feat(linear_algebra/pi_tensor_product): define the tensor product of …

…an indexed family of semimodules (#5311)

This PR defines the tensor product of an indexed family `s : ι → Type*` of semimodules over commutative semirings. We denote this space by `⨂[R] i, s i` and define it as `free_add_monoid (R × Π i, s i)` quotiented by the appropriate equivalence relation. The treatment follows very closely that of the binary tensor product in `linear_algebra/tensor_product.lean`.



Co-authored-by: Frédéric Dupuis <31101893+dupuisf@users.noreply.github.com>
Co-authored-by: Eric Wieser <wieser.eric@gmail.com>

src/linear_algebra/pi_tensor_product.lean

@@ -0,0 +1,382 @@
+/-
+Copyright (c) 2020 Frédéric Dupuis. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Frédéric Dupuis
+-/
+
+import group_theory.congruence
+import linear_algebra.multilinear
+
+/-!
+# Tensor product of an indexed family of semimodules over commutative semirings
+
+We define the tensor product of an indexed family `s : ι → Type*` of semimodules over commutative
+semirings. We denote this space by `⨂[R] i, s i` and define it as `free_add_monoid (R × Π i, s i)`
+quotiented by the appropriate equivalence relation. The treatment follows very closely that of the
+binary tensor product in `linear_algebra/tensor_product.lean`.
+
+## Main definitions
+
+* `pi_tensor_product R s` with `R` a commutative semiring and `s : ι → Type*` is the tensor product
+  of all the `s i`'s. This is denoted by `⨂[R] i, s i`.
+* `tprod R f` with `f : Π i, s i` is the tensor product of the vectors `f i` over all `i : ι`.
+  This is bundled as a multilinear map from `Π i, s i` to `⨂[R] i, s i`.
+* `lift_add_hom` constructs an `add_monoid_hom` from `(⨂[R] i, s i)` to some space `F` from a
+  function `φ : (R × Π i, s i) → F` with the appropriate properties.
+* `lift φ` with `φ : multilinear_map R s E` is the corresponding linear map
+  `(⨂[R] i, s i) →ₗ[R] E`. This is bundled as a linear equivalence.
+
+## Notations
+
+* `⨂[R] i, s i` is defined as localized notation in locale `tensor_product`
+* `⨂ₜ[R] i, f i` with `f : Π i, f i` is defined globally as the tensor product of all the `f i`'s.
+
+## Implementation notes
+
+* We define it via `free_add_monoid (R × Π i, s i)` with the `R` representing a "hidden" tensor
+  factor, rather than `free_add_monoid (Π i, s i)` to ensure that, if `ι` is an empty type,
+  the space is isomorphic to the base ring `R`.
+* We have not restricted the index type `ι` to be a `fintype`, as nothing we do here strictly
+  requires it. However, problems may arise in the case where `ι` is infinite; use at your own
+  caution.
+
+## TODO
+
+* Define tensor powers, symmetric subspace, etc.
+* API for the various ways `ι` can be split into subsets; connect this with the binary
+  tensor product.
+* Include connection with holors.
+* Port more of the API from the binary tensor product over to this case.
+
+## Tags
+
+multilinear, tensor, tensor product
+-/
+
+noncomputable theory
+open_locale classical
+open function
+
+section semiring
+
+variables {ι : Type*} {R : Type*} [comm_semiring R]
+variables {R' : Type*} [comm_semiring R'] [algebra R' R]
+variables {s : ι → Type*} [∀ i, add_comm_monoid (s i)] [∀ i, semimodule R (s i)]
+variables {E : Type*} [add_comm_monoid E] [semimodule R E]
+variables {F : Type*} [add_comm_monoid F]
+
+namespace pi_tensor_product
+include R
+variables (R) (s)
+
+/-- The relation on `free_add_monoid (R × Π i, s i)` that generates a congruence whose quotient is
+the tensor product. -/
+inductive eqv : free_add_monoid (R × Π i, s i) → free_add_monoid (R × Π i, s i) → Prop
+| of_zero : ∀ (r : R) (f : Π i, s i) (i : ι) (hf : f i = 0), eqv (free_add_monoid.of (r, f)) 0
+| of_zero_scalar : ∀ (f : Π i, s i), eqv (free_add_monoid.of (0, f)) 0
+| of_add : ∀ (r : R) (f : Π i, s i) (i : ι) (m₁ m₂ : s i), eqv
+    (free_add_monoid.of (r, update f i m₁) + free_add_monoid.of (r, update f i m₂))
+    (free_add_monoid.of (r, update f i (m₁ + m₂)))
+| of_add_scalar : ∀ (r r' : R) (f : Π i, s i), eqv
+    (free_add_monoid.of (r, f) + free_add_monoid.of (r', f))
+    (free_add_monoid.of (r + r', f))
+| of_smul : ∀ (r : R) (f : Π i, s i) (i : ι) (r' : R), eqv
+    (free_add_monoid.of (r, update f i (r' • (f i))))
+    (free_add_monoid.of (r' * r, f))
+| add_comm : ∀ x y, eqv (x + y) (y + x)
+
+end pi_tensor_product
+
+variables (R) (s)
+
+/-- `pi_tensor_product R s` with `R` a commutative semiring and `s : ι → Type*` is the tensor
+  product of all the `s i`'s. This is denoted by `⨂[R] i, s i`. -/
+def pi_tensor_product : Type* :=
+(add_con_gen (pi_tensor_product.eqv R s)).quotient
+
+variables {R}
+
+/- This enables the notation `⨂[R] i : ι, s i` for the pi tensor product, given `s : ι → Type*`. -/
+localized "notation `⨂[`:100 R `] ` binders `, ` r:(scoped:67 f, pi_tensor_product R f) := r"
+  in tensor_product
+
+open_locale tensor_product
+
+namespace pi_tensor_product
+
+section module
+
+instance : add_comm_monoid (⨂[R] i, s i) :=
+{ add_comm := λ x y, add_con.induction_on₂ x y $ λ x y, quotient.sound' $
+    add_con_gen.rel.of _ _ $ eqv.add_comm _ _,
+  .. (add_con_gen (pi_tensor_product.eqv R s)).add_monoid }
+
+instance : inhabited (⨂[R] i, s i) := ⟨0⟩
+
+variables (R) {s}
+
+/-- `tprod_coeff R r f` with `r : R` and `f : Π i, s i` is the tensor product of the vectors `f i`
+over all `i : ι`, multiplied by the coefficient `r`. Note that this is meant as an auxiliary
+definition for this file alone, and that one should use `tprod` defined below for most purposes. -/
+def tprod_coeff (r : R) (f : Π i, s i) : ⨂[R] i, s i := add_con.mk' _ $ free_add_monoid.of (r, f)
+
+variables {R}
+
+lemma zero_tprod_coeff (f : Π i, s i) : tprod_coeff R 0 f = 0 :=
+quotient.sound' $ add_con_gen.rel.of _ _ $ eqv.of_zero_scalar _
+
+lemma zero_tprod_coeff' (z : R) (f : Π i, s i) (i : ι) (hf: f i = 0) : tprod_coeff R z f = 0 :=
+quotient.sound' $ add_con_gen.rel.of _ _ $ eqv.of_zero _ _ i hf
+
+lemma add_tprod_coeff (z : R) (f : Π i, s i) (i : ι) (m₁ m₂ : s i) :
+  tprod_coeff R z (update f i m₁) + tprod_coeff R z (update f i m₂) =
+    tprod_coeff R z (update f i (m₁ + m₂)) :=
+quotient.sound' $ add_con_gen.rel.of _ _ (eqv.of_add z f i m₁ m₂)
+
+lemma add_tprod_coeff' (z₁ z₂ : R) (f : Π i, s i) :
+  tprod_coeff R z₁ f + tprod_coeff R z₂ f = tprod_coeff R (z₁ + z₂) f :=
+quotient.sound' $ add_con_gen.rel.of _ _ (eqv.of_add_scalar z₁ z₂ f)
+
+lemma smul_tprod_coeff_aux (z : R) (f : Π i, s i) (i : ι) (r : R) :
+  tprod_coeff R z (update f i (r • f i)) = tprod_coeff R (r * z) f :=
+ quotient.sound' $ add_con_gen.rel.of _ _ $ eqv.of_smul _ _ _ _
+
+lemma smul_tprod_coeff (z : R) (f : Π i, s i) (i : ι) (r : R')
+  [semimodule R' (s i)] [is_scalar_tower R' R (s i)] :
+  tprod_coeff R z (update f i (r • f i)) = tprod_coeff R (r • z) f :=
+begin
+  have h₁ : r • z = (r • (1 : R)) * z := by simp,
+  have h₂ : r • (f i) = (r • (1 : R)) • f i := by simp,
+  rw [h₁, h₂],
+  exact smul_tprod_coeff_aux z f i _,
+end
+
+/-- Construct an `add_monoid_hom` from `(⨂[R] i, s i)` to some space `F` from a function
+`φ : (R × Π i, s i) → F` with the appropriate properties. -/
+def lift_add_hom (φ : (R × Π i, s i) → F)
+  (C0 : ∀ (r : R) (f : Π i, s i) (i : ι) (hf : f i = 0), φ (r, f) = 0)
+  (C0' : ∀ (f : Π i, s i), φ (0, f) = 0)
+  (C_add : ∀ (r : R) (f : Π i, s i) (i : ι) (m₁ m₂ : s i),
+    φ (r, update f i m₁) + φ (r, update f i m₂) = φ (r, update f i (m₁ + m₂)))
+  (C_add_scalar : ∀ (r r' : R) (f : Π i, s i),
+    φ (r , f) + φ (r', f) = φ (r + r', f))
+  (C_smul : ∀ (r : R) (f : Π i, s i) (i : ι) (r' : R),
+    φ (r, update f i (r' • (f i))) = φ (r' * r, f))
+: (⨂[R] i, s i) →+ F :=
+(add_con_gen (pi_tensor_product.eqv R s)).lift (free_add_monoid.lift φ) $ add_con.add_con_gen_le $
+λ x y hxy, match x, y, hxy with
+| _, _, (eqv.of_zero r' f i hf)        := (add_con.ker_rel _).2 $
+    by simp [free_add_monoid.lift_eval_of, C0 r' f i hf]
+| _, _, (eqv.of_zero_scalar f)        := (add_con.ker_rel _).2 $
+    by simp [free_add_monoid.lift_eval_of, C0']
+| _, _, (eqv.of_add z f i m₁ m₂)      := (add_con.ker_rel _).2 $
+    by simp [free_add_monoid.lift_eval_of, C_add]
+| _, _, (eqv.of_add_scalar z₁ z₂ f)      := (add_con.ker_rel _).2 $
+    by simp [free_add_monoid.lift_eval_of, C_add_scalar]
+| _, _, (eqv.of_smul z f i r')     := (add_con.ker_rel _).2 $
+    by simp [free_add_monoid.lift_eval_of, C_smul]
+| _, _, (eqv.add_comm x y)         := (add_con.ker_rel _).2 $
+    by simp_rw [add_monoid_hom.map_add, add_comm]
+end
+
+-- Most of the time we want the instance below this one, which is easier for typeclass resolution
+-- to find.
+instance has_scalar' : has_scalar R' (⨂[R] i, s i) :=
+⟨λ r, lift_add_hom (λ f : R × Π i, s i, tprod_coeff R (r • f.1) f.2)
+  (λ r' f i hf, by simp_rw [zero_tprod_coeff' _ f i hf])
+  (λ f, by simp [zero_tprod_coeff])
+  (λ r' f i m₁ m₂, by simp [add_tprod_coeff])
+  (λ r' r'' f, by simp [add_tprod_coeff', mul_add])
+  (λ z f i r', by simp [smul_tprod_coeff])⟩
+
+instance : has_scalar R (⨂[R] i, s i) := pi_tensor_product.has_scalar'
+
+lemma smul_tprod_coeff' (r : R') (z : R) (f : Π i, s i) :
+  r • (tprod_coeff R z f) = tprod_coeff R (r • z) f := rfl
+
+protected theorem smul_add (r : R') (x y : ⨂[R] i, s i) :
+  r • (x + y) = r • x + r • y :=
+add_monoid_hom.map_add _ _ _
+
+@[elab_as_eliminator]
+protected theorem induction_on'
+  {C : (⨂[R] i, s i) → Prop}
+  (z : ⨂[R] i, s i)
+  (C1 : ∀ {r : R} {f : Π i, s i}, C (tprod_coeff R r f))
+  (Cp : ∀ {x y}, C x → C y → C (x + y)) : C z :=
+begin
+  have C0 : C 0,
+  { have h₁ := @C1 0 0,
+    rwa [zero_tprod_coeff] at h₁ },
+  refine add_con.induction_on z (λ x, free_add_monoid.rec_on x C0 _),
+  simp_rw add_con.coe_add,
+  refine λ f y ih, Cp _ ih,
+  convert @C1 f.1 f.2,
+  simp only [prod.mk.eta],
+end
+
+-- Most of the time we want the instance below this one, which is easier for typeclass resolution
+-- to find.
+instance semimodule' : semimodule R' (⨂[R] i, s i) :=
+{ smul := (•),
+  smul_add := λ r x y, pi_tensor_product.smul_add r x y,
+  mul_smul := λ r r' x,
+    begin
+      refine pi_tensor_product.induction_on' x _ _,
+      { intros r'' f,
+        simp [smul_tprod_coeff', smul_smul] },
+      { intros x y ihx ihy,
+        simp [pi_tensor_product.smul_add, ihx, ihy] }
+    end,
+  one_smul := λ x, pi_tensor_product.induction_on' x
+    (λ f, by simp [smul_tprod_coeff' _ _])
+    (λ z y ihz ihy, by simp_rw [pi_tensor_product.smul_add, ihz, ihy]),
+  add_smul := λ r r' x,
+    begin
+      refine pi_tensor_product.induction_on' x _ _,
+      { intros r f,
+        simp [smul_tprod_coeff' _ _, add_smul, add_tprod_coeff'] },
+      { intros x y ihx ihy,
+        simp [pi_tensor_product.smul_add, ihx, ihy, add_add_add_comm] }
+    end,
+  smul_zero := λ r, add_monoid_hom.map_zero _,
+  zero_smul := λ x,
+    begin
+      refine pi_tensor_product.induction_on' x _ _,
+      { intros r f,
+        simp_rw [smul_tprod_coeff' _ _, zero_smul],
+        exact zero_tprod_coeff _ },
+      { intros x y ihx ihy,
+        rw [pi_tensor_product.smul_add, ihx, ihy, add_zero] },
+    end }
+
+instance : semimodule R' (⨂[R] i, s i) := pi_tensor_product.semimodule'
+
+variables {R}
+
+variables (R)
+/-- The canonical `multilinear_map R s (⨂[R] i, s i)`. -/
+def tprod : multilinear_map R s (⨂[R] i, s i) :=
+{ to_fun := tprod_coeff R 1,
+  map_add' := λ f i x y, (add_tprod_coeff (1 : R) f i x y).symm,
+  map_smul' := λ f i r x,
+    by simp_rw [smul_tprod_coeff', ←smul_tprod_coeff (1 : R) _ i, update_idem, update_same] }
+
+variables {R}
+
+notation `⨂ₜ[`:100 R`] ` binders `, ` r:(scoped:67 f, tprod R f) := r
+
+@[simp]
+lemma tprod_coeff_eq_smul_tprod (z : R) (f : Π i, s i) : tprod_coeff R z f = z • tprod R f :=
+begin
+  have : z = z • (1 : R) := by simp only [mul_one, algebra.id.smul_eq_mul],
+  conv_lhs { rw this },
+  rw ←smul_tprod_coeff',
+  refl,
+end
+
+@[elab_as_eliminator]
+protected theorem induction_on
+  {C : (⨂[R] i, s i) → Prop}
+  (z : ⨂[R] i, s i)
+  (C1 : ∀ {r : R} {f : Π i, s i}, C (r • (tprod R f)))
+  (Cp : ∀ {x y}, C x → C y → C (x + y)) : C z :=
+begin
+  simp_rw ←tprod_coeff_eq_smul_tprod at C1,
+  exact pi_tensor_product.induction_on' z @C1 @Cp,
+end
+
+@[ext]
+theorem ext {φ₁ φ₂ : (⨂[R] i, s i) →ₗ[R] E}
+  (H : φ₁.comp_multilinear_map (tprod R) = φ₂.comp_multilinear_map (tprod R)) : φ₁ = φ₂ :=
+begin
+  refine linear_map.ext _,
+  refine λ z,
+    (pi_tensor_product.induction_on' z _ (λ x y hx hy, by rw [φ₁.map_add, φ₂.map_add, hx, hy])),
+  { intros r f,
+    rw [tprod_coeff_eq_smul_tprod, φ₁.map_smul, φ₂.map_smul],
+    apply _root_.congr_arg,
+    exact multilinear_map.congr_fun H f }
+end
+
+end module
+
+section multilinear
+open multilinear_map
+variables {s}
+
+/-- Auxiliary function to constructing a linear map `(⨂[R] i, s i) → E` given a
+`multilinear map R s E` with the property that its composition with the canonical
+`multilinear_map R s (⨂[R] i, s i)` is the given multilinear map. -/
+def lift_aux (φ : multilinear_map R s E) : (⨂[R] i, s i) →+ E :=
+  lift_add_hom (λ (p : R × Π i, s i), p.1 • (φ p.2))
+    (λ z f i hf, by rw [map_coord_zero φ i hf, smul_zero])
+    (λ f, by rw [zero_smul])
+    (λ z f i m₁ m₂, by rw [←smul_add, φ.map_add])
+    (λ z₁ z₂ f, by rw [←add_smul])
+    (λ z f i r, by simp [φ.map_smul, smul_smul, mul_comm])
+
+lemma lift_aux_tprod (φ : multilinear_map R s E) (f : Π i, s i) : lift_aux φ (tprod R f) = φ f :=
+by simp only [lift_aux, lift_add_hom, tprod, multilinear_map.coe_mk, tprod_coeff,
+              free_add_monoid.lift_eval_of, one_smul, add_con.lift_mk']
+
+lemma lift_aux_tprod_coeff (φ : multilinear_map R s E) (z : R) (f : Π i, s i) :
+  lift_aux φ (tprod_coeff R z f) = z • φ f :=
+by simp [lift_aux, lift_add_hom, tprod_coeff, free_add_monoid.lift_eval_of]
+
+lemma lift_aux.smul {φ : multilinear_map R s E} (r : R) (x : ⨂[R] i, s i) :
+  lift_aux φ (r • x) = r • lift_aux φ x :=
+begin
+  refine pi_tensor_product.induction_on' x _ _,
+  { intros z f,
+    rw [smul_tprod_coeff' r z f, lift_aux_tprod_coeff, lift_aux_tprod_coeff, smul_assoc] },
+  { intros z y ihz ihy,
+    rw [smul_add, (lift_aux φ).map_add, ihz, ihy, (lift_aux φ).map_add, smul_add] }
+end
+
+/-- Constructing a linear map `(⨂[R] i, s i) → E` given a `multilinear_map R s E` with the
+property that its composition with the canonical `multilinear_map R s E` is
+the given multilinear map `φ`. -/
+def lift : (multilinear_map R s E) ≃ₗ[R] ((⨂[R] i, s i) →ₗ[R] E) :=
+{ to_fun := λ φ, { map_smul' := lift_aux.smul, .. lift_aux φ },
+  inv_fun := λ φ', φ'.comp_multilinear_map (tprod R),
+  left_inv := λ φ, by { ext, simp [lift_aux_tprod, linear_map.comp_multilinear_map] },
+  right_inv := λ φ, by { ext, simp [lift_aux_tprod] },
+  map_add' := λ φ₁ φ₂, by { ext, simp [lift_aux_tprod] },
+  map_smul' := λ r φ₂, by { ext, simp [lift_aux_tprod] } }
+
+variables {φ : multilinear_map R s E}
+
+@[simp] lemma lift.tprod (f : Π i, s i) : lift φ (tprod R f) = φ f := lift_aux_tprod φ f
+theorem lift.unique' {φ' : (⨂[R] i, s i) →ₗ[R] E} (H : φ'.comp_multilinear_map (tprod R) = φ) :
+  φ' = lift φ :=
+ext $ H.symm ▸ (lift.symm_apply_apply φ).symm
+
+theorem lift.unique {φ' : (⨂[R] i, s i) →ₗ[R] E} (H : ∀ f, φ' (tprod R f) = φ f) :
+  φ' = lift φ :=
+lift.unique' (multilinear_map.ext H)
+
+theorem lift_tprod : lift (tprod R : multilinear_map R s _) = linear_map.id :=
+eq.symm $ lift.unique' rfl
+
+end multilinear
+
+end pi_tensor_product
+
+end semiring
+
+section ring
+namespace pi_tensor_product
+
+open pi_tensor_product
+open_locale tensor_product
+
+variables {ι : Type*} {R : Type*} [comm_ring R]
+variables {s : ι → Type*} [∀ i, add_comm_group (s i)] [∀ i, module R (s i)]
+
+/- Unlike for the binary tensor product, we require `R` to be a `comm_ring` here, otherwise
+this is false in the case where `ι` is empty. -/
+instance : add_comm_group (⨂[R] i, s i) := semimodule.add_comm_monoid_to_add_comm_group R
+
+end pi_tensor_product
+end ring