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feat: Add multiplicative unit

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jstoobysmith 2024-07-18 08:36:56 -04:00
parent 73df7d24a7
commit 0d2d4ffc1d
6 changed files with 264 additions and 14 deletions
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57
HepLean/SpaceTime/LorentzTensor/Real/Multiplication.lean
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@ -15,7 +15,6 @@ marked index. This is equivalent to contracting two Lorentz tensors.
We prove various results about this multiplication.
-/
/-! TODO: Add unit to the multiplication. -/
/-! TODO: Generalize to contracting any marked index of a marked tensor. -/
/-! TODO: Set up a good notation for the multiplication. -/
@ -39,6 +38,54 @@ def mul {X Y : Type} (T : Marked d X 1) (S : Marked d Y 1)
S.coord (splitIndexValue.symm ((indexValueSumEquiv i).2,
oneMarkedIndexValue $ colorsIndexDualCast h x))
/-!
## Expressions for multiplication
-/
/-! TODO: Where appropriate write these expresions in terms of `indexValueDualIso`. -/
lemma mul_colorsIndex_right {X Y : Type} (T : Marked d X 1) (S : Marked d Y 1)
(h : T.markedColor 0 = τ (S.markedColor 0)) :
(mul T S h).coord = fun i => ∑ x,
T.coord (splitIndexValue.symm ((indexValueSumEquiv i).1,
oneMarkedIndexValue $ colorsIndexDualCast (color_eq_dual_symm h) x)) *
S.coord (splitIndexValue.symm ((indexValueSumEquiv i).2, oneMarkedIndexValue x)) := by
funext i
rw [← Equiv.sum_comp (colorsIndexDualCast h)]
apply Finset.sum_congr rfl (fun x _ => ?_)
congr
rw [← colorsIndexDualCast_symm]
exact (Equiv.apply_eq_iff_eq_symm_apply (colorsIndexDualCast h)).mp rfl
lemma mul_indexValue_left {X Y : Type} (T : Marked d X 1) (S : Marked d Y 1)
(h : T.markedColor 0 = τ (S.markedColor 0)) :
(mul T S h).coord = fun i => ∑ j,
T.coord (splitIndexValue.symm ((indexValueSumEquiv i).1, j)) *
S.coord (splitIndexValue.symm ((indexValueSumEquiv i).2,
(oneMarkedIndexValue $ (colorsIndexDualCast h) $ oneMarkedIndexValue.symm j))) := by
funext i
rw [← Equiv.sum_comp (oneMarkedIndexValue)]
rfl
lemma mul_indexValue_right {X Y : Type} (T : Marked d X 1) (S : Marked d Y 1)
(h : T.markedColor 0 = τ (S.markedColor 0)) :
(mul T S h).coord = fun i => ∑ j,
T.coord (splitIndexValue.symm ((indexValueSumEquiv i).1,
oneMarkedIndexValue $ (colorsIndexDualCast h).symm $ oneMarkedIndexValue.symm j)) *
S.coord (splitIndexValue.symm ((indexValueSumEquiv i).2, j)) := by
funext i
rw [mul_colorsIndex_right]
rw [← Equiv.sum_comp (oneMarkedIndexValue)]
apply Finset.sum_congr rfl (fun x _ => ?_)
congr
exact Eq.symm (colorsIndexDualCast_symm h)
/-!
## Properties of multiplication
-/
/-- Multiplication is well behaved with regard to swapping tensors. -/
lemma mul_symm {X Y : Type} (T : Marked d X 1) (S : Marked d Y 1)
(h : T.markedColor 0 = τ (S.markedColor 0)) :
@ -49,14 +96,12 @@ lemma mul_symm {X Y : Type} (T : Marked d X 1) (S : Marked d Y 1)
| inl _ => rfl
| inr _ => rfl
· funext i
rw [mapIso_apply_coord, mul_coord, mul_coord]
erw [← Equiv.sum_comp (colorsIndexDualCast h).symm]
rw [mul_colorsIndex_right]
refine Fintype.sum_congr _ _ (fun x => ?_)
rw [mul_comm]
congr
· exact Equiv.apply_symm_apply (colorsIndexDualCast h) x
· exact colorsIndexDualCast_symm h
rfl
/-- Multiplication commutes with `mapIso`. -/
lemma mul_mapIso {X Y Z : Type} (T : Marked d X 1) (S : Marked d Y 1) (f : X ≃ W)
(g : Y ≃ Z) (h : T.markedColor 0 = τ (S.markedColor 0)) :
mapIso d (Equiv.sumCongr f g) (mul T S h) = mul (mapIso d (Equiv.sumCongr f (Equiv.refl _)) T)