feat: Properties of lists
This commit is contained in:
jstoobysmith
parent
db9f7ebfa9
commit
7ee877af55
2 changed files with 326 additions and 0 deletions
|
|
@ -8,6 +8,7 @@ import HepLean.Lorentz.RealVector.Basic
|
|||
import HepLean.Mathematics.Fin
|
||||
import HepLean.SpaceTime.Basic
|
||||
import HepLean.Mathematics.SuperAlgebra.Basic
|
||||
import HepLean.Mathematics.List
|
||||
import HepLean.Meta.Notes.Basic
|
||||
import Init.Data.List.Sort.Basic
|
||||
/-!
|
||||
|
|
@ -319,6 +320,7 @@ def listToConstDestList : (l : List (index S)) →
|
|||
| i :: l, f =>
|
||||
(f ⟨0, Nat.zero_lt_succ l.length⟩, i.1, i.2) :: listToConstDestList l (f ∘ Fin.succ)
|
||||
|
||||
@[simp]
|
||||
lemma listToConstDestList_length (l : List (index S)) (f : Fin l.length → Fin 2) :
|
||||
(listToConstDestList l f).length = l.length := by
|
||||
induction l with
|
||||
|
|
@ -327,6 +329,26 @@ lemma listToConstDestList_length (l : List (index S)) (f : Fin l.length → Fin
|
|||
simp only [listToConstDestList, List.length_cons, Fin.zero_eta, Prod.mk.eta, add_left_inj]
|
||||
rw [ih]
|
||||
|
||||
lemma listToConstDestList_get (l : List (index S)) (f : Fin l.length → Fin 2) :
|
||||
(listToConstDestList l f).get = (fun i => (f i, l.get i)) ∘ Fin.cast (by simp) := by
|
||||
induction l with
|
||||
| nil =>
|
||||
funext i
|
||||
exact Fin.elim0 i
|
||||
| cons i l ih =>
|
||||
simp [listToConstDestList]
|
||||
funext x
|
||||
match x with
|
||||
| ⟨0, h⟩ => rfl
|
||||
| ⟨x + 1, h⟩ =>
|
||||
simp
|
||||
change (listToConstDestList l _).get _ = _
|
||||
rw [ih]
|
||||
simp
|
||||
|
||||
|
||||
|
||||
|
||||
lemma toConstDestAlgebra_single (x : ℂ) : (l : FreeMonoid (index S)) →
|
||||
toConstDestAlgebra (FreeAlgebra.equivMonoidAlgebraFreeMonoid.symm (MonoidAlgebra.single l x))
|
||||
= ∑ (f : Fin l.length → Fin 2), FreeAlgebra.equivMonoidAlgebraFreeMonoid.symm
|
||||
|
|
@ -429,6 +451,127 @@ instance : DecidableRel (@timeOrderRel S) :=
|
|||
def timeOrder (q : index S → Fin 2) : S.FieldAlgebra →ₗ[ℂ] S.FieldAlgebra :=
|
||||
koszulOrder timeOrderRel q
|
||||
|
||||
lemma listToConstDestList_insertionSortEquiv (l : List (index S))
|
||||
(f : Fin l.length → Fin 2) :
|
||||
(HepLean.List.insertionSortEquiv ConstDestAlgebra.timeOrderRel (listToConstDestList l f))
|
||||
= (Fin.castOrderIso (by simp)).toEquiv.trans ((HepLean.List.insertionSortEquiv timeOrderRel l).trans
|
||||
(Fin.castOrderIso (by simp)).toEquiv) := by
|
||||
induction l with
|
||||
| nil =>
|
||||
simp [listToConstDestList, HepLean.List.insertionSortEquiv]
|
||||
| cons i l ih =>
|
||||
simp [listToConstDestList]
|
||||
conv_lhs => simp [HepLean.List.insertionSortEquiv]
|
||||
have h1 (l' : List (ConstDestAlgebra.index S)) :
|
||||
(HepLean.List.insertEquiv ConstDestAlgebra.timeOrderRel (f ⟨0, by simp⟩, i.1, i.2) l') =
|
||||
(Fin.castOrderIso (by simp)).toEquiv.trans
|
||||
((HepLean.List.insertEquiv timeOrderRel (i.1, i.2) (l'.unzip).2).trans
|
||||
(Fin.castOrderIso (by simp [List.orderedInsert_length])).toEquiv) := by
|
||||
induction l' with
|
||||
| nil =>
|
||||
simp [HepLean.List.insertEquiv]
|
||||
rfl
|
||||
| cons j l' ih' =>
|
||||
by_cases hr : ConstDestAlgebra.timeOrderRel (f ⟨0, by simp⟩, i) j
|
||||
· rw [HepLean.List.insertEquiv_cons_pos]
|
||||
· erw [HepLean.List.insertEquiv_cons_pos]
|
||||
· rfl
|
||||
· exact hr
|
||||
· exact hr
|
||||
· rw [HepLean.List.insertEquiv_cons_neg]
|
||||
· erw [HepLean.List.insertEquiv_cons_neg]
|
||||
· simp
|
||||
erw [ih']
|
||||
ext x
|
||||
simp
|
||||
congr 2
|
||||
match x with
|
||||
| ⟨0, h⟩ => rfl
|
||||
| ⟨1, h⟩ => rfl
|
||||
| ⟨Nat.succ (Nat.succ x), h⟩ => rfl
|
||||
· exact hr
|
||||
· exact hr
|
||||
erw [h1]
|
||||
rw [ih]
|
||||
simp
|
||||
ext x
|
||||
conv_rhs => simp [HepLean.List.insertionSortEquiv]
|
||||
simp
|
||||
have h2' (i : ConstDestAlgebra.index S) (l' : List (ConstDestAlgebra.index S)) :
|
||||
(List.orderedInsert ConstDestAlgebra.timeOrderRel i l').unzip.2 =
|
||||
List.orderedInsert timeOrderRel i.2 l'.unzip.2 := by
|
||||
induction l' with
|
||||
| nil =>
|
||||
simp [HepLean.List.insertEquiv]
|
||||
| cons j l' ih' =>
|
||||
by_cases hij : ConstDestAlgebra.timeOrderRel i j
|
||||
· rw [List.orderedInsert_of_le]
|
||||
· erw [List.orderedInsert_of_le]
|
||||
· simp
|
||||
· exact hij
|
||||
· exact hij
|
||||
· simp [hij]
|
||||
have hn : ¬ timeOrderRel i.2 j.2 := hij
|
||||
simp [hn]
|
||||
simpa using ih'
|
||||
have h2 (l' : List (ConstDestAlgebra.index S)) :
|
||||
(List.insertionSort ConstDestAlgebra.timeOrderRel l').unzip.2 =
|
||||
List.insertionSort timeOrderRel l'.unzip.2 := by
|
||||
induction l' with
|
||||
| nil =>
|
||||
simp [HepLean.List.insertEquiv]
|
||||
| cons i l' ih' =>
|
||||
simp
|
||||
simp at h2'
|
||||
rw [h2']
|
||||
congr
|
||||
simpa using ih'
|
||||
rw [HepLean.List.insertEquiv_congr _ _ _ (h2 _)]
|
||||
simp
|
||||
have h3 : (List.insertionSort timeOrderRel (listToConstDestList l (f ∘ Fin.succ)).unzip.2) =
|
||||
List.insertionSort timeOrderRel l := by
|
||||
congr
|
||||
have h3' (l : List (index S)) (f : Fin l.length → Fin 2) :
|
||||
(listToConstDestList l (f)).unzip.2 = l := by
|
||||
induction l with
|
||||
| nil => rfl
|
||||
| cons i l ih' =>
|
||||
simp [listToConstDestList]
|
||||
simpa using ih' (f ∘ Fin.succ)
|
||||
rw [h3']
|
||||
rw [HepLean.List.insertEquiv_congr _ _ _ h3]
|
||||
simp
|
||||
rfl
|
||||
|
||||
lemma listToConstDestList_timeOrder (l : List (index S))
|
||||
(f : Fin l.length → Fin 2) :
|
||||
List.insertionSort ConstDestAlgebra.timeOrderRel (listToConstDestList l f) =
|
||||
listToConstDestList (List.insertionSort timeOrderRel l)
|
||||
(f ∘ (HepLean.List.insertionSortEquiv (timeOrderRel) l).symm) := by
|
||||
let l1 := List.insertionSort (ConstDestAlgebra.timeOrderRel) (listToConstDestList l f)
|
||||
let l2 := listToConstDestList (List.insertionSort timeOrderRel l)
|
||||
(f ∘ (HepLean.List.insertionSortEquiv (timeOrderRel) l).symm)
|
||||
change l1 = l2
|
||||
have hlen : l1.length = l2.length := by
|
||||
simp [l1, l2]
|
||||
have hget : l1.get = l2.get ∘ Fin.cast hlen := by
|
||||
rw [← HepLean.List.insertionSortEquiv_get]
|
||||
rw [listToConstDestList_get]
|
||||
rw [listToConstDestList_get]
|
||||
rw [← HepLean.List.insertionSortEquiv_get]
|
||||
funext i
|
||||
simp
|
||||
congr 2
|
||||
· rw [listToConstDestList_insertionSortEquiv]
|
||||
simp
|
||||
· rw [listToConstDestList_insertionSortEquiv]
|
||||
simp
|
||||
apply List.ext_get hlen
|
||||
rw [hget]
|
||||
simp
|
||||
|
||||
|
||||
/-f ∘ (HepLean.List.insertionSortEquiv (timeOrder q) l).symm.toFun-/
|
||||
/-
|
||||
lemma timeOrder_comm_toConstDestAlgebra (q : index S → Fin 2)
|
||||
(q' : ConstDestAlgebra.index S → Fin 2) :
|
||||
|
|
|
|||
Loading…
Add table
Add a link
Reference in a new issue