Type of functions with finite support #
For any type α and any 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, which is defined in
algebra/big_operators/finsupp.
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.
Main declarations #
finsupp: The type of finitely supported functions fromαtoβ.finsupp.single: Thefinsuppwhich is nonzero in exactly one point.finsupp.update: Changes one value of afinsupp.finsupp.erase: Replaces one value of afinsuppby0.finsupp.on_finset: The restriction of a function to afinsetas afinsupp.finsupp.map_range: Composition of azero_homwith afinsupp.finsupp.emb_domain: Maps the domain of afinsuppby an embedding.finsupp.zip_with: Postcomposition of twofinsupps with a functionfsuch thatf 0 0 = 0.
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.
Implementation notes #
This file is a noncomputable theory and uses classical logic throughout.
TODO #
- Expand the list of definitions and important lemmas to the module docstring.
- support : finset α
- to_fun : α → M
- mem_support_to_fun : ∀ (a : α), a ∈ self.support ↔ self.to_fun a ≠ 0
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.
Instances for finsupp
- finsupp.has_sizeof_inst
- finsupp.fun_like
- finsupp.has_coe_to_fun
- finsupp.has_zero
- finsupp.inhabited
- finsupp.nontrivial
- finsupp.can_lift
- finsupp.has_add
- finsupp.add_zero_class
- finsupp.has_nat_scalar
- finsupp.add_monoid
- finsupp.add_comm_monoid
- finsupp.has_neg
- finsupp.has_sub
- finsupp.has_int_scalar
- finsupp.add_group
- finsupp.add_comm_group
- finsupp.has_smul
- finsupp.has_faithful_smul
- finsupp.distrib_mul_action
- finsupp.is_scalar_tower
- finsupp.smul_comm_class
- finsupp.is_central_scalar
- finsupp.module
- finsupp.no_zero_smul_divisors
- finsupp.unique_of_right
- finsupp.unique_of_left
- finsupp.has_le
- finsupp.preorder
- finsupp.partial_order
- finsupp.semilattice_inf
- finsupp.semilattice_sup
- finsupp.lattice
- finsupp.ordered_add_comm_monoid
- finsupp.ordered_cancel_add_comm_monoid
- finsupp.has_le.le.contravariant_class
- finsupp.order_bot
- finsupp.tsub
- finsupp.has_ordered_sub
- finsupp.canonically_ordered_add_monoid
- finite_dimensional_finsupp
- module.free.finsupp
- finsupp.locally_finite_order
- finsupp.has_mul
- finsupp.mul_zero_class
- finsupp.semigroup_with_zero
- finsupp.non_unital_non_assoc_semiring
- finsupp.non_unital_semiring
- finsupp.non_unital_comm_semiring
- finsupp.non_unital_non_assoc_ring
- finsupp.non_unital_ring
- finsupp.non_unital_comm_ring
- finsupp.pointwise_scalar
- finsupp.pointwise_module
- slim_check.total_function.finsupp.sampleable_ext
@[protected, instance]
Equations
- finsupp.fun_like = {coe := finsupp.to_fun _inst_1, coe_injective' := _}
@[protected, instance]
def
finsupp.has_coe_to_fun
{α : Type u_1}
{M : Type u_5}
[has_zero M] :
has_coe_to_fun (α →₀ M) (λ (_x : α →₀ M), α → M)
Helper instance for when there are too many metavariables to apply fun_like.has_coe_to_fun
directly.
Equations
Deprecated. Use fun_like.ext_iff instead.
Deprecated. Use fun_like.coe_fn_eq instead.
Deprecated. Use fun_like.coe_injective instead.
theorem
finsupp.congr_fun
{α : Type u_1}
{M : Type u_5}
[has_zero M]
{f g : α →₀ M}
(h : f = g)
(a : α) :
Deprecated. Use fun_like.congr_fun instead.
@[protected, instance]
Equations
- finsupp.has_zero = {zero := {support := ∅, to_fun := 0, mem_support_to_fun := _}}
@[protected, instance]
Equations
- finsupp.inhabited = {default := 0}
@[simp, norm_cast]
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)
@[protected, instance]
def
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)
@[simp]
theorem
finsupp.equiv_fun_on_fintype_symm_apply_to_fun
{α : Type u_1}
{M : Type u_5}
[has_zero M]
[fintype α]
(f : α → M)
(ᾰ : α) :
⇑(⇑(finsupp.equiv_fun_on_fintype.symm) f) ᾰ = f ᾰ
@[simp]
theorem
finsupp.equiv_fun_on_fintype_apply
{α : Type u_1}
{M : Type u_5}
[has_zero M]
[fintype α]
(f : α →₀ M)
(a : α) :
⇑finsupp.equiv_fun_on_fintype f a = ⇑f a
@[simp]
theorem
finsupp.equiv_fun_on_fintype_symm_apply_support
{α : Type u_1}
{M : Type u_5}
[has_zero M]
[fintype α]
(f : α → M) :
(⇑(finsupp.equiv_fun_on_fintype.symm) f).support = finset.filter (λ (a : α), f a ≠ 0) finset.univ
noncomputable
def
finsupp.equiv_fun_on_fintype
{α : Type u_1}
{M : Type u_5}
[has_zero M]
[fintype α] :
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 := _}
@[simp]
theorem
finsupp.equiv_fun_on_fintype_symm_coe
{M : Type u_5}
[has_zero M]
{α : Type u_1}
[fintype α]
(f : α →₀ M) :
@[simp]
theorem
equiv.finsupp_unique_symm_apply_to_fun
{M : Type u_5}
[has_zero M]
{ι : Type u_1}
[unique ι]
(ᾰ : M)
(ᾰ_1 : ι) :
⇑(⇑(equiv.finsupp_unique.symm) ᾰ) ᾰ_1 = ᾰ
If α has a unique term, the type of finitely supported functions α →₀ β is equivalent to β.
Equations
@[simp]
theorem
equiv.finsupp_unique_symm_apply_support_val
{M : Type u_5}
[has_zero M]
{ι : Type u_1}
[unique ι]
(ᾰ : M) :
(⇑(equiv.finsupp_unique.symm) ᾰ).support.val = multiset.filter (λ (a : ι), ¬ᾰ = 0) {inhabited.default}
@[simp]
theorem
equiv.finsupp_unique_apply
{M : Type u_5}
[has_zero M]
{ι : Type u_1}
[unique ι]
(ᾰ : ι →₀ M) :
Declarations about single #
noncomputable
def
finsupp.single
{α : Type u_1}
{M : Type u_5}
[has_zero M]
(a : α)
(b : M) :
α →₀ M
single a b is the finitely supported function with 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}
[decidable (a = a')] :
⇑(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]
[decidable_eq α]
(a : α)
(b : M) :
⇑(finsupp.single a b) = function.update 0 a b
theorem
finsupp.single_eq_pi_single
{α : Type u_1}
{M : Type u_5}
[has_zero M]
[decidable_eq α]
(a : α)
(b : M) :
⇑(finsupp.single a b) = pi.single 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]
{b : M}
(a : α)
(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}
finsupp.single a b is injective in b. For the statement that it is injective in a, see
finsupp.single_left_injective
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_injective
{α : Type u_1}
{M : Type u_5}
[has_zero M]
{b : M}
(h : b ≠ 0) :
function.injective (λ (a : α), finsupp.single a b)
finsupp.single a b is injective in a. For the statement that it is injective in b, see
finsupp.single_injective
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'
theorem
finsupp.support_single_ne_bot
{α : Type u_1}
{M : Type u_5}
[has_zero M]
{b : M}
(i : α)
(h : b ≠ 0) :
(finsupp.single i b).support ≠ ⊥
theorem
finsupp.support_single_disjoint
{α : Type u_1}
{M : Type u_5}
[has_zero M]
{b : M}
[decidable_eq α]
{b' : M}
(hb : b ≠ 0)
(hb' : b' ≠ 0)
{i j : α} :
disjoint (finsupp.single i b).support (finsupp.single j b').support ↔ i ≠ j
@[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
@[protected, instance]
def
finsupp.nontrivial
{α : Type u_1}
{M : Type u_5}
[has_zero M]
[nonempty α]
[nontrivial M] :
nontrivial (α →₀ M)
@[ext]
theorem
finsupp.unique_ext
{α : Type u_1}
{M : Type u_5}
[has_zero M]
[unique α]
{f g : α →₀ M}
(h : ⇑f inhabited.default = ⇑g inhabited.default) :
f = g
theorem
finsupp.unique_ext_iff
{α : Type u_1}
{M : Type u_5}
[has_zero M]
[unique α]
{f g : α →₀ M} :
f = g ↔ ⇑f inhabited.default = ⇑g inhabited.default
@[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} :
theorem
finsupp.support_subset_singleton
{α : Type u_1}
{M : Type u_5}
[has_zero M]
{f : α →₀ M}
{a : α} :
theorem
finsupp.support_subset_singleton'
{α : Type u_1}
{M : Type u_5}
[has_zero M]
{f : α →₀ M}
{a : α} :
f.support ⊆ {a} ↔ ∃ (b : M), f = finsupp.single a b
@[simp]
theorem
finsupp.equiv_fun_on_fintype_single
{α : Type u_1}
{M : Type u_5}
[has_zero M]
[decidable_eq α]
[fintype α]
(x : α)
(m : M) :
@[simp]
theorem
finsupp.equiv_fun_on_fintype_symm_single
{α : Type u_1}
{M : Type u_5}
[has_zero M]
[decidable_eq α]
[fintype α]
(x : α)
(m : M) :
⇑(finsupp.equiv_fun_on_fintype.symm) (pi.single x m) = finsupp.single x m
Declarations about update #
noncomputable
def
finsupp.update
{α : Type u_1}
{M : Type u_5}
[has_zero M]
(f : α →₀ M)
(a : α)
(b : M) :
α →₀ M
Replace the value of a α →₀ M at a given point a : α by a given value b : M.
If b = 0, this amounts to removing a from the finsupp.support.
Otherwise, if a was not in the finsupp.support, it is added to it.
This is the finitely-supported version of function.update.
Equations
- f.update a b = {support := ite (b = 0) (f.support.erase a) (has_insert.insert a f.support), to_fun := function.update ⇑f a b, mem_support_to_fun := _}
@[simp]
theorem
finsupp.coe_update
{α : Type u_1}
{M : Type u_5}
[has_zero M]
(f : α →₀ M)
(a : α)
(b : M)
[decidable_eq α] :
⇑(f.update a b) = function.update ⇑f a b
@[simp]
theorem
finsupp.zero_update
{α : Type u_1}
{M : Type u_5}
[has_zero M]
(a : α)
(b : M) :
0.update a b = finsupp.single a b
theorem
finsupp.support_update
{α : Type u_1}
{M : Type u_5}
[has_zero M]
(f : α →₀ M)
(a : α)
(b : M)
[decidable_eq α] :
@[simp]
theorem
finsupp.support_update_zero
{α : Type u_1}
{M : Type u_5}
[has_zero M]
(f : α →₀ M)
(a : α)
[decidable_eq α] :
theorem
finsupp.support_update_ne_zero
{α : Type u_1}
{M : Type u_5}
[has_zero M]
(f : α →₀ M)
(a : α)
{b : M}
[decidable_eq α]
(h : b ≠ 0) :
(f.update a b).support = has_insert.insert a f.support
Declarations about erase #
noncomputable
def
finsupp.erase
{α : Type u_1}
{M : Type u_5}
[has_zero M]
(a : α)
(f : α →₀ M) :
α →₀ M
erase a f is the finitely supported function equal to f except at a where it is equal to 0.
If a is not in the support of f then erase a f = f.
@[simp]
theorem
finsupp.support_erase
{α : Type u_1}
{M : Type u_5}
[has_zero M]
[decidable_eq α]
{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_of_not_mem_support
{α : Type u_1}
{M : Type u_5}
[has_zero M]
{f : α →₀ M}
{a : α}
(haf : a ∉ f.support) :
finsupp.erase a f = f
@[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 on_finset #
noncomputable
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 must 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
noncomputable
def
finsupp.of_support_finite
{α : Type u_1}
{M : Type u_5}
[has_zero M]
(f : α → M)
(hf : (function.support f).finite) :
α →₀ M
The natural finsupp induced by the function f given that it has finite support.
Equations
- finsupp.of_support_finite f hf = {support := hf.to_finset, to_fun := f, mem_support_to_fun := _}
theorem
finsupp.of_support_finite_coe
{α : Type u_1}
{M : Type u_5}
[has_zero M]
{f : α → M}
{hf : (function.support f).finite} :
⇑(finsupp.of_support_finite f hf) = f
@[protected, instance]
Equations
- finsupp.can_lift = {coe := coe_fn finsupp.has_coe_to_fun, cond := λ (f : α → M), (function.support f).finite, prf := _}
Declarations about map_range #
noncomputable
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,
which is well-defined when f 0 = 0.
This preserves the structure on f, and exists in various bundled forms for when f is itself
bundled (defined in data/finsupp/basic):
finsupp.map_range.equivfinsupp.map_range.zero_homfinsupp.map_range.add_monoid_homfinsupp.map_range.add_equivfinsupp.map_range.linear_mapfinsupp.map_range.linear_equiv
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
@[simp]
theorem
finsupp.map_range_id
{α : Type u_1}
{M : Type u_5}
[has_zero M]
(g : α →₀ M) :
finsupp.map_range id rfl g = g
theorem
finsupp.map_range_comp
{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 : N → P)
(hf : f 0 = 0)
(f₂ : M → N)
(hf₂ : f₂ 0 = 0)
(h : (f ∘ f₂) 0 = 0)
(g : α →₀ M) :
finsupp.map_range (f ∘ f₂) h g = finsupp.map_range f hf (finsupp.map_range f₂ hf₂ g)
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)
theorem
finsupp.support_map_range_of_injective
{ι : Type u_4}
{M : Type u_5}
{N : Type u_7}
[has_zero M]
[has_zero N]
{e : M → N}
(he0 : e 0 = 0)
(f : ι →₀ M)
(he : function.injective e) :
(finsupp.map_range e he0 f).support = f.support
Declarations about emb_domain #
noncomputable
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
@[simp]
theorem
finsupp.emb_domain_single
{α : Type u_1}
{β : Type u_2}
{M : Type u_5}
[has_zero M]
(f : α ↪ β)
(a : α)
(m : M) :
finsupp.emb_domain f (finsupp.single a m) = finsupp.single (⇑f a) m
Declarations about zip_with #
noncomputable
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
Given finitely supported functions g₁ : α →₀ M and g₂ : α →₀ N and function f : M → N → P,
zip_with f hf g₁ g₂ is the finitely supported function α →₀ P satisfying
zip_with f hf g₁ g₂ a = f (g₁ a) (g₂ a), which 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)) _
theorem
finsupp.support_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]
[D : decidable_eq α]
{f : M → N → P}
{hf : f 0 0 = 0}
{g₁ : α →₀ M}
{g₂ : α →₀ N} :
Additive monoid structure on α →₀ M #
@[protected, instance]
Equations
- finsupp.has_add = {add := finsupp.zip_with has_add.add finsupp.has_add._proof_1}
@[simp]
theorem
finsupp.add_apply
{α : Type u_1}
{M : Type u_5}
[add_zero_class M]
(g₁ g₂ : α →₀ M)
(a : α) :
theorem
finsupp.support_add
{α : Type u_1}
{M : Type u_5}
[add_zero_class M]
[decidable_eq α]
{g₁ g₂ : α →₀ M} :
theorem
finsupp.support_add_eq
{α : Type u_1}
{M : Type u_5}
[add_zero_class M]
[decidable_eq α]
{g₁ g₂ : α →₀ M}
(h : disjoint g₁.support g₂.support) :
@[simp]
theorem
finsupp.single_add
{α : Type u_1}
{M : Type u_5}
[add_zero_class M]
(a : α)
(b₁ b₂ : M) :
finsupp.single a (b₁ + b₂) = finsupp.single a b₁ + finsupp.single a b₂
@[protected, instance]
noncomputable
def
finsupp.add_zero_class
{α : Type u_1}
{M : Type u_5}
[add_zero_class M] :
add_zero_class (α →₀ M)
Equations
- finsupp.add_zero_class = function.injective.add_zero_class coe_fn finsupp.add_zero_class._proof_1 finsupp.add_zero_class._proof_2 finsupp.coe_add
finsupp.single as an add_monoid_hom.
See finsupp.lsingle in linear_algebra/finsupp 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_zero_class 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 in linear_algebra/finsupp for the stronger version as a linear map.
@[simp]
theorem
finsupp.apply_add_hom_apply
{α : Type u_1}
{M : Type u_5}
[add_zero_class M]
(a : α)
(g : α →₀ M) :
⇑(finsupp.apply_add_hom a) g = ⇑g a
@[simp]
theorem
finsupp.coe_fn_add_hom_apply
{α : Type u_1}
{M : Type u_5}
[add_zero_class M]
(x : α →₀ M)
(ᾰ : α) :
⇑finsupp.coe_fn_add_hom x ᾰ = ⇑x ᾰ
Coercion from a finsupp to a function type is an add_monoid_hom.
Equations
- finsupp.coe_fn_add_hom = {to_fun := coe_fn finsupp.has_coe_to_fun, map_zero' := _, map_add' := _}
theorem
finsupp.update_eq_single_add_erase
{α : Type u_1}
{M : Type u_5}
[add_zero_class M]
(f : α →₀ M)
(a : α)
(b : M) :
f.update a b = finsupp.single a b + finsupp.erase a f
theorem
finsupp.update_eq_erase_add_single
{α : Type u_1}
{M : Type u_5}
[add_zero_class M]
(f : α →₀ M)
(a : α)
(b : M) :
f.update a b = finsupp.erase a f + finsupp.single a b
theorem
finsupp.single_add_erase
{α : Type u_1}
{M : Type u_5}
[add_zero_class 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_zero_class 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_zero_class M]
(a : α)
(f f' : α →₀ M) :
finsupp.erase a (f + f') = finsupp.erase a f + finsupp.erase a f'
@[simp]
theorem
finsupp.erase_add_hom_apply
{α : Type u_1}
{M : Type u_5}
[add_zero_class M]
(a : α)
(f : α →₀ M) :
⇑(finsupp.erase_add_hom a) f = finsupp.erase a f
finsupp.erase as an add_monoid_hom.
Equations
- finsupp.erase_add_hom a = {to_fun := finsupp.erase a, map_zero' := _, map_add' := _}
@[protected]
theorem
finsupp.induction
{α : Type u_1}
{M : Type u_5}
[add_zero_class 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_zero_class 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
theorem
finsupp.induction_linear
{α : Type u_1}
{M : Type u_5}
[add_zero_class M]
{p : (α →₀ M) → Prop}
(f : α →₀ M)
(h0 : p 0)
(hadd : ∀ (f g : α →₀ M), p f → p g → p (f + g))
(hsingle : ∀ (a : α) (b : M), p (finsupp.single a b)) :
p f
@[simp]
theorem
finsupp.add_closure_set_of_eq_single
{α : Type u_1}
{M : Type u_5}
[add_zero_class M] :
add_submonoid.closure {f : α →₀ M | ∃ (a : α) (b : M), f = finsupp.single a b} = ⊤
theorem
finsupp.add_hom_ext
{α : Type u_1}
{M : Type u_5}
{N : Type u_7}
[add_zero_class M]
[add_zero_class 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_zero_class M]
[add_zero_class 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_zero_class M]
[mul_one_class 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_zero_class M]
[mul_one_class 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_zero_class M]
[add_zero_class 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
@[simp]
theorem
finsupp.emb_domain.add_monoid_hom_apply
{α : Type u_1}
{β : Type u_2}
{M : Type u_5}
[add_zero_class M]
(f : α ↪ β)
(v : α →₀ M) :
noncomputable
def
finsupp.emb_domain.add_monoid_hom
{α : Type u_1}
{β : Type u_2}
{M : Type u_5}
[add_zero_class M]
(f : α ↪ β) :
Bundle emb_domain f as an additive map from α →₀ M to β →₀ M.
Equations
- finsupp.emb_domain.add_monoid_hom f = {to_fun := λ (v : α →₀ M), finsupp.emb_domain f v, map_zero' := _, map_add' := _}
@[simp]
theorem
finsupp.emb_domain_add
{α : Type u_1}
{β : Type u_2}
{M : Type u_5}
[add_zero_class M]
(f : α ↪ β)
(v w : α →₀ M) :
finsupp.emb_domain f (v + w) = finsupp.emb_domain f v + finsupp.emb_domain f w
@[protected, instance]
Note the general finsupp.has_smul instance doesn't apply as ℕ is not distributive
unless β i's addition is commutative.
Equations
- finsupp.has_nat_scalar = {smul := λ (n : ℕ) (v : α →₀ M), finsupp.map_range (has_smul.smul n) _ v}
@[protected, instance]
noncomputable
def
finsupp.add_monoid
{α : Type u_1}
{M : Type u_5}
[add_monoid M] :
add_monoid (α →₀ M)
Equations
- finsupp.add_monoid = function.injective.add_monoid coe_fn finsupp.add_monoid._proof_1 finsupp.add_monoid._proof_2 finsupp.add_monoid._proof_3 finsupp.add_monoid._proof_4
@[protected, instance]
noncomputable
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 = function.injective.add_comm_monoid coe_fn finsupp.add_comm_monoid._proof_1 finsupp.add_comm_monoid._proof_2 finsupp.add_comm_monoid._proof_3 finsupp.add_comm_monoid._proof_4
@[protected, instance]
Equations
- finsupp.has_neg = {neg := finsupp.map_range has_neg.neg finsupp.has_neg._proof_1}
@[protected, instance]
Equations
- finsupp.has_sub = {sub := finsupp.zip_with has_sub.sub finsupp.has_sub._proof_1}
@[protected, instance]
Note the general finsupp.has_smul instance doesn't apply as ℤ is not distributive
unless β i's addition is commutative.
Equations
- finsupp.has_int_scalar = {smul := λ (n : ℤ) (v : α →₀ G), finsupp.map_range (has_smul.smul n) _ v}
@[protected, instance]
Equations
- finsupp.add_group = function.injective.add_group coe_fn finsupp.add_group._proof_1 finsupp.add_group._proof_2 finsupp.add_group._proof_3 finsupp.coe_neg finsupp.coe_sub finsupp.add_group._proof_4 finsupp.add_group._proof_5
@[protected, instance]
noncomputable
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 = function.injective.add_comm_group coe_fn finsupp.add_comm_group._proof_1 finsupp.add_comm_group._proof_2 finsupp.add_comm_group._proof_3 finsupp.add_comm_group._proof_4 finsupp.add_comm_group._proof_5 finsupp.add_comm_group._proof_6 finsupp.add_comm_group._proof_7
theorem
finsupp.single_add_single_eq_single_add_single
{α : Type u_1}
{M : Type u_5}
[add_comm_monoid M]
{k l m n : α}
{u v : M}
(hu : u ≠ 0)
(hv : v ≠ 0) :
theorem
finsupp.erase_eq_sub_single
{α : Type u_1}
{G : Type u_9}
[add_group G]
(f : α →₀ G)
(a : α) :
finsupp.erase a f = f - finsupp.single a (⇑f a)
theorem
finsupp.update_eq_sub_add_single
{α : Type u_1}
{G : Type u_9}
[add_group G]
(f : α →₀ G)
(a : α)
(b : G) :
f.update a b = f - finsupp.single a (⇑f a) + finsupp.single a b