feat: Einstein notation

HepLean.lean

@@ -72,6 +72,7 @@ import HepLean.SpaceTime.LorentzGroup.Restricted
 import HepLean.SpaceTime.LorentzGroup.Rotations
 import HepLean.SpaceTime.LorentzTensor.Basic
 import HepLean.SpaceTime.LorentzTensor.Contraction
+import HepLean.SpaceTime.LorentzTensor.EinsteinNotation.Basic
 import HepLean.SpaceTime.LorentzTensor.IndexNotation.AreDual
 import HepLean.SpaceTime.LorentzTensor.IndexNotation.Basic
 import HepLean.SpaceTime.LorentzTensor.IndexNotation.Color

HepLean/SpaceTime/LorentzTensor/EinsteinNotation/Basic.lean

@@ -0,0 +1,89 @@
+/-
+Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Joseph Tooby-Smith
+-/
+import Mathlib.LinearAlgebra.StdBasis
+import HepLean.SpaceTime.LorentzTensor.Basic
+import HepLean.SpaceTime.LorentzTensor.IndexNotation.Basic
+/-!
+
+# Einstein notation for real tensors
+
+Einstein notation is a specific example of index notation, with only one color.
+
+In this file we define Einstein notation for generic ring `R`.
+
+-/
+
+open TensorProduct
+
+/-- Einstein tensors have only one color, corresponding to a `down` index. . -/
+def einsteinTensorColor : TensorColor where
+  Color := Unit
+  τ a := a
+  τ_involutive μ := by rfl
+
+instance : Fintype einsteinTensorColor.Color := Unit.fintype
+
+instance : DecidableEq einsteinTensorColor.Color := instDecidableEqPUnit
+
+variable {R : Type} [CommSemiring R]
+
+
+/-- The `TensorStructure` associated with `n`-dimensional tensors. -/
+noncomputable def einsteinTensor (R : Type) [CommSemiring R] (n : ℕ) : TensorStructure R where
+  toTensorColor := einsteinTensorColor
+  ColorModule _ := Fin n → R
+  colorModule_addCommMonoid _ := Pi.addCommMonoid
+  colorModule_module _ := Pi.Function.module (Fin n) R R
+  contrDual _ := TensorProduct.lift (Fintype.total R R)
+  contrDual_symm a x y := by
+    simp only [lift.tmul, Fintype.total_apply, smul_eq_mul, mul_comm, Equiv.cast_refl,
+      Equiv.refl_apply]
+  unit a := ∑ i, Pi.basisFun R (Fin n) i ⊗ₜ[R] Pi.basisFun R (Fin n) i
+  unit_rid a x:= by
+    simp only [Pi.basisFun_apply]
+    rw [tmul_sum, map_sum]
+    trans ∑ i, x i • Pi.basisFun R (Fin n) i
+    · refine Finset.sum_congr rfl (fun i _ => ?_)
+      simp only [TensorStructure.contrLeftAux, LinearEquiv.refl_toLinearMap, LinearMap.coe_comp,
+        LinearEquiv.coe_coe, Function.comp_apply, assoc_symm_tmul, map_tmul, lift.tmul,
+        Fintype.total_apply, LinearMap.stdBasis_apply', smul_eq_mul, ite_mul, one_mul, zero_mul,
+        Finset.sum_ite_eq, Finset.mem_univ, ↓reduceIte, LinearMap.id_coe, id_eq, lid_tmul,
+        Pi.basisFun_apply]
+    · funext a
+      simp only [Pi.basisFun_apply, Finset.sum_apply, Pi.smul_apply, LinearMap.stdBasis_apply',
+        smul_eq_mul, mul_ite, mul_one, mul_zero, Finset.sum_ite_eq', Finset.mem_univ, ↓reduceIte]
+  metric a := ∑ i, Pi.basisFun R (Fin n) i ⊗ₜ[R] Pi.basisFun R (Fin n) i
+  metric_dual a := by
+    simp only [Pi.basisFun_apply, map_sum, comm_tmul]
+    rw [tmul_sum, map_sum]
+    refine Finset.sum_congr rfl (fun i _ => ?_)
+    rw [sum_tmul, map_sum, Fintype.sum_eq_single i]
+    · simp only [TensorStructure.contrMidAux, LinearEquiv.refl_toLinearMap,
+        TensorStructure.contrLeftAux, LinearMap.coe_comp, LinearEquiv.coe_coe, Function.comp_apply,
+        assoc_tmul, map_tmul, LinearMap.id_coe, id_eq, assoc_symm_tmul, lift.tmul,
+        Fintype.total_apply, LinearMap.stdBasis_apply', smul_eq_mul, mul_ite, mul_one, mul_zero,
+        Finset.sum_ite_eq, Finset.mem_univ, ↓reduceIte, lid_tmul, one_smul]
+    · intro x hi
+      simp only [TensorStructure.contrMidAux, LinearEquiv.refl_toLinearMap,
+        TensorStructure.contrLeftAux, LinearMap.coe_comp, LinearEquiv.coe_coe, Function.comp_apply,
+        assoc_tmul, map_tmul, LinearMap.id_coe, id_eq, assoc_symm_tmul, lift.tmul,
+        Fintype.total_apply, LinearMap.stdBasis_apply', smul_eq_mul, mul_ite, mul_one, mul_zero,
+        Finset.sum_ite_eq, Finset.mem_univ, ↓reduceIte, lid_tmul, ite_smul, one_smul, zero_smul]
+      rw [if_neg]
+      simp only [tmul_zero]
+      exact id (Ne.symm hi)
+
+instance : IndexNotation einsteinTensorColor.Color where
+  charList := {'ᵢ'}
+  notaEquiv :=
+    ⟨fun _ => ⟨'ᵢ', Finset.mem_singleton.mpr rfl⟩,
+    fun _ => Unit.unit,
+    fun _ => rfl,
+    by
+      intro c
+      have hc2 := c.2
+      simp only [↓Char.isValue, Finset.mem_singleton] at hc2
+      exact SetCoe.ext (id (Eq.symm hc2))⟩

HepLean/SpaceTime/LorentzTensor/Real/Basic.lean

@@ -86,7 +86,7 @@ instance : Fintype realTensorColor.Color := realTensorColor.instFintypeColorType
 instance : DecidableEq realTensorColor.Color := realTensorColor.instDecidableEqColorType
 
 /-! TODO: Set up the notation `𝓛𝓣ℝ` or similar. -/
-/-- The `LorentzTensorStructure` associated with real Lorentz tensors. -/
+/-- The `TensorStructure` associated with real Lorentz tensors. -/
 def realLorentzTensor (d : ℕ) : TensorStructure ℝ where
   toTensorColor := realTensorColor
   ColorModule μ :=
@@ -146,4 +146,5 @@ instance : MulActionTensor (LorentzGroup d) (realLorentzTensor d) where
     match μ with
     | .up => asTenProd_invariant g
     | .down => asCoTenProd_invariant g
+
 end