# Conversion between Finsupp and homogeneous DFinsupp#

This module provides conversions between Finsupp and DFinsupp. It is in its own file since neither Finsupp or DFinsupp depend on each other.

## Main definitions #

• "identity" maps between Finsupp and DFinsupp:
• Finsupp.toDFinsupp : (ι →₀ M) → (Π₀ i : ι, M)
• DFinsupp.toFinsupp : (Π₀ i : ι, M) → (ι →₀ M)
• Bundled equiv versions of the above:
• finsuppEquivDFinsupp : (ι →₀ M) ≃ (Π₀ i : ι, M)
• finsuppAddEquivDFinsupp : (ι →₀ M) ≃+ (Π₀ i : ι, M)
• finsuppLequivDFinsupp R : (ι →₀ M) ≃ₗ[R] (Π₀ i : ι, M)
• stronger versions of Finsupp.split:
• sigmaFinsuppEquivDFinsupp : ((Σ i, η i) →₀ N) ≃ (Π₀ i, (η i →₀ N))
• sigmaFinsuppAddEquivDFinsupp : ((Σ i, η i) →₀ N) ≃+ (Π₀ i, (η i →₀ N))
• sigmaFinsuppLequivDFinsupp : ((Σ i, η i) →₀ N) ≃ₗ[R] (Π₀ i, (η i →₀ N))

## Theorems #

The defining features of these operations is that they preserve the function and support:

• Finsupp.toDFinsupp_coe
• Finsupp.toDFinsupp_support
• DFinsupp.toFinsupp_coe
• DFinsupp.toFinsupp_support

and therefore map Finsupp.single to DFinsupp.single and vice versa:

• Finsupp.toDFinsupp_single
• DFinsupp.toFinsupp_single

as well as preserving arithmetic operations.

For the bundled equivalences, we provide lemmas that they reduce to Finsupp.toDFinsupp:

• finsupp_add_equiv_dfinsupp_apply
• finsupp_lequiv_dfinsupp_apply
• finsupp_add_equiv_dfinsupp_symm_apply
• finsupp_lequiv_dfinsupp_symm_apply

## Implementation notes #

We provide DFinsupp.toFinsupp and finsuppEquivDFinsupp computably by adding [DecidableEq ι] and [Π m : M, Decidable (m ≠ 0)] arguments. To aid with definitional unfolding, these arguments are also present on the noncomputable equivs.

### Basic definitions and lemmas #

def Finsupp.toDFinsupp {ι : Type u_1} {M : Type u_3} [Zero M] (f : ι →₀ M) :
Π₀ (x : ι), M

Interpret a Finsupp as a homogeneous DFinsupp.

Equations
• f.toDFinsupp = { toFun := f, support' := Trunc.mk f.support.val, }
Instances For
@[simp]
theorem Finsupp.toDFinsupp_coe {ι : Type u_1} {M : Type u_3} [Zero M] (f : ι →₀ M) :
f.toDFinsupp = f
@[simp]
theorem Finsupp.toDFinsupp_single {ι : Type u_1} {M : Type u_3} [] [Zero M] (i : ι) (m : M) :
(Finsupp.single i m).toDFinsupp =
@[simp]
theorem toDFinsupp_support {ι : Type u_1} {M : Type u_3} [] [Zero M] [(m : M) → Decidable (m 0)] (f : ι →₀ M) :
f.toDFinsupp.support = f.support
def DFinsupp.toFinsupp {ι : Type u_1} {M : Type u_3} [] [Zero M] [(m : M) → Decidable (m 0)] (f : Π₀ (x : ι), M) :
ι →₀ M

Interpret a homogeneous DFinsupp as a Finsupp.

Note that the elaborator has a lot of trouble with this definition - it is often necessary to write (DFinsupp.toFinsupp f : ι →₀ M) instead of f.toFinsupp, as for some unknown reason using dot notation or omitting the type ascription prevents the type being resolved correctly.

Equations
• f.toFinsupp = { support := f.support, toFun := f, mem_support_toFun := }
Instances For
@[simp]
theorem DFinsupp.toFinsupp_coe {ι : Type u_1} {M : Type u_3} [] [Zero M] [(m : M) → Decidable (m 0)] (f : Π₀ (x : ι), M) :
f.toFinsupp = f
@[simp]
theorem DFinsupp.toFinsupp_support {ι : Type u_1} {M : Type u_3} [] [Zero M] [(m : M) → Decidable (m 0)] (f : Π₀ (x : ι), M) :
f.toFinsupp.support = f.support
@[simp]
theorem DFinsupp.toFinsupp_single {ι : Type u_1} {M : Type u_3} [] [Zero M] [(m : M) → Decidable (m 0)] (i : ι) (m : M) :
(DFinsupp.single i m).toFinsupp =
@[simp]
theorem Finsupp.toDFinsupp_toFinsupp {ι : Type u_1} {M : Type u_3} [] [Zero M] [(m : M) → Decidable (m 0)] (f : ι →₀ M) :
f.toDFinsupp.toFinsupp = f
@[simp]
theorem DFinsupp.toFinsupp_toDFinsupp {ι : Type u_1} {M : Type u_3} [] [Zero M] [(m : M) → Decidable (m 0)] (f : Π₀ (x : ι), M) :
f.toFinsupp.toDFinsupp = f

### Lemmas about arithmetic operations #

@[simp]
theorem Finsupp.toDFinsupp_zero {ι : Type u_1} {M : Type u_3} [Zero M] :
@[simp]
theorem Finsupp.toDFinsupp_add {ι : Type u_1} {M : Type u_3} [] (f : ι →₀ M) (g : ι →₀ M) :
(f + g).toDFinsupp = f.toDFinsupp + g.toDFinsupp
@[simp]
theorem Finsupp.toDFinsupp_neg {ι : Type u_1} {M : Type u_3} [] (f : ι →₀ M) :
(-f).toDFinsupp = -f.toDFinsupp
@[simp]
theorem Finsupp.toDFinsupp_sub {ι : Type u_1} {M : Type u_3} [] (f : ι →₀ M) (g : ι →₀ M) :
(f - g).toDFinsupp = f.toDFinsupp - g.toDFinsupp
@[simp]
theorem Finsupp.toDFinsupp_smul {ι : Type u_1} {R : Type u_2} {M : Type u_3} [] [] [] (r : R) (f : ι →₀ M) :
(r f).toDFinsupp = r f.toDFinsupp
@[simp]
theorem DFinsupp.toFinsupp_zero {ι : Type u_1} {M : Type u_3} [] [Zero M] [(m : M) → Decidable (m 0)] :
@[simp]
theorem DFinsupp.toFinsupp_add {ι : Type u_1} {M : Type u_3} [] [] [(m : M) → Decidable (m 0)] (f : Π₀ (x : ι), M) (g : Π₀ (x : ι), M) :
(f + g).toFinsupp = f.toFinsupp + g.toFinsupp
@[simp]
theorem DFinsupp.toFinsupp_neg {ι : Type u_1} {M : Type u_3} [] [] [(m : M) → Decidable (m 0)] (f : Π₀ (x : ι), M) :
(-f).toFinsupp = -f.toFinsupp
@[simp]
theorem DFinsupp.toFinsupp_sub {ι : Type u_1} {M : Type u_3} [] [] [(m : M) → Decidable (m 0)] (f : Π₀ (x : ι), M) (g : Π₀ (x : ι), M) :
(f - g).toFinsupp = f.toFinsupp - g.toFinsupp
@[simp]
theorem DFinsupp.toFinsupp_smul {ι : Type u_1} {R : Type u_2} {M : Type u_3} [] [] [] [] [(m : M) → Decidable (m 0)] (r : R) (f : Π₀ (x : ι), M) :
(r f).toFinsupp = r f.toFinsupp

### Bundled Equivs #

@[simp]
theorem finsuppEquivDFinsupp_apply {ι : Type u_1} {M : Type u_3} [] [Zero M] [(m : M) → Decidable (m 0)] :
finsuppEquivDFinsupp = Finsupp.toDFinsupp
@[simp]
theorem finsuppEquivDFinsupp_symm_apply {ι : Type u_1} {M : Type u_3} [] [Zero M] [(m : M) → Decidable (m 0)] :
finsuppEquivDFinsupp.symm = DFinsupp.toFinsupp
def finsuppEquivDFinsupp {ι : Type u_1} {M : Type u_3} [] [Zero M] [(m : M) → Decidable (m 0)] :
(ι →₀ M) Π₀ (x : ι), M

Finsupp.toDFinsupp and DFinsupp.toFinsupp together form an equiv.

Equations
• finsuppEquivDFinsupp = { toFun := Finsupp.toDFinsupp, invFun := DFinsupp.toFinsupp, left_inv := , right_inv := }
Instances For
@[simp]
theorem finsuppAddEquivDFinsupp_symm_apply {ι : Type u_1} {M : Type u_3} [] [] [(m : M) → Decidable (m 0)] :
@[simp]
theorem finsuppAddEquivDFinsupp_apply {ι : Type u_1} {M : Type u_3} [] [] [(m : M) → Decidable (m 0)] :
def finsuppAddEquivDFinsupp {ι : Type u_1} {M : Type u_3} [] [] [(m : M) → Decidable (m 0)] :
(ι →₀ M) ≃+ Π₀ (x : ι), M

The additive version of finsupp.toFinsupp. Note that this is noncomputable because Finsupp.add is noncomputable.

Equations
• finsuppAddEquivDFinsupp = { toFun := Finsupp.toDFinsupp, invFun := DFinsupp.toFinsupp, left_inv := , right_inv := , map_add' := }
Instances For
def finsuppLequivDFinsupp {ι : Type u_1} (R : Type u_2) {M : Type u_3} [] [] [] [(m : M) → Decidable (m 0)] [Module R M] :
(ι →₀ M) ≃ₗ[R] Π₀ (x : ι), M

The additive version of Finsupp.toFinsupp. Note that this is noncomputable because Finsupp.add is noncomputable.

Equations
• = { toFun := Finsupp.toDFinsupp, map_add' := , map_smul' := , invFun := DFinsupp.toFinsupp, left_inv := , right_inv := }
Instances For
@[simp]
theorem finsuppLequivDFinsupp_apply_apply {ι : Type u_1} (R : Type u_2) {M : Type u_3} [] [] [] [(m : M) → Decidable (m 0)] [Module R M] :
= Finsupp.toDFinsupp
@[simp]
theorem finsuppLequivDFinsupp_symm_apply {ι : Type u_1} (R : Type u_2) {M : Type u_3} [] [] [] [(m : M) → Decidable (m 0)] [Module R M] :
.symm = DFinsupp.toFinsupp

### Stronger versions of Finsupp.split#

def sigmaFinsuppEquivDFinsupp {ι : Type u_1} {η : ιType u_4} {N : Type u_5} [Zero N] :
((i : ι) × η i →₀ N) Π₀ (i : ι), η i →₀ N

Finsupp.split is an equivalence between (Σ i, η i) →₀ N and Π₀ i, (η i →₀ N).

Equations
• One or more equations did not get rendered due to their size.
Instances For
@[simp]
theorem sigmaFinsuppEquivDFinsupp_apply {ι : Type u_1} {η : ιType u_4} {N : Type u_5} [Zero N] (f : (i : ι) × η i →₀ N) :
(sigmaFinsuppEquivDFinsupp f) = f.split
@[simp]
theorem sigmaFinsuppEquivDFinsupp_symm_apply {ι : Type u_1} {η : ιType u_4} {N : Type u_5} [Zero N] (f : Π₀ (i : ι), η i →₀ N) (s : (i : ι) × η i) :
(sigmaFinsuppEquivDFinsupp.symm f) s = (f s.fst) s.snd
@[simp]
theorem sigmaFinsuppEquivDFinsupp_support {ι : Type u_1} {η : ιType u_4} {N : Type u_5} [] [Zero N] [(i : ι) → (x : η i →₀ N) → Decidable (x 0)] (f : (i : ι) × η i →₀ N) :
(sigmaFinsuppEquivDFinsupp f).support = f.splitSupport
@[simp]
theorem sigmaFinsuppEquivDFinsupp_single {ι : Type u_1} {η : ιType u_4} {N : Type u_5} [] [Zero N] (a : (i : ι) × η i) (n : N) :
sigmaFinsuppEquivDFinsupp (Finsupp.single a n) = DFinsupp.single a.fst (Finsupp.single a.snd n)
@[simp]
theorem sigmaFinsuppEquivDFinsupp_add {ι : Type u_1} {η : ιType u_4} {N : Type u_5} [] (f : (i : ι) × η i →₀ N) (g : (i : ι) × η i →₀ N) :
sigmaFinsuppEquivDFinsupp (f + g) = sigmaFinsuppEquivDFinsupp f + sigmaFinsuppEquivDFinsupp g
@[simp]
theorem sigmaFinsuppAddEquivDFinsupp_apply {ι : Type u_1} {η : ιType u_4} {N : Type u_5} [] (a : (i : ι) × η i →₀ N) :
@[simp]
theorem sigmaFinsuppAddEquivDFinsupp_symm_apply {ι : Type u_1} {η : ιType u_4} {N : Type u_5} [] (a : Π₀ (i : ι), η i →₀ N) :
def sigmaFinsuppAddEquivDFinsupp {ι : Type u_1} {η : ιType u_4} {N : Type u_5} [] :
((i : ι) × η i →₀ N) ≃+ Π₀ (i : ι), η i →₀ N

Finsupp.split is an additive equivalence between (Σ i, η i) →₀ N and Π₀ i, (η i →₀ N).

Equations
• sigmaFinsuppAddEquivDFinsupp = { toFun := sigmaFinsuppEquivDFinsupp, invFun := sigmaFinsuppEquivDFinsupp.symm, left_inv := , right_inv := , map_add' := }
Instances For
@[simp]
theorem sigmaFinsuppEquivDFinsupp_smul {ι : Type u_1} {η : ιType u_4} {N : Type u_5} {R : Type u_6} [] [] [] (r : R) (f : (i : ι) × η i →₀ N) :
sigmaFinsuppEquivDFinsupp (r f) = r sigmaFinsuppEquivDFinsupp f
@[simp]
theorem sigmaFinsuppLequivDFinsupp_symm_apply {ι : Type u_1} (R : Type u_2) {η : ιType u_4} {N : Type u_5} [] [] [Module R N] (a : Π₀ (i : ι), η i →₀ N) :
.symm a = sigmaFinsuppEquivDFinsupp.symm a
@[simp]
theorem sigmaFinsuppLequivDFinsupp_apply {ι : Type u_1} (R : Type u_2) {η : ιType u_4} {N : Type u_5} [] [] [Module R N] (a : (i : ι) × η i →₀ N) :
= sigmaFinsuppEquivDFinsupp a
def sigmaFinsuppLequivDFinsupp {ι : Type u_1} (R : Type u_2) {η : ιType u_4} {N : Type u_5} [] [] [Module R N] :
((i : ι) × η i →₀ N) ≃ₗ[R] Π₀ (i : ι), η i →₀ N

Finsupp.split is a linear equivalence between (Σ i, η i) →₀ N and Π₀ i, (η i →₀ N).

Equations
• = { toFun := sigmaFinsuppEquivDFinsupp, map_add' := , map_smul' := , invFun := sigmaFinsuppEquivDFinsupp.symm, left_inv := , right_inv := }
Instances For