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.
@[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)
Equations
- finsupp.has_coe_to_fun = {coe := finsupp.to_fun _inst_1}
@[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) :
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 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}
[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]
{a : α}
{b : M}
[decidable_eq α] :
⇑(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]
{a : α}
{b : M}
[decidable_eq α] :
⇑(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]
{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}
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)
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 (⇑x 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.
@[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.support_update_zero
{α : Type u_1}
{M : Type u_5}
[has_zero M]
(f : α →₀ M)
(a : α)
[decidable_eq α] :
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 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
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, well-defined when f 0 = 0.
This preserves the structure on f, and exists in various bundled forms for when f is itself
bundled:
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)
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
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)) _
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} :
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.
@[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 sum and prod #
In most of this section, the domain β is assumed to be an add_monoid.
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.
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.
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]
[decidable_eq α]
(f : α →₀ M)
(a : α)
(b : α → M → N) :
@[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]
[decidable_eq α]
(f : α →₀ M)
(a : α)
(b : α → M → N) :
@[simp]
theorem
finsupp.sum_ite_self_eq
{α : Type u_1}
[decidable_eq α]
{N : Type u_2}
[add_comm_monoid N]
(f : α →₀ N)
(a : α) :
@[simp]
theorem
finsupp.prod_ite_eq'
{α : Type u_1}
{M : Type u_5}
{N : Type u_7}
[has_zero M]
[comm_monoid N]
[decidable_eq α]
(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]
[decidable_eq α]
(f : α →₀ M)
(a : α)
(b : α → M → N) :
A restatement of sum_ite_eq with the equality test reversed.
@[simp]
theorem
finsupp.sum_ite_self_eq'
{α : Type u_1}
[decidable_eq α]
{N : Type u_2}
[add_comm_monoid N]
(f : α →₀ N)
(a : α) :
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.
theorem
finsupp.mul_prod_erase
{α : Type u_1}
{M : Type u_5}
{N : Type u_7}
[has_zero M]
[comm_monoid N]
(f : α →₀ M)
(y : α)
(g : α → M → N)
(hyf : y ∈ f.support) :
Taking a product over f : α →₀ M is the same as multiplying the value on a single element
y ∈ f.support by the product over erase y f.
theorem
finsupp.add_sum_erase
{α : Type u_1}
{M : Type u_5}
{N : Type u_7}
[has_zero M]
[add_comm_monoid N]
(f : α →₀ M)
(y : α)
(g : α → M → N)
(hyf : y ∈ f.support) :
Taking a sum over over f : α →₀ M is the same as adding the value on a
single element y ∈ f.support to the sum over erase y f.
theorem
finsupp.mul_prod_erase'
{α : Type u_1}
{M : Type u_5}
{N : Type u_7}
[has_zero M]
[comm_monoid N]
(f : α →₀ M)
(y : α)
(g : α → M → N)
(hg : ∀ (i : α), g i 0 = 1) :
Generalization of finsupp.mul_prod_erase: if g maps a second argument of 0 to 1,
then its product over f : α →₀ M is the same as multiplying the value on any element
y : α by the product over erase y f.
theorem
finsupp.add_sum_erase'
{α : Type u_1}
{M : Type u_5}
{N : Type u_7}
[has_zero M]
[add_comm_monoid N]
(f : α →₀ M)
(y : α)
(g : α → M → N)
(hg : ∀ (i : α), g i 0 = 0) :
Generalization of finsupp.add_sum_erase: if g maps a second argument of 0
to 0, then its sum over f : α →₀ M is the same as adding the value on any element
y : α to the sum over erase y f.
theorem
add_submonoid.finsupp_sum_mem
{α : Type u_1}
{M : Type u_5}
{N : Type u_7}
[has_zero M]
[add_comm_monoid N]
(S : add_submonoid N)
(f : α →₀ M)
(g : α → M → N)
(h : ∀ (c : α), ⇑f c ≠ 0 → g c (⇑f c) ∈ S) :
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 = {zero := 0, add := has_add.add finsupp.has_add, 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_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 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]
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 = {add := has_add.add finsupp.has_add, add_assoc := _, zero := 0, zero_add := _, add_zero := _, nsmul := λ (n : ℕ) (v : α →₀ M), finsupp.map_range (has_scalar.smul n) _ v, nsmul_zero' := _, nsmul_succ' := _}
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) :
theorem
add_monoid_hom.coe_finsupp_sum
{α : Type u_1}
{β : Type u_2}
{N : Type u_7}
{P : Type u_8}
[has_zero β]
[add_monoid N]
[add_comm_monoid P]
(f : α →₀ β)
(g : α → β → N →+ P) :
@[simp]
theorem
add_monoid_hom.finsupp_sum_apply
{α : Type u_1}
{β : Type u_2}
{N : Type u_7}
{P : Type u_8}
[has_zero β]
[add_monoid N]
[add_comm_monoid P]
(f : α →₀ β)
(g : α → β → N →+ P)
(x : N) :
@[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 = {add := add_monoid.add finsupp.add_monoid, add_assoc := _, zero := add_monoid.zero finsupp.add_monoid, zero_add := _, add_zero := _, nsmul := add_monoid.nsmul finsupp.add_monoid, nsmul_zero' := _, nsmul_succ' := _, add_comm := _}
@[protected, instance]
Equations
- finsupp.has_sub = {sub := finsupp.zip_with has_sub.sub finsupp.has_sub._proof_1}
@[protected, 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 := _, nsmul := add_monoid.nsmul finsupp.add_monoid, nsmul_zero' := _, nsmul_succ' := _, neg := finsupp.map_range has_neg.neg neg_zero, sub := has_sub.sub finsupp.has_sub, sub_eq_add_neg := _, zsmul := λ (n : ℤ) (v : α →₀ G), finsupp.map_range (has_scalar.smul n) _ v, zsmul_zero' := _, zsmul_succ' := _, zsmul_neg' := _, add_left_neg := _}
@[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 = {add := add_group.add finsupp.add_group, add_assoc := _, zero := add_group.zero finsupp.add_group, zero_add := _, add_zero := _, nsmul := add_group.nsmul finsupp.add_group, nsmul_zero' := _, nsmul_succ' := _, neg := add_group.neg finsupp.add_group, sub := add_group.sub finsupp.add_group, sub_eq_add_neg := _, zsmul := add_group.zsmul finsupp.add_group, zsmul_zero' := _, zsmul_succ' := _, zsmul_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))
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
theorem
finsupp.finset_sum_apply
{α : Type u_1}
{ι : Type u_4}
{N : Type u_7}
[add_comm_monoid N]
(S : finset ι)
(f : ι → α →₀ N)
(a : α) :
theorem
finsupp.coe_finset_sum
{α : Type u_1}
{ι : Type u_4}
{N : Type u_7}
[add_comm_monoid N]
(S : finset ι)
(f : ι → α →₀ N) :
theorem
finsupp.support_sum
{α : Type u_1}
{β : Type u_2}
{M : Type u_5}
{N : Type u_7}
[decidable_eq β]
[has_zero M]
[add_comm_monoid N]
{f : α →₀ M}
{g : α → M → β →₀ N} :
theorem
finsupp.support_finset_sum
{α : Type u_1}
{β : Type u_2}
{M : Type u_5}
[decidable_eq β]
[add_comm_monoid M]
{s : finset α}
{f : α → β →₀ M} :
@[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))
noncomputable
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))
theorem
finsupp.prod_add_index_of_disjoint
{α : Type u_1}
{M : Type u_5}
[add_comm_monoid M]
{f1 f2 : α →₀ M}
(hd : disjoint f1.support f2.support)
{β : Type u_2}
[comm_monoid β]
(g : α → M → β) :
For disjoint f1 and f2, and function g, the product of the products of g
over f1 and f2 equals the product of g over f1 + f2
noncomputable
def
finsupp.map_range.equiv
{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)
(hf' : ⇑(f.symm) 0 = 0) :
finsupp.map_range as an equiv.
Equations
- finsupp.map_range.equiv f hf hf' = {to_fun := finsupp.map_range ⇑f hf, inv_fun := finsupp.map_range ⇑(f.symm) hf', left_inv := _, right_inv := _}
@[simp]
theorem
finsupp.map_range.equiv_refl
{α : Type u_1}
{M : Type u_5}
[has_zero M] :
finsupp.map_range.equiv (equiv.refl M) rfl rfl = equiv.refl (α →₀ M)
theorem
finsupp.map_range.equiv_trans
{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)
(hf : ⇑f 0 = 0)
(hf' : ⇑(f.symm) 0 = 0)
(f₂ : N ≃ P)
(hf₂ : ⇑f₂ 0 = 0)
(hf₂' : ⇑(f₂.symm) 0 = 0) :
finsupp.map_range.equiv (f.trans f₂) _ _ = (finsupp.map_range.equiv f hf hf').trans (finsupp.map_range.equiv f₂ hf₂ hf₂')
@[simp]
theorem
finsupp.map_range.equiv_symm
{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)
(hf' : ⇑(f.symm) 0 = 0) :
(finsupp.map_range.equiv f hf hf').symm = finsupp.map_range.equiv f.symm hf' hf
noncomputable
def
finsupp.map_range.zero_hom
{α : Type u_1}
{M : Type u_5}
{N : Type u_7}
[has_zero M]
[has_zero N]
(f : zero_hom M N) :
Composition with a fixed zero-preserving homomorphism is itself an zero-preserving homomorphism on functions.
Equations
- finsupp.map_range.zero_hom f = {to_fun := finsupp.map_range ⇑f _, map_zero' := _}
@[simp]
theorem
finsupp.map_range.zero_hom_apply
{α : Type u_1}
{M : Type u_5}
{N : Type u_7}
[has_zero M]
[has_zero N]
(f : zero_hom M N)
(g : α →₀ M) :
⇑(finsupp.map_range.zero_hom f) g = finsupp.map_range ⇑f _ g
@[simp]
theorem
finsupp.map_range.zero_hom_id
{α : Type u_1}
{M : Type u_5}
[has_zero M] :
finsupp.map_range.zero_hom (zero_hom.id M) = zero_hom.id (α →₀ M)
noncomputable
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' := _}
@[simp]
theorem
finsupp.map_range.add_monoid_hom_apply
{α : Type u_1}
{M : Type u_5}
{N : Type u_7}
[add_comm_monoid M]
[add_comm_monoid N]
(f : M →+ N)
(g : α →₀ M) :
⇑(finsupp.map_range.add_monoid_hom f) g = finsupp.map_range ⇑f _ g
@[simp]
theorem
finsupp.map_range.add_monoid_hom_comp
{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]
(f : N →+ P)
(f₂ : M →+ N) :
@[simp]
theorem
finsupp.map_range.add_monoid_hom_to_zero_hom
{α : Type u_1}
{M : Type u_5}
{N : Type u_7}
[add_comm_monoid M]
[add_comm_monoid N]
(f : M →+ N) :
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)
noncomputable
def
finsupp.map_range.add_equiv
{α : Type u_1}
{M : Type u_5}
{N : Type u_7}
[add_comm_monoid M]
[add_comm_monoid N]
(f : M ≃+ N) :
finsupp.map_range.add_monoid_hom as an equiv.
Equations
- finsupp.map_range.add_equiv f = {to_fun := finsupp.map_range ⇑f _, inv_fun := finsupp.map_range ⇑(f.symm) _, left_inv := _, right_inv := _, map_add' := _}
@[simp]
theorem
finsupp.map_range.add_equiv_apply
{α : Type u_1}
{M : Type u_5}
{N : Type u_7}
[add_comm_monoid M]
[add_comm_monoid N]
(f : M ≃+ N)
(g : α →₀ M) :
⇑(finsupp.map_range.add_equiv f) g = finsupp.map_range ⇑f _ g
@[simp]
theorem
finsupp.map_range.add_equiv_trans
{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]
(f : M ≃+ N)
(f₂ : N ≃+ P) :
@[simp]
theorem
finsupp.map_range.add_equiv_symm
{α : Type u_1}
{M : Type u_5}
{N : Type u_7}
[add_comm_monoid M]
[add_comm_monoid N]
(f : M ≃+ N) :
@[simp]
theorem
finsupp.map_range.add_equiv_to_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) :
@[simp]
theorem
finsupp.map_range.add_equiv_to_equiv
{α : Type u_1}
{M : Type u_5}
{N : Type u_7}
[add_comm_monoid M]
[add_comm_monoid N]
(f : M ≃+ N) :
Declarations about map_domain #
noncomputable
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
@[simp]
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)
@[simp]
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
@[simp]
theorem
finsupp.map_domain_equiv_apply
{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) :
@[simp]
theorem
finsupp.map_domain.add_monoid_hom_apply
{α : Type u_1}
{β : Type u_2}
{M : Type u_5}
[add_comm_monoid M]
(f : α → β)
(v : α →₀ M) :
noncomputable
def
finsupp.map_domain.add_monoid_hom
{α : Type u_1}
{β : Type u_2}
{M : Type u_5}
[add_comm_monoid M]
(f : α → β) :
finsupp.map_domain is an add_monoid_hom.
Equations
- finsupp.map_domain.add_monoid_hom f = {to_fun := finsupp.map_domain f, map_zero' := _, map_add' := _}
@[simp]
theorem
finsupp.map_domain.add_monoid_hom_comp
{α : Type u_1}
{β : Type u_2}
{γ : Type u_3}
{M : Type u_5}
[add_comm_monoid M]
(f : β → γ)
(g : α → β) :
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]
[decidable_eq β]
{f : α → β}
{s : α →₀ M} :
(finsupp.map_domain f s).support ⊆ finset.image f s.support
theorem
finsupp.map_domain_apply'
{α : Type u_1}
{β : Type u_2}
{M : Type u_5}
[add_comm_monoid M]
(S : set α)
{f : α → β}
(x : α →₀ M)
(hS : ↑(x.support) ⊆ S)
(hf : set.inj_on f S)
{a : α}
(ha : a ∈ S) :
⇑(finsupp.map_domain f x) (f a) = ⇑x a
theorem
finsupp.map_domain_support_of_inj_on
{α : Type u_1}
{β : Type u_2}
{M : Type u_5}
[add_comm_monoid M]
[decidable_eq β]
{f : α → β}
(s : α →₀ M)
(hf : set.inj_on f ↑(s.support)) :
(finsupp.map_domain f s).support = finset.image f s.support
theorem
finsupp.map_domain_support_of_injective
{α : Type u_1}
{β : Type u_2}
{M : Type u_5}
[add_comm_monoid M]
[decidable_eq β]
{f : α → β}
(hf : function.injective f)
(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 : ∀ (b : β), h b 0 = 0)
(h_add : ∀ (b : β) (m₁ m₂ : M), h b (m₁ + m₂) = h b m₁ + h b m₂) :
(finsupp.map_domain f s).sum h = s.sum (λ (a : α) (m : M), h (f a) m)
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 : ∀ (b : β), h b 0 = 1)
(h_add : ∀ (b : β) (m₁ m₂ : M), h b (m₁ + m₂) = (h b m₁) * h b m₂) :
(finsupp.map_domain f s).prod h = s.prod (λ (a : α) (m : M), h (f a) m)
@[simp]
theorem
finsupp.sum_map_domain_index_add_monoid_hom
{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) :
A version of sum_map_domain_index that takes a bundled add_monoid_hom,
rather than separate linearity hypotheses.
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) :
When f is an embedding we have an embedding (α →₀ ℕ) ↪ (β →₀ ℕ) given by map_domain.
Equations
- finsupp.map_domain_embedding f = {to_fun := finsupp.map_domain ⇑f, inj' := _}
@[simp]
theorem
finsupp.map_domain.add_monoid_hom_comp_map_range
{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)
(g : M →+ N) :
theorem
finsupp.map_domain_map_range
{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)
(v : α →₀ M)
(g : M → N)
(h0 : g 0 = 0)
(hadd : ∀ (x y : M), g (x + y) = g x + g y) :
finsupp.map_domain f (finsupp.map_range g h0 v) = finsupp.map_range g h0 (finsupp.map_domain f v)
When g preserves addition, map_range and map_domain commute.
theorem
finsupp.sum_update_add
{α : Type u_1}
{β : Type u_2}
{ι : Type u_4}
[add_comm_monoid α]
[add_comm_monoid β]
(f : ι →₀ α)
(i : ι)
(a : α)
(g : ι → α → β)
(hg : ∀ (i : ι), g i 0 = 0)
(hgg : ∀ (j : ι) (a₁ a₂ : α), g j (a₁ + a₂) = g j a₁ + g j a₂) :
theorem
finsupp.map_domain_inj_on
{α : Type u_1}
{β : Type u_2}
{M : Type u_5}
[add_comm_monoid M]
(S : set α)
{f : α → β}
(hf : set.inj_on f S) :
set.inj_on (finsupp.map_domain f) {w : α →₀ M | ↑(w.support) ⊆ S}
Declarations about comap_domain #
noncomputable
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
@[simp]
theorem
finsupp.some_single_none
{α : Type u_1}
{M : Type u_5}
[has_zero M]
(m : M) :
(finsupp.single none m).some = 0
@[simp]
theorem
finsupp.some_single_some
{α : Type u_1}
{M : Type u_5}
[has_zero M]
(a : α)
(m : M) :
(finsupp.single (some a) m).some = finsupp.single a m
theorem
finsupp.prod_option_index
{α : Type u_1}
{M : Type u_5}
{N : Type u_7}
[add_comm_monoid M]
[comm_monoid N]
(f : option α →₀ M)
(b : option α → M → N)
(h_zero : ∀ (o : option α), b o 0 = 1)
(h_add : ∀ (o : option α) (m₁ m₂ : M), b o (m₁ + m₂) = (b o m₁) * b o m₂) :
theorem
finsupp.sum_option_index
{α : Type u_1}
{M : Type u_5}
{N : Type u_7}
[add_comm_monoid M]
[add_comm_monoid N]
(f : option α →₀ M)
(b : option α → M → N)
(h_zero : ∀ (o : option α), b o 0 = 0)
(h_add : ∀ (o : option α) (m₁ m₂ : M), b o (m₁ + m₂) = b o m₁ + b o m₂) :
Declarations about equiv_congr_left #
def
finsupp.equiv_map_domain
{α : Type u_1}
{β : Type u_2}
{M : Type u_5}
[has_zero M]
(f : α ≃ β)
(l : α →₀ M) :
β →₀ M
Given f : α ≃ β, we can map l : α →₀ M to equiv_map_domain f l : β →₀ M (computably)
by mapping the support forwards and the function backwards.
Equations
- finsupp.equiv_map_domain f l = {support := finset.map f.to_embedding l.support, to_fun := λ (a : β), ⇑l (⇑(f.symm) a), mem_support_to_fun := _}
@[simp]
theorem
finsupp.equiv_map_domain_refl
{α : Type u_1}
{M : Type u_5}
[has_zero M]
(l : α →₀ M) :
finsupp.equiv_map_domain (equiv.refl α) l = l
theorem
finsupp.equiv_map_domain_trans
{α : Type u_1}
{β : Type u_2}
{γ : Type u_3}
{M : Type u_5}
[has_zero M]
(f : α ≃ β)
(g : β ≃ γ)
(l : α →₀ M) :
finsupp.equiv_map_domain (f.trans g) l = finsupp.equiv_map_domain g (finsupp.equiv_map_domain f l)
theorem
finsupp.equiv_map_domain_trans'
{α : Type u_1}
{β : Type u_2}
{γ : Type u_3}
{M : Type u_5}
[has_zero M]
(f : α ≃ β)
(g : β ≃ γ) :
@[simp]
theorem
finsupp.equiv_map_domain_single
{α : Type u_1}
{β : Type u_2}
{M : Type u_5}
[has_zero M]
(f : α ≃ β)
(a : α)
(b : M) :
finsupp.equiv_map_domain f (finsupp.single a b) = finsupp.single (⇑f a) b
@[simp]
theorem
finsupp.equiv_map_domain_zero
{α : Type u_1}
{β : Type u_2}
{M : Type u_5}
[has_zero M]
{f : α ≃ β} :
finsupp.equiv_map_domain f 0 = 0
theorem
finsupp.equiv_map_domain_eq_map_domain
{α : Type u_1}
{β : Type u_2}
{M : Type u_3}
[add_comm_monoid M]
(f : α ≃ β)
(l : α →₀ M) :
def
finsupp.equiv_congr_left
{α : Type u_1}
{β : Type u_2}
{M : Type u_5}
[has_zero M]
(f : α ≃ β) :
Given f : α ≃ β, the finitely supported function spaces are also in bijection:
(α →₀ M) ≃ (β →₀ M).
This is the finitely-supported version of equiv.Pi_congr_left.
Equations
- finsupp.equiv_congr_left f = {to_fun := finsupp.equiv_map_domain f, inv_fun := finsupp.equiv_map_domain f.symm, left_inv := _, right_inv := _}
@[simp]
theorem
finsupp.equiv_congr_left_apply
{α : Type u_1}
{β : Type u_2}
{M : Type u_5}
[has_zero M]
(f : α ≃ β)
(l : α →₀ M) :
@[simp]
theorem
finsupp.equiv_congr_left_symm
{α : Type u_1}
{β : Type u_2}
{M : Type u_5}
[has_zero M]
(f : α ≃ β) :
noncomputable
def
finsupp.filter
{α : Type u_1}
{M : Type u_5}
[has_zero M]
(p : α → Prop)
(f : α →₀ M) :
α →₀ M
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)
[D : decidable (p a)] :
⇑(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)
[D : decidable_pred p] :
(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_zero_class 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 #
noncomputable
def
finsupp.subtype_domain
{α : Type u_1}
{M : Type u_5}
[has_zero M]
(p : α → Prop)
(f : α →₀ M) :
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}
[D : decidable_pred p]
{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_zero_class 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'
noncomputable
def
finsupp.subtype_domain_add_monoid_hom
{α : Type u_1}
{M : Type u_5}
[add_zero_class M]
{p : α → Prop} :
subtype_domain but as an add_monoid_hom.
Equations
- finsupp.subtype_domain_add_monoid_hom = {to_fun := finsupp.subtype_domain p, map_zero' := _, map_add' := _}
noncomputable
def
finsupp.filter_add_hom
{α : Type u_1}
{M : Type u_5}
[add_zero_class 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_zero_class 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)
[D : decidable_pred p]
(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'
@[simp]
theorem
finsupp.single_neg
{α : Type u_1}
{G : Type u_9}
[add_group G]
{a : α}
{b : G} :
finsupp.single a (-b) = -finsupp.single a b
@[simp]