Datasets:
Tasks:
Text Generation
Modalities:
Text
Sub-tasks:
language-modeling
Languages:
English
Size:
100K - 1M
License:
File size: 3,904 Bytes
4365a98 |
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 |
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import algebra.direct_sum.finsupp
import linear_algebra.finsupp
import linear_algebra.direct_sum.tensor_product
import data.finsupp.to_dfinsupp
/-!
# Results on finitely supported functions.
The tensor product of ι →₀ M and κ →₀ N is linearly equivalent to (ι × κ) →₀ (M ⊗ N).
-/
universes u v w
noncomputable theory
open_locale direct_sum
open set linear_map submodule
variables {R : Type u} {M : Type v} {N : Type w} [ring R] [add_comm_group M] [module R M]
[add_comm_group N] [module R N]
section tensor_product
open tensor_product
open_locale tensor_product classical
/-- The tensor product of ι →₀ M and κ →₀ N is linearly equivalent to (ι × κ) →₀ (M ⊗ N). -/
def finsupp_tensor_finsupp (R M N ι κ : Sort*) [comm_ring R]
[add_comm_group M] [module R M] [add_comm_group N] [module R N] :
(ι →₀ M) ⊗[R] (κ →₀ N) ≃ₗ[R] (ι × κ) →₀ (M ⊗[R] N) :=
(tensor_product.congr (finsupp_lequiv_direct_sum R M ι) (finsupp_lequiv_direct_sum R N κ))
≪≫ₗ ((tensor_product.direct_sum R ι κ (λ _, M) (λ _, N))
≪≫ₗ (finsupp_lequiv_direct_sum R (M ⊗[R] N) (ι × κ)).symm)
@[simp] theorem finsupp_tensor_finsupp_single (R M N ι κ : Sort*) [comm_ring R]
[add_comm_group M] [module R M] [add_comm_group N] [module R N]
(i : ι) (m : M) (k : κ) (n : N) :
finsupp_tensor_finsupp R M N ι κ (finsupp.single i m ⊗ₜ finsupp.single k n) =
finsupp.single (i, k) (m ⊗ₜ n) :=
by simp [finsupp_tensor_finsupp]
@[simp] theorem finsupp_tensor_finsupp_apply (R M N ι κ : Sort*) [comm_ring R]
[add_comm_group M] [module R M] [add_comm_group N] [module R N]
(f : ι →₀ M) (g : κ →₀ N) (i : ι) (k : κ) :
finsupp_tensor_finsupp R M N ι κ (f ⊗ₜ g) (i, k) = f i ⊗ₜ g k :=
begin
apply finsupp.induction_linear f,
{ simp, },
{ intros f₁ f₂ hf₁ hf₂, simp [add_tmul, hf₁, hf₂], },
{ intros i' m,
apply finsupp.induction_linear g,
{ simp, },
{ intros g₁ g₂ hg₁ hg₂, simp [tmul_add, hg₁, hg₂], },
{ intros k' n,
simp only [finsupp_tensor_finsupp_single],
simp only [finsupp.single, finsupp.coe_mk],
-- split_ifs; finish can close the goal from here
by_cases h1 : (i', k') = (i, k),
{ simp only [prod.mk.inj_iff] at h1, simp [h1] },
{ simp only [h1, if_false],
simp only [prod.mk.inj_iff, not_and_distrib] at h1,
cases h1; simp [h1] } } }
end
@[simp] theorem finsupp_tensor_finsupp_symm_single (R M N ι κ : Sort*) [comm_ring R]
[add_comm_group M] [module R M] [add_comm_group N] [module R N]
(i : ι × κ) (m : M) (n : N) :
(finsupp_tensor_finsupp R M N ι κ).symm (finsupp.single i (m ⊗ₜ n)) =
(finsupp.single i.1 m ⊗ₜ finsupp.single i.2 n) :=
prod.cases_on i $ λ i k, (linear_equiv.symm_apply_eq _).2
(finsupp_tensor_finsupp_single _ _ _ _ _ _ _ _ _).symm
variables (S : Type*) [comm_ring S] (α β : Type*)
/--
A variant of `finsupp_tensor_finsupp` where both modules are the ground ring.
-/
def finsupp_tensor_finsupp' : ((α →₀ S) ⊗[S] (β →₀ S)) ≃ₗ[S] (α × β →₀ S) :=
(finsupp_tensor_finsupp S S S α β).trans (finsupp.lcongr (equiv.refl _) (tensor_product.lid S S))
@[simp] lemma finsupp_tensor_finsupp'_apply_apply (f : α →₀ S) (g : β →₀ S) (a : α) (b : β) :
finsupp_tensor_finsupp' S α β (f ⊗ₜ[S] g) (a, b) = f a * g b :=
by simp [finsupp_tensor_finsupp']
@[simp] lemma finsupp_tensor_finsupp'_single_tmul_single (a : α) (b : β) (r₁ r₂ : S) :
finsupp_tensor_finsupp' S α β (finsupp.single a r₁ ⊗ₜ[S] finsupp.single b r₂) =
finsupp.single (a, b) (r₁ * r₂) :=
by { ext ⟨a', b'⟩, simp [finsupp.single, ite_and] }
end tensor_product
|