u_9}
[group G]
[mul_action G α]
[add_comm_monoid M] :
has_scalar G (α →₀ M)
Scalar multiplication by a group element g, given by precomposition with the action of g⁻¹ on the domain.
Equations
- finsupp.comap_has_scalar = {smul := λ (g : G) (f : α →₀ M), finsupp.comap_domain (λ (a : α), g⁻¹ • a) f _}
def
finsupp.comap_mul_action
{α : Type u_1}
{M : Type u_5}
{G : Type u_9}
[group G]
[mul_action G α]
[add_comm_monoid M] :
mul_action G (α →₀ M)
Scalar multiplication by a group element, given by precomposition with the action of g⁻¹ on the domain, is multiplicative in g.
Equations
- finsupp.comap_mul_action = {to_has_scalar := finsupp.comap_has_scalar _inst_3, one_smul := _, mul_smul := _}
def
finsupp.comap_distrib_mul_action
{α : Type u_1}
{M : Type u_5}
{G : Type u_9}
[group G]
[mul_action G α]
[add_comm_monoid M] :
distrib_mul_action G (α →₀ M)
Scalar multiplication by a group element, given by precomposition with the action of g⁻¹ on the domain, is additive in the second argument.
Equations
- finsupp.comap_distrib_mul_action = {to_mul_action := finsupp.comap_mul_action _inst_3, smul_add := _, smul_zero := _}
def
finsupp.comap_distrib_mul_action_self
{M : Type u_5}
{G : Type u_9}
[group G]
[add_comm_monoid M] :
distrib_mul_action G (G →₀ M)
Scalar multiplication by a group element on finitely supported functions on a group, given by precomposition with the action of g⁻¹.
@[simp]
theorem
finsupp.comap_smul_single
{α : Type u_1}
{M : Type u_5}
{G : Type u_9}
[group G]
[mul_action G α]
[add_comm_monoid M]
(g : G)
(a : α)
(b : M) :
g • finsupp.single a b = finsupp.single (g • a) b
@[simp]
theorem
finsupp.comap_smul_apply
{α : Type u_1}
{M : Type u_5}
{G : Type u_9}
[group G]
[mul_action G α]
[add_comm_monoid M]
(g : G)
(f : α →₀ M)
(a : α) :
@[instance]
def
finsupp.has_scalar
{α : Type u_1}
{M : Type u_5}
{R : Type u_11}
[semiring R]
[add_comm_monoid M]
[semimodule R M] :
has_scalar R (α →₀ M)
Equations
- finsupp.has_scalar = {smul := λ (a : R) (v : α →₀ M), finsupp.map_range (has_scalar.smul a) _ v}
Throughout this section, some semiring arguments are specified with {} instead of [].
See note [implicit instance arguments].
@[simp]
theorem
finsupp.smul_apply'
(α : Type u_1)
(M : Type u_5)
{R : Type u_11}
{_x : semiring R}
[add_comm_monoid M]
[semimodule R M]
{a : α}
{b : R}
{v : α →₀ M} :
@[instance]
def
finsupp.semimodule
(α : Type u_1)
(M : Type u_5)
{R : Type u_11}
[semiring R]
[add_comm_monoid M]
[semimodule R M] :
semimodule R (α →₀ M)
Equations
- finsupp.semimodule α M = {to_distrib_mul_action := {to_mul_action := {to_has_scalar := {smul := has_scalar.smul finsupp.has_scalar}, one_smul := _, mul_smul := _}, smul_add := _, smul_zero := _}, add_smul := _, zero_smul := _}
theorem
finsupp.support_smul
{α : Type u_1}
{M : Type u_5}
{R : Type u_11}
{_x : semiring R}
[add_comm_monoid M]
[semimodule R M]
{b : R}
{g : α →₀ M} :
@[simp]
theorem
finsupp.filter_smul
{α : Type u_1}
{M : Type u_5}
{R : Type u_11}
{p : α → Prop}
{_x : semiring R}
[add_comm_monoid M]
[semimodule R M]
{b : R}
{v : α →₀ M} :
finsupp.filter p (b • v) = b • finsupp.filter p v
theorem
finsupp.map_domain_smul
{α : Type u_1}
{β : Type u_2}
{M : Type u_5}
{R : Type u_11}
{_x : semiring R}
[add_comm_monoid M]
[semimodule R M]
{f : α → β}
(b : R)
(v : α →₀ M) :
finsupp.map_domain f (b • v) = b • finsupp.map_domain f v
@[simp]
theorem
finsupp.smul_single
{α : Type u_1}
{M : Type u_5}
{R : Type u_11}
{_x : semiring R}
[add_comm_monoid M]
[semimodule R M]
(c : R)
(a : α)
(b : M) :
c • finsupp.single a b = finsupp.single a (c • b)
@[simp]
theorem
finsupp.smul_single'
{α : Type u_1}
{R : Type u_11}
{_x : semiring R}
(c : R)
(a : α)
(b : R) :
c • finsupp.single a b = finsupp.single a (c * b)
theorem
finsupp.smul_single_one
{α : Type u_1}
{R : Type u_11}
[semiring R]
(a : α)
(b : R) :
b • finsupp.single a 1 = finsupp.single a b
theorem
finsupp.sum_smul_index'
{α : Type u_1}
{M : Type u_5}
{N : Type u_7}
{R : Type u_11}
[semiring R]
[add_comm_monoid M]
[semimodule R M]
[add_comm_monoid N]
{g : α →₀ M}
{b : R}
{h : α → M → N}
(h0 : ∀ (i : α), h i 0 = 0) :
@[instance]
Equations
- finsupp.unique_of_right = {to_inhabited := {default := default (α →₀ R) finsupp.inhabited}, uniq := _}
Given an add_comm_monoid M and s : set α, restrict_support_equiv s M is the equiv
between the subtype of finitely supported functions with support contained in s and
the type of finitely supported functions from s.
Equations
- finsupp.restrict_support_equiv s M = {to_fun := λ (f : {f // ↑(f.support) ⊆ s}), finsupp.subtype_domain (λ (x : α), x ∈ s) f.val, inv_fun := λ (f : ↥s →₀ M), ⟨finsupp.map_domain subtype.val f, _⟩, left_inv := _, right_inv := _}
def
finsupp.dom_congr
{α : Type u_1}
{β : Type u_2}
{M : Type u_5}
[add_comm_monoid M]
(e : α ≃ β) :
Given add_comm_monoid M and e : α ≃ β, dom_congr e is the corresponding equiv between
α →₀ M and β →₀ M.
Equations
- finsupp.dom_congr e = {to_fun := finsupp.map_domain ⇑e, inv_fun := finsupp.map_domain ⇑(e.symm), left_inv := _, right_inv := _, map_add' := _}
Declarations about sigma types
def
finsupp.split
{ι : Type u_4}
{M : Type u_5}
{αs : ι → Type u_13}
[has_zero M]
(l : (Σ (i : ι), αs i) →₀ M)
(i : ι) :
αs i →₀ M
Given l, a finitely supported function from the sigma type Σ (i : ι), αs i to M and
an index element i : ι, split l i is the ith component of l,
a finitely supported function from as i to M.
Equations
- l.split i = finsupp.comap_domain (sigma.mk i) l _
def
finsupp.split_support
{ι : Type u_4}
{M : Type u_5}
{αs : ι → Type u_13}
[has_zero M]
(l : (Σ (i : ι), αs i) →₀ M) :
finset ι
Given l, a finitely supported function from the sigma type Σ (i : ι), αs i to β,
split_support l is the finset of indices in ι that appear in the support of l.
Equations
theorem
finsupp.mem_split_support_iff_nonzero
{ι : Type u_4}
{M : Type u_5}
{αs : ι → Type u_13}
[has_zero M]
(l : (Σ (i : ι), αs i) →₀ M)
(i : ι) :
i ∈ l.split_support ↔ l.split i ≠ 0
def
finsupp.split_comp
{ι : Type u_4}
{M : Type u_5}
{N : Type u_7}
{αs : ι → Type u_13}
[has_zero M]
(l : (Σ (i : ι), αs i) →₀ M)
[has_zero N]
(g : Π (i : ι), (αs i →₀ M) → N)
(hg : ∀ (i : ι) (x : αs i →₀ M), x = 0 ↔ g i x = 0) :
ι →₀ N
Given l, a finitely supported function from the sigma type Σ i, αs i to β and
an ι-indexed family g of functions from (αs i →₀ β) to γ, split_comp defines a
finitely supported function from the index type ι to γ given by composing g i with
split l i.
Equations
- l.split_comp g hg = {support := l.split_support, to_fun := λ (i : ι), g i (l.split i), mem_support_to_fun := _}
theorem
finsupp.sigma_support
{ι : Type u_4}
{M : Type u_5}
{αs : ι → Type u_13}
[has_zero M]
(l : (Σ (i : ι), αs i) →₀ M) :
theorem
finsupp.sigma_sum
{ι : Type u_4}
{M : Type u_5}
{N : Type u_7}
{αs : ι → Type u_13}
[has_zero M]
(l : (Σ (i : ι), αs i) →₀ M)
[add_comm_monoid N]
(f : (Σ (i : ι), αs i) → M → N) :
Given a multiset s, s.to_finsupp returns the finitely supported function on ℕ given by
the multiplicities of the elements of s.
Equations
@[simp]
@[simp]
theorem
multiset.to_finsupp_apply
{α : Type u_1}
(s : multiset α)
(a : α) :
⇑(⇑multiset.to_finsupp s) a = multiset.count a s
@[simp]
theorem
multiset.to_finsupp_singleton
{α : Type u_1}
(a : α) :
⇑multiset.to_finsupp (a ::ₘ 0) = finsupp.single a 1
@[simp]
theorem
multiset.to_finsupp_eq_iff
{α : Type u_1}
{s : multiset α}
{f : α →₀ ℕ} :
⇑multiset.to_finsupp s = f ↔ s = ⇑finsupp.to_multiset f
@[simp]
Declarations about order(ed) instances on finsupp
@[instance]
@[instance]
def
finsupp.partial_order
{α : Type u_1}
{M : Type u_5}
[partial_order M]
[has_zero M] :
partial_order (α →₀ M)
Equations
- finsupp.partial_order = {le := preorder.le finsupp.preorder, lt := preorder.lt finsupp.preorder, le_refl := _, le_trans := _, lt_iff_le_not_le := _, le_antisymm := _}
@[instance]
def
finsupp.add_left_cancel_semigroup
{α : Type u_1}
{M : Type u_5}
[ordered_cancel_add_comm_monoid M] :
add_left_cancel_semigroup (α →₀ M)
Equations
@[instance]
def
finsupp.add_right_cancel_semigroup
{α : Type u_1}
{M : Type u_5}
[ordered_cancel_add_comm_monoid M] :
add_right_cancel_semigroup (α →₀ M)
Equations
@[instance]
def
finsupp.ordered_cancel_add_comm_monoid
{α : Type u_1}
{M : Type u_5}
[ordered_cancel_add_comm_monoid M] :
Equations
- finsupp.ordered_cancel_add_comm_monoid = {add := add_comm_monoid.add finsupp.add_comm_monoid, add_assoc := _, add_left_cancel := _, zero := add_comm_monoid.zero finsupp.add_comm_monoid, zero_add := _, add_zero := _, add_comm := _, add_right_cancel := _, le := partial_order.le finsupp.partial_order, lt := partial_order.lt finsupp.partial_order, le_refl := _, le_trans := _, lt_iff_le_not_le := _, le_antisymm := _, add_le_add_left := _, le_of_add_le_add_left := _}
@[simp]
theorem
finsupp.add_eq_zero_iff
{α : Type u_1}
{M : Type u_5}
[canonically_ordered_add_monoid M]
(f g : α →₀ M) :
finsupp.to_multiset as an order isomorphism.
Equations
@[simp]
@[simp]
The order on σ →₀ ℕ is well-founded.
@[instance]
Equations
- finsupp.decidable_le α = λ (m n : α →₀ ℕ), _.mpr finset.decidable_dforall_finset
@[instance]
Equations
- finsupp.canonically_ordered_add_monoid = {add := ordered_add_comm_monoid.add infer_instance, add_assoc := _, zero := ordered_add_comm_monoid.zero infer_instance, zero_add := _, add_zero := _, add_comm := _, le := ordered_add_comm_monoid.le infer_instance, lt := ordered_add_comm_monoid.lt infer_instance, le_refl := _, le_trans := _, lt_iff_le_not_le := _, le_antisymm := _, add_le_add_left := _, lt_of_add_lt_add_left := _, bot := 0, bot_le := _, le_iff_exists_add := _}
The finsupp counterpart of multiset.antidiagonal: the antidiagonal of
s : α →₀ ℕ consists of all pairs (t₁, t₂) : (α →₀ ℕ) × (α →₀ ℕ) such that t₁ + t₂ = s.
The finitely supported function antidiagonal s is equal to the multiplicities of these pairs.
@[simp]
The set {m : α →₀ ℕ | m ≤ n} as a finset.
Equations
@[simp]
@[simp]