Documentation

Mathlib.Data.Finsupp.ToDFinsupp

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 #

Theorems #

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

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

as well as preserving arithmetic operations.

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

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
    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} [DecidableEq ι] [Zero M] (i : ι) (m : M) :
      @[simp]
      theorem toDFinsupp_support {ι : Type u_1} {M : Type u_3} [DecidableEq ι] [Zero M] [(m : M) → Decidable (m 0)] (f : ι →₀ M) :
      def DFinsupp.toFinsupp {ι : Type u_1} {M : Type u_3} [DecidableEq ι] [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
        Instances For
          @[simp]
          theorem DFinsupp.toFinsupp_coe {ι : Type u_1} {M : Type u_3} [DecidableEq ι] [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} [DecidableEq ι] [Zero M] [(m : M) → Decidable (m 0)] (f : Π₀ (x : ι), M) :
          @[simp]
          theorem DFinsupp.toFinsupp_single {ι : Type u_1} {M : Type u_3} [DecidableEq ι] [Zero M] [(m : M) → Decidable (m 0)] (i : ι) (m : M) :
          @[simp]
          theorem Finsupp.toDFinsupp_toFinsupp {ι : Type u_1} {M : Type u_3} [DecidableEq ι] [Zero M] [(m : M) → Decidable (m 0)] (f : ι →₀ M) :
          @[simp]
          theorem DFinsupp.toFinsupp_toDFinsupp {ι : Type u_1} {M : Type u_3} [DecidableEq ι] [Zero M] [(m : M) → Decidable (m 0)] (f : Π₀ (x : ι), M) :

          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} [AddZeroClass M] (f g : ι →₀ M) :
          @[simp]
          theorem Finsupp.toDFinsupp_neg {ι : Type u_1} {M : Type u_3} [AddGroup M] (f : ι →₀ M) :
          @[simp]
          theorem Finsupp.toDFinsupp_sub {ι : Type u_1} {M : Type u_3} [AddGroup M] (f g : ι →₀ M) :
          @[simp]
          theorem Finsupp.toDFinsupp_smul {ι : Type u_1} {R : Type u_2} {M : Type u_3} [Monoid R] [AddMonoid M] [DistribMulAction R M] (r : R) (f : ι →₀ M) :
          @[simp]
          theorem DFinsupp.toFinsupp_zero {ι : Type u_1} {M : Type u_3} [DecidableEq ι] [Zero M] [(m : M) → Decidable (m 0)] :
          @[simp]
          theorem DFinsupp.toFinsupp_add {ι : Type u_1} {M : Type u_3} [DecidableEq ι] [AddZeroClass M] [(m : M) → Decidable (m 0)] (f g : Π₀ (x : ι), M) :
          @[simp]
          theorem DFinsupp.toFinsupp_neg {ι : Type u_1} {M : Type u_3} [DecidableEq ι] [AddGroup M] [(m : M) → Decidable (m 0)] (f : Π₀ (x : ι), M) :
          @[simp]
          theorem DFinsupp.toFinsupp_sub {ι : Type u_1} {M : Type u_3} [DecidableEq ι] [AddGroup M] [(m : M) → Decidable (m 0)] (f g : Π₀ (x : ι), M) :
          @[simp]
          theorem DFinsupp.toFinsupp_smul {ι : Type u_1} {R : Type u_2} {M : Type u_3} [DecidableEq ι] [Monoid R] [AddMonoid M] [DistribMulAction R M] [(m : M) → Decidable (m 0)] (r : R) (f : Π₀ (x : ι), M) :

          Bundled Equivs #

          def finsuppEquivDFinsupp {ι : Type u_1} {M : Type u_3} [DecidableEq ι] [Zero M] [(m : M) → Decidable (m 0)] :
          (ι →₀ M) Π₀ (x : ι), M

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

          Equations
            Instances For
              @[simp]
              @[simp]
              theorem finsuppEquivDFinsupp_apply {ι : Type u_1} {M : Type u_3} [DecidableEq ι] [Zero M] [(m : M) → Decidable (m 0)] :
              def finsuppAddEquivDFinsupp {ι : Type u_1} {M : Type u_3} [DecidableEq ι] [AddZeroClass M] [(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
                Instances For
                  def finsuppLequivDFinsupp {ι : Type u_1} (R : Type u_2) {M : Type u_3} [DecidableEq ι] [Semiring R] [AddCommMonoid M] [(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
                    Instances For
                      @[simp]
                      theorem finsuppLequivDFinsupp_apply_apply {ι : Type u_1} (R : Type u_2) {M : Type u_3} [DecidableEq ι] [Semiring R] [AddCommMonoid M] [(m : M) → Decidable (m 0)] [Module R M] :
                      @[simp]
                      theorem finsuppLequivDFinsupp_symm_apply {ι : Type u_1} (R : Type u_2) {M : Type u_3} [DecidableEq ι] [Semiring R] [AddCommMonoid M] [(m : M) → Decidable (m 0)] [Module R M] :

                      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
                        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) :
                          @[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) :
                          @[simp]
                          theorem sigmaFinsuppEquivDFinsupp_support {ι : Type u_1} {η : ιType u_4} {N : Type u_5} [DecidableEq ι] [Zero N] [(i : ι) → (x : η i →₀ N) → Decidable (x 0)] (f : (i : ι) × η i →₀ N) :
                          @[simp]
                          theorem sigmaFinsuppEquivDFinsupp_single {ι : Type u_1} {η : ιType u_4} {N : Type u_5} [DecidableEq ι] [Zero N] (a : (i : ι) × η i) (n : N) :
                          @[simp]
                          theorem sigmaFinsuppEquivDFinsupp_add {ι : Type u_1} {η : ιType u_4} {N : Type u_5} [AddZeroClass N] (f g : (i : ι) × η i →₀ N) :
                          def sigmaFinsuppAddEquivDFinsupp {ι : Type u_1} {η : ιType u_4} {N : Type u_5} [AddZeroClass N] :
                          ((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
                            Instances For
                              @[simp]
                              theorem sigmaFinsuppAddEquivDFinsupp_apply {ι : Type u_1} {η : ιType u_4} {N : Type u_5} [AddZeroClass N] (a : (i : ι) × η i →₀ N) :
                              @[simp]
                              @[simp]
                              theorem sigmaFinsuppEquivDFinsupp_smul {ι : Type u_1} {η : ιType u_4} {N : Type u_5} {R : Type u_6} [Monoid R] [AddMonoid N] [DistribMulAction R N] (r : R) (f : (i : ι) × η i →₀ N) :
                              def sigmaFinsuppLequivDFinsupp {ι : Type u_1} (R : Type u_2) {η : ιType u_4} {N : Type u_5} [Semiring R] [AddCommMonoid N] [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
                                Instances For
                                  @[simp]
                                  theorem sigmaFinsuppLequivDFinsupp_symm_apply {ι : Type u_1} (R : Type u_2) {η : ιType u_4} {N : Type u_5} [Semiring R] [AddCommMonoid N] [Module R N] (a✝ : Π₀ (i : ι), η i →₀ N) :
                                  @[simp]
                                  theorem sigmaFinsuppLequivDFinsupp_apply {ι : Type u_1} (R : Type u_2) {η : ιType u_4} {N : Type u_5} [Semiring R] [AddCommMonoid N] [Module R N] (a✝ : (i : ι) × η i →₀ N) :