refactor: Partial major refactor wip
This commit is contained in:
jstoobysmith
parent
782f4929d1
commit
f90fa1ac1a
6 changed files with 636 additions and 188 deletions
|
|
@ -9,6 +9,9 @@ import Mathlib.Data.Fintype.BigOperators
|
|||
import Mathlib.Logic.Equiv.Fin
|
||||
import Mathlib.Tactic.FinCases
|
||||
import Mathlib.Logic.Equiv.Fintype
|
||||
import Mathlib.Algebra.Module.Pi
|
||||
import Mathlib.Algebra.Module.Equiv
|
||||
import Mathlib.Algebra.Module.LinearMap.Basic
|
||||
/-!
|
||||
|
||||
# Real Lorentz Tensors
|
||||
|
|
@ -53,12 +56,24 @@ instance (d : ℕ) (μ : RealLorentzTensor.Colors) : DecidableEq (RealLorentzTen
|
|||
def RealLorentzTensor.IndexValue {X : Type} (d : ℕ) (c : X → RealLorentzTensor.Colors) :
|
||||
Type 0 := (x : X) → RealLorentzTensor.ColorsIndex d (c x)
|
||||
|
||||
def RealLorentzTensor.ColorFiber {X : Type} (d : ℕ) (c : X → RealLorentzTensor.Colors) : Type :=
|
||||
RealLorentzTensor.IndexValue d c → ℝ
|
||||
|
||||
instance {X : Type} (d : ℕ) (c : X → RealLorentzTensor.Colors) :
|
||||
AddCommMonoid (RealLorentzTensor.ColorFiber d c) := Pi.addCommMonoid
|
||||
|
||||
instance {X : Type} (d : ℕ) (c : X → RealLorentzTensor.Colors) :
|
||||
Module ℝ (RealLorentzTensor.ColorFiber d c) := Pi.module _ _ _
|
||||
|
||||
instance {X : Type} (d : ℕ) (c : X → RealLorentzTensor.Colors) :
|
||||
AddCommGroup (RealLorentzTensor.ColorFiber d c) := Pi.addCommGroup
|
||||
|
||||
/-- A Lorentz Tensor defined by its coordinate map. -/
|
||||
structure RealLorentzTensor (d : ℕ) (X : Type) where
|
||||
/-- The color associated to each index of the tensor. -/
|
||||
color : X → RealLorentzTensor.Colors
|
||||
/-- The coordinate map for the tensor. -/
|
||||
coord : RealLorentzTensor.IndexValue d color → ℝ
|
||||
coord : RealLorentzTensor.ColorFiber d color
|
||||
|
||||
namespace RealLorentzTensor
|
||||
open Matrix
|
||||
|
|
@ -85,6 +100,9 @@ lemma τ_involutive : Function.Involutive τ := by
|
|||
lemma color_eq_dual_symm {μ ν : Colors} (h : μ = τ ν) : ν = τ μ :=
|
||||
(Function.Involutive.eq_iff τ_involutive).mp h.symm
|
||||
|
||||
lemma color_comp_τ_symm {c1 c2 : X → Colors} (h : c1 = τ ∘ c2) : c2 = τ ∘ c1 :=
|
||||
funext (fun x => color_eq_dual_symm (congrFun h x))
|
||||
|
||||
/-- The color associated with an element of `x ∈ X` for a tensor `T`. -/
|
||||
def ch {X : Type} (x : X) (T : RealLorentzTensor d X) : Colors := T.color x
|
||||
|
||||
|
|
@ -93,6 +111,11 @@ def colorsIndexCast {d : ℕ} {μ₁ μ₂ : RealLorentzTensor.Colors} (h : μ
|
|||
ColorsIndex d μ₁ ≃ ColorsIndex d μ₂ :=
|
||||
Equiv.cast (congrArg (ColorsIndex d) h)
|
||||
|
||||
@[simp]
|
||||
lemma colorsIndexCast_symm {d : ℕ} {μ₁ μ₂ : RealLorentzTensor.Colors} (h : μ₁ = μ₂) :
|
||||
(@colorsIndexCast d _ _ h).symm = colorsIndexCast h.symm := by
|
||||
rfl
|
||||
|
||||
/-- An equivalence of `ColorsIndex` between that of a color and that of its dual. -/
|
||||
def colorsIndexDualCastSelf {d : ℕ} {μ : RealLorentzTensor.Colors}:
|
||||
ColorsIndex d μ ≃ ColorsIndex d (τ μ) where
|
||||
|
|
@ -112,6 +135,7 @@ def colorsIndexDualCast {μ ν : Colors} (h : μ = τ ν) :
|
|||
ColorsIndex d μ ≃ ColorsIndex d ν :=
|
||||
(colorsIndexCast h).trans colorsIndexDualCastSelf.symm
|
||||
|
||||
@[simp]
|
||||
lemma colorsIndexDualCast_symm {μ ν : Colors} (h : μ = τ ν) :
|
||||
(colorsIndexDualCast h).symm =
|
||||
@colorsIndexDualCast d _ _ ((Function.Involutive.eq_iff τ_involutive).mp h.symm) := by
|
||||
|
|
@ -119,6 +143,31 @@ lemma colorsIndexDualCast_symm {μ ν : Colors} (h : μ = τ ν) :
|
|||
| Colors.up, Colors.down => rfl
|
||||
| Colors.down, Colors.up => rfl
|
||||
|
||||
@[simp]
|
||||
lemma colorsIndexDualCast_trans {μ ν ξ : Colors} (h : μ = τ ν) (h' : ν = τ ξ) :
|
||||
(@colorsIndexDualCast d _ _ h).trans (colorsIndexDualCast h') =
|
||||
colorsIndexCast (by rw [h, h', τ_involutive]) := by
|
||||
match μ, ν, ξ with
|
||||
| Colors.up, Colors.down, Colors.up => rfl
|
||||
| Colors.down, Colors.up, Colors.down => rfl
|
||||
|
||||
@[simp]
|
||||
lemma colorsIndexDualCast_trans_colorsIndexCast {μ ν ξ : Colors} (h : μ = τ ν) (h' : ν = ξ) :
|
||||
(@colorsIndexDualCast d _ _ h).trans (colorsIndexCast h') =
|
||||
colorsIndexDualCast (by rw [h, h']) := by
|
||||
match μ, ν, ξ with
|
||||
| Colors.down, Colors.up, Colors.up => rfl
|
||||
| Colors.up, Colors.down, Colors.down => rfl
|
||||
|
||||
@[simp]
|
||||
lemma colorsIndexCast_trans_colorsIndexDualCast {μ ν ξ : Colors} (h : μ = ν) (h' : ν = τ ξ) :
|
||||
(colorsIndexCast h).trans (@colorsIndexDualCast d _ _ h') =
|
||||
colorsIndexDualCast (by rw [h, h']) := by
|
||||
match μ, ν, ξ with
|
||||
| Colors.up, Colors.up, Colors.down => rfl
|
||||
| Colors.down, Colors.down, Colors.up => rfl
|
||||
|
||||
|
||||
/-!
|
||||
|
||||
## Index values
|
||||
|
|
@ -137,12 +186,12 @@ instance [Fintype X] : DecidableEq (IndexValue d c) :=
|
|||
-/
|
||||
|
||||
/-- An isomorphism of the type of index values given an isomorphism of sets. -/
|
||||
@[simps!]
|
||||
def indexValueIso (d : ℕ) (f : X ≃ Y) {i : X → Colors} {j : Y → Colors} (h : i = j ∘ f) :
|
||||
IndexValue d i ≃ IndexValue d j :=
|
||||
(Equiv.piCongrRight (fun μ => colorsIndexCast (congrFun h μ))).trans $
|
||||
Equiv.piCongrLeft (fun y => RealLorentzTensor.ColorsIndex d (j y)) f
|
||||
|
||||
@[simp]
|
||||
lemma indexValueIso_symm_apply' (d : ℕ) (f : X ≃ Y) {i : X → Colors} {j : Y → Colors}
|
||||
(h : i = j ∘ f) (y : IndexValue d j) (x : X) :
|
||||
(indexValueIso d f h).symm y x = (colorsIndexCast (congrFun h x)).symm (y (f x)) := by
|
||||
|
|
@ -159,9 +208,10 @@ lemma indexValueIso_trans (d : ℕ) (f : X ≃ Y) (g : Y ≃ Z) {i : X → Color
|
|||
exact Equiv.coe_inj.mp rfl
|
||||
simpa only [Equiv.symm_symm] using congrArg (fun e => e.symm) h1
|
||||
|
||||
@[simp]
|
||||
lemma indexValueIso_symm (d : ℕ) (f : X ≃ Y) (h : i = j ∘ f) :
|
||||
(indexValueIso d f h).symm =
|
||||
indexValueIso d f.symm ((Equiv.eq_comp_symm f j i).mpr (id (Eq.symm h))) := by
|
||||
indexValueIso d f.symm ((Equiv.eq_comp_symm f j i).mpr h.symm) := by
|
||||
ext i : 1
|
||||
rw [← Equiv.symm_apply_eq]
|
||||
funext y
|
||||
|
|
@ -172,7 +222,7 @@ lemma indexValueIso_symm (d : ℕ) (f : X ≃ Y) (h : i = j ∘ f) :
|
|||
|
||||
lemma indexValueIso_eq_symm (d : ℕ) (f : X ≃ Y) (h : i = j ∘ f) :
|
||||
indexValueIso d f h =
|
||||
(indexValueIso d f.symm ((Equiv.eq_comp_symm f j i).mpr (id (Eq.symm h)))).symm := by
|
||||
(indexValueIso d f.symm ((Equiv.eq_comp_symm f j i).mpr h.symm)).symm := by
|
||||
rw [indexValueIso_symm]
|
||||
rfl
|
||||
|
||||
|
|
@ -188,10 +238,161 @@ lemma indexValueIso_refl (d : ℕ) (i : X → Colors) :
|
|||
-/
|
||||
|
||||
/-- The isomorphism between the index values of a color map and its dual. -/
|
||||
def indexValueDualIso (d : ℕ) {i : X → Colors} {j : Y → Colors} (e : X ≃ Y)
|
||||
(h : i = τ ∘ j ∘ e) : IndexValue d i ≃ IndexValue d j :=
|
||||
(Equiv.piCongrRight (fun μ => colorsIndexDualCast (congrFun h μ))).trans $
|
||||
Equiv.piCongrLeft (fun y => ColorsIndex d (j y)) e
|
||||
|
||||
lemma indexValueDualIso_symm_apply' (d : ℕ) {i : X → Colors} {j : Y → Colors} (e : X ≃ Y)
|
||||
(h : i = τ ∘ j ∘ e) (y : IndexValue d j) (x : X) :
|
||||
(indexValueDualIso d e h).symm y x = (colorsIndexDualCast (congrFun h x)).symm (y (e x)) := by
|
||||
rfl
|
||||
|
||||
lemma indexValueDualIso_cond_symm {i : X → Colors} {j : Y → Colors} {e : X ≃ Y}
|
||||
(h : i = τ ∘ j ∘ e) : j = τ ∘ i ∘ e.symm := by
|
||||
subst h
|
||||
simp only [Function.comp.assoc, Equiv.self_comp_symm, CompTriple.comp_eq]
|
||||
rw [← Function.comp.assoc]
|
||||
funext a
|
||||
simp only [τ_involutive, Function.Involutive.comp_self, Function.comp_apply, id_eq]
|
||||
|
||||
@[simp]
|
||||
lemma indexValueDualIso_symm {d : ℕ} {i : X → Colors} {j : Y → Colors} (e : X ≃ Y)
|
||||
(h : i = τ ∘ j ∘ e) : (indexValueDualIso d e h).symm =
|
||||
indexValueDualIso d e.symm (indexValueDualIso_cond_symm h) := by
|
||||
ext i : 1
|
||||
rw [← Equiv.symm_apply_eq]
|
||||
funext a
|
||||
rw [indexValueDualIso_symm_apply', indexValueDualIso_symm_apply']
|
||||
erw [← Equiv.trans_apply, colorsIndexDualCast_symm, colorsIndexDualCast_symm,
|
||||
colorsIndexDualCast_trans]
|
||||
simp only [Function.comp_apply, colorsIndexCast, Equiv.cast_apply]
|
||||
apply cast_eq_iff_heq.mpr
|
||||
rw [Equiv.apply_symm_apply]
|
||||
|
||||
lemma indexValueDualIso_eq_symm {d : ℕ} {i : X → Colors} {j : Y → Colors} (e : X ≃ Y)
|
||||
(h : i = τ ∘ j ∘ e) :
|
||||
indexValueDualIso d e h = (indexValueDualIso d e.symm (indexValueDualIso_cond_symm h)).symm := by
|
||||
rw [indexValueDualIso_symm]
|
||||
simp
|
||||
|
||||
lemma indexValueDualIso_cond_trans {i : X → Colors} {j : Y → Colors} {k : Z → Colors}
|
||||
{e : X ≃ Y} {f : Y ≃ Z} (h : i = τ ∘ j ∘ e) (h' : j = τ ∘ k ∘ f) :
|
||||
i = k ∘ (e.trans f) := by
|
||||
rw [h, h']
|
||||
funext a
|
||||
simp only [Function.comp_apply, Equiv.coe_trans]
|
||||
rw [τ_involutive]
|
||||
|
||||
@[simp]
|
||||
lemma indexValueDualIso_trans {d : ℕ} {i : X → Colors} {j : Y → Colors} {k : Z → Colors}
|
||||
(e : X ≃ Y) (f : Y ≃ Z) (h : i = τ ∘ j ∘ e) (h' : j = τ ∘ k ∘ f) :
|
||||
(indexValueDualIso d e h).trans (indexValueDualIso d f h') =
|
||||
indexValueIso d (e.trans f) (indexValueDualIso_cond_trans h h') := by
|
||||
ext i
|
||||
rw [Equiv.trans_apply]
|
||||
rw [← Equiv.eq_symm_apply, ← Equiv.eq_symm_apply, indexValueIso_eq_symm]
|
||||
funext a
|
||||
rw [indexValueDualIso_symm_apply', indexValueDualIso_symm_apply',
|
||||
indexValueIso_symm_apply']
|
||||
erw [← Equiv.trans_apply]
|
||||
rw [colorsIndexDualCast_symm, colorsIndexDualCast_symm, colorsIndexDualCast_trans]
|
||||
simp only [Function.comp_apply, colorsIndexCast, Equiv.symm_trans_apply, Equiv.cast_symm,
|
||||
Equiv.cast_apply, cast_cast]
|
||||
symm
|
||||
apply cast_eq_iff_heq.mpr
|
||||
rw [Equiv.symm_apply_apply, Equiv.symm_apply_apply]
|
||||
|
||||
lemma indexValueDualIso_cond_trans_indexValueIso {i : X → Colors} {j : Y → Colors} {k : Z → Colors}
|
||||
{e : X ≃ Y} {f : Y ≃ Z} (h : i = τ ∘ j ∘ e) (h' : j = k ∘ f) :
|
||||
i = τ ∘ k ∘ (e.trans f) := by
|
||||
rw [h, h']
|
||||
funext a
|
||||
simp only [Function.comp_apply, Equiv.coe_trans]
|
||||
|
||||
@[simp]
|
||||
lemma indexValueDualIso_trans_indexValueIso {d : ℕ} {i : X → Colors} {j : Y → Colors} {k : Z → Colors}
|
||||
(e : X ≃ Y) (f : Y ≃ Z) (h : i = τ ∘ j ∘ e) (h' : j = k ∘ f) :
|
||||
(indexValueDualIso d e h).trans (indexValueIso d f h') =
|
||||
indexValueDualIso d (e.trans f) (indexValueDualIso_cond_trans_indexValueIso h h') := by
|
||||
ext i
|
||||
rw [Equiv.trans_apply, ← Equiv.eq_symm_apply]
|
||||
funext a
|
||||
rw [ indexValueDualIso_eq_symm, indexValueDualIso_symm_apply',
|
||||
indexValueIso_symm_apply',indexValueDualIso_eq_symm, indexValueDualIso_symm_apply']
|
||||
rw [Equiv.symm_apply_eq]
|
||||
erw [← Equiv.trans_apply, ← Equiv.trans_apply]
|
||||
simp only [Function.comp_apply, Equiv.symm_trans_apply, colorsIndexDualCast_symm,
|
||||
colorsIndexCast_symm, colorsIndexCast_trans_colorsIndexDualCast, colorsIndexDualCast_trans]
|
||||
simp only [colorsIndexCast, Equiv.cast_apply]
|
||||
symm
|
||||
apply cast_eq_iff_heq.mpr
|
||||
rw [Equiv.symm_apply_apply]
|
||||
|
||||
lemma indexValueIso_trans_indexValueDualIso_cond {i : X → Colors} {j : Y → Colors} {k : Z → Colors}
|
||||
{e : X ≃ Y} {f : Y ≃ Z} (h : i = j ∘ e) (h' : j = τ ∘ k ∘ f) :
|
||||
i = τ ∘ k ∘ (e.trans f) := by
|
||||
rw [h, h']
|
||||
funext a
|
||||
simp only [Function.comp_apply, Equiv.coe_trans]
|
||||
|
||||
@[simp]
|
||||
lemma indexValueIso_trans_indexValueDualIso {d : ℕ} {i : X → Colors} {j : Y → Colors} {k : Z → Colors}
|
||||
(e : X ≃ Y) (f : Y ≃ Z) (h : i = j ∘ e) (h' : j = τ ∘ k ∘ f) :
|
||||
(indexValueIso d e h).trans (indexValueDualIso d f h') =
|
||||
indexValueDualIso d (e.trans f) (indexValueIso_trans_indexValueDualIso_cond h h') := by
|
||||
ext i
|
||||
rw [Equiv.trans_apply, ← Equiv.eq_symm_apply, ← Equiv.eq_symm_apply]
|
||||
funext a
|
||||
rw [indexValueIso_symm_apply', indexValueDualIso_symm_apply',
|
||||
indexValueDualIso_eq_symm, indexValueDualIso_symm_apply']
|
||||
erw [← Equiv.trans_apply, ← Equiv.trans_apply]
|
||||
simp only [Function.comp_apply, Equiv.symm_trans_apply, colorsIndexDualCast_symm,
|
||||
colorsIndexCast_symm, colorsIndexDualCast_trans_colorsIndexCast, colorsIndexDualCast_trans]
|
||||
simp only [colorsIndexCast, Equiv.cast_apply]
|
||||
symm
|
||||
apply cast_eq_iff_heq.mpr
|
||||
rw [Equiv.symm_apply_apply, Equiv.symm_apply_apply]
|
||||
|
||||
|
||||
|
||||
/-!
|
||||
|
||||
## Mapping isomorphisms on fibers
|
||||
|
||||
-/
|
||||
|
||||
@[simps!]
|
||||
def indexValueDualIso (d : ℕ) {i j : X → Colors} (h : i = τ ∘ j) :
|
||||
IndexValue d i ≃ IndexValue d j :=
|
||||
(Equiv.piCongrRight (fun μ => colorsIndexDualCast (congrFun h μ)))
|
||||
def mapIsoFiber (d : ℕ) (f : X ≃ Y) {i : X → Colors} {j : Y → Colors} (h : i = j ∘ f) :
|
||||
ColorFiber d i ≃ₗ[ℝ] ColorFiber d j where
|
||||
toFun F := F ∘ (indexValueIso d f h).symm
|
||||
invFun F := F ∘ indexValueIso d f h
|
||||
map_add' F G := by rfl
|
||||
map_smul' a F := by rfl
|
||||
left_inv F := by
|
||||
funext x
|
||||
simp only [Function.comp_apply]
|
||||
exact congrArg _ $ Equiv.symm_apply_apply (indexValueIso d f h) x
|
||||
right_inv F := by
|
||||
funext x
|
||||
simp only [Function.comp_apply]
|
||||
exact congrArg _ $ Equiv.apply_symm_apply (indexValueIso d f h) x
|
||||
|
||||
@[simp]
|
||||
lemma mapIsoFiber_trans (d : ℕ) (f : X ≃ Y) (g : Y ≃ Z) {i : X → Colors}
|
||||
{j : Y → Colors} {k : Z → Colors} (h : i = j ∘ f) (h' : j = k ∘ g) :
|
||||
(mapIsoFiber d f h).trans (mapIsoFiber d g h') =
|
||||
mapIsoFiber d (f.trans g) (by rw [h, h', Function.comp.assoc]; rfl) := by
|
||||
refine LinearEquiv.toEquiv_inj.mp (Equiv.ext (fun x => (funext (fun a => ?_))))
|
||||
simp only [LinearEquiv.coe_toEquiv, LinearEquiv.trans_apply, mapIsoFiber_apply,
|
||||
indexValueIso_symm, ← Equiv.trans_apply, indexValueIso_trans]
|
||||
rfl
|
||||
|
||||
lemma mapIsoFiber_symm (d : ℕ) (f : X ≃ Y) (h : i = j ∘ f) :
|
||||
(mapIsoFiber d f h).symm = mapIsoFiber d f.symm ((Equiv.eq_comp_symm f j i).mpr h.symm) := by
|
||||
refine LinearEquiv.toEquiv_inj.mp (Equiv.ext (fun x => (funext (fun a => ?_))))
|
||||
simp only [LinearEquiv.coe_toEquiv, mapIsoFiber_symm_apply, mapIsoFiber_apply, indexValueIso_symm,
|
||||
Equiv.symm_symm]
|
||||
|
||||
/-!
|
||||
|
||||
|
|
@ -200,62 +401,40 @@ def indexValueDualIso (d : ℕ) {i j : X → Colors} (h : i = τ ∘ j) :
|
|||
-/
|
||||
|
||||
lemma ext {T₁ T₂ : RealLorentzTensor d X} (h : T₁.color = T₂.color)
|
||||
(h' : T₁.coord = fun i => T₂.coord (indexValueIso d (Equiv.refl X) h i)) :
|
||||
T₁ = T₂ := by
|
||||
(h' : T₁.coord = mapIsoFiber d (Equiv.refl X) h.symm T₂.coord) : T₁ = T₂ := by
|
||||
cases T₁
|
||||
cases T₂
|
||||
simp_all only [IndexValue, mk.injEq]
|
||||
apply And.intro h
|
||||
simp only at h
|
||||
subst h
|
||||
simp only [Equiv.cast_refl, Equiv.coe_refl, CompTriple.comp_eq] at h'
|
||||
rfl
|
||||
|
||||
/-!
|
||||
|
||||
## Mapping isomorphisms.
|
||||
## Mapping isomorphisms
|
||||
|
||||
-/
|
||||
|
||||
/-- An equivalence of Tensors given an equivalence of underlying sets. -/
|
||||
@[simps!]
|
||||
def mapIso (d : ℕ) (f : X ≃ Y) : RealLorentzTensor d X ≃ RealLorentzTensor d Y where
|
||||
toFun T := {
|
||||
color := T.color ∘ f.symm,
|
||||
coord := T.coord ∘ (indexValueIso d f (by simp : T.color = T.color ∘ f.symm ∘ f)).symm}
|
||||
invFun T := {
|
||||
color := T.color ∘ f,
|
||||
coord := T.coord ∘ (indexValueIso d f.symm (by simp : T.color = T.color ∘ f ∘ f.symm)).symm}
|
||||
toFun T := ⟨T.color ∘ f.symm, mapIsoFiber d f (by funext x; simp) T.coord⟩
|
||||
invFun T := ⟨T.color ∘ f, (mapIsoFiber d f (by funext x; simp)).symm T.coord⟩
|
||||
left_inv T := by
|
||||
refine ext ?_ ?_
|
||||
· simp [Function.comp.assoc]
|
||||
· funext i
|
||||
simp only [IndexValue, Function.comp_apply, Function.comp_id]
|
||||
apply congrArg
|
||||
funext x
|
||||
erw [indexValueIso_symm_apply', indexValueIso_symm_apply', indexValueIso_eq_symm,
|
||||
indexValueIso_symm_apply']
|
||||
rw [← Equiv.apply_eq_iff_eq_symm_apply]
|
||||
simp only [Equiv.refl_symm, Equiv.coe_refl, Function.comp_apply, id_eq, colorsIndexCast,
|
||||
Equiv.cast_symm, Equiv.cast_apply, cast_cast, Equiv.refl_apply]
|
||||
apply cast_eq_iff_heq.mpr
|
||||
· simp only [Function.comp_apply, ← LinearEquiv.trans_apply, mapIsoFiber_symm,
|
||||
mapIsoFiber_trans]
|
||||
congr
|
||||
exact Equiv.symm_apply_apply f x
|
||||
exact Equiv.self_trans_symm f
|
||||
right_inv T := by
|
||||
refine ext ?_ ?_
|
||||
· simp [Function.comp.assoc]
|
||||
· funext i
|
||||
simp only [IndexValue, Function.comp_apply, Function.comp_id]
|
||||
apply congrArg
|
||||
funext x
|
||||
erw [indexValueIso_symm_apply', indexValueIso_symm_apply', indexValueIso_eq_symm,
|
||||
indexValueIso_symm_apply']
|
||||
rw [← Equiv.apply_eq_iff_eq_symm_apply]
|
||||
simp only [Equiv.refl_symm, Equiv.coe_refl, Function.comp_apply, id_eq, colorsIndexCast,
|
||||
Equiv.cast_symm, Equiv.cast_apply, cast_cast, Equiv.refl_apply]
|
||||
apply cast_eq_iff_heq.mpr
|
||||
· simp only [Function.comp_apply, ← LinearEquiv.trans_apply, mapIsoFiber_symm,
|
||||
mapIsoFiber_trans]
|
||||
congr
|
||||
exact Equiv.apply_symm_apply f x
|
||||
exact Equiv.symm_trans_self f
|
||||
|
||||
@[simp]
|
||||
lemma mapIso_trans (f : X ≃ Y) (g : Y ≃ Z) :
|
||||
|
|
@ -263,16 +442,17 @@ lemma mapIso_trans (f : X ≃ Y) (g : Y ≃ Z) :
|
|||
refine Equiv.coe_inj.mp ?_
|
||||
funext T
|
||||
refine ext rfl ?_
|
||||
simp only [Equiv.trans_apply, IndexValue, mapIso_apply_color, Equiv.symm_trans_apply,
|
||||
indexValueIso_refl, Equiv.refl_apply, mapIso_apply_coord]
|
||||
funext i
|
||||
rw [mapIso_apply_coord, mapIso_apply_coord]
|
||||
apply congrArg
|
||||
rw [← indexValueIso_trans]
|
||||
simp
|
||||
erw [← LinearEquiv.trans_apply, ← LinearEquiv.trans_apply, mapIsoFiber_trans, mapIsoFiber_trans]
|
||||
rfl
|
||||
exact (Equiv.comp_symm_eq f (T.color ∘ ⇑f.symm) T.color).mp rfl
|
||||
|
||||
lemma mapIso_symm (f : X ≃ Y) : (mapIso d f).symm = mapIso d f.symm := rfl
|
||||
lemma mapIso_symm (f : X ≃ Y) : (mapIso d f).symm = mapIso d f.symm := by
|
||||
refine Equiv.coe_inj.mp ?_
|
||||
funext T
|
||||
refine ext rfl ?_
|
||||
erw [← LinearEquiv.trans_apply]
|
||||
rw [mapIso_symm_apply_coord, mapIsoFiber_trans, mapIsoFiber_symm]
|
||||
rfl
|
||||
|
||||
lemma mapIso_refl : mapIso d (Equiv.refl X) = Equiv.refl _ := rfl
|
||||
|
||||
|
|
@ -302,6 +482,16 @@ def indexValueSumComm {X Y : Type} (Tc : X → Colors) (Sc : Y → Colors) :
|
|||
IndexValue d (Sum.elim Tc Sc) ≃ IndexValue d (Sum.elim Sc Tc) :=
|
||||
indexValueIso d (Equiv.sumComm X Y) (by aesop)
|
||||
|
||||
def indexValueFinOne {c : Fin 1 → Colors} :
|
||||
ColorsIndex d (c 0) ≃ IndexValue d c where
|
||||
toFun x := fun i => match i with
|
||||
| 0 => x
|
||||
invFun i := i 0
|
||||
left_inv x := rfl
|
||||
right_inv i := by
|
||||
funext x
|
||||
fin_cases x
|
||||
rfl
|
||||
/-!
|
||||
|
||||
## Marked Lorentz tensors
|
||||
|
|
|
|||
Loading…
Add table
Add a link
Reference in a new issue