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refactor: Simplify some notation

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jstoobysmith 2025-01-24 05:19:04 +00:00
parent 7d053695dd
commit 6701ee7b37
8 changed files with 106 additions and 111 deletions
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46
HepLean/PerturbationTheory/WickContraction/InsertList.lean
View file

@ -29,7 +29,7 @@ open HepLean.Fin
`j : Option (c.uncontracted)` of `c`.
The Wick contraction associated with `(φs.insertIdx i φ).length` formed by 'inserting' `φ`
into `φs` after the first `i` elements and contracting it optionally with j. -/
def insertList (φ : 𝓕.States) (φs : List 𝓕.States)
def insertList (φ : 𝓕.States) {φs : List 𝓕.States}
(c : WickContraction φs.length) (i : Fin φs.length.succ) (j : Option (c.uncontracted)) :
WickContraction (φs.insertIdx i φ).length :=
congr (by simp) (c.insert i j)
@ -37,7 +37,7 @@ def insertList (φ : 𝓕.States) (φs : List 𝓕.States)
@[simp]
lemma insertList_fstFieldOfContract (φ : 𝓕.States) (φs : List 𝓕.States)
(c : WickContraction φs.length) (i : Fin φs.length.succ) (j : Option (c.uncontracted))
(a : c.1) : (insertList φ φs c i j).fstFieldOfContract
(a : c.1) : (c.insertList φ i j).fstFieldOfContract
(congrLift (insertIdx_length_fin φ φs i).symm (insertLift i j a)) =
finCongr (insertIdx_length_fin φ φs i).symm (i.succAbove (c.fstFieldOfContract a)) := by
simp [insertList]
@ -45,7 +45,7 @@ lemma insertList_fstFieldOfContract (φ : 𝓕.States) (φs : List 𝓕.States)
@[simp]
lemma insertList_sndFieldOfContract (φ : 𝓕.States) (φs : List 𝓕.States)
(c : WickContraction φs.length) (i : Fin φs.length.succ) (j : Option (c.uncontracted))
(a : c.1) : (insertList φ φs c i j).sndFieldOfContract
(a : c.1) : (c.insertList φ i j).sndFieldOfContract
(congrLift (insertIdx_length_fin φ φs i).symm (insertLift i j a)) =
finCongr (insertIdx_length_fin φ φs i).symm (i.succAbove (c.sndFieldOfContract a)) := by
simp [insertList]
@ -53,14 +53,14 @@ lemma insertList_sndFieldOfContract (φ : 𝓕.States) (φs : List 𝓕.States)
@[simp]
lemma insertList_fstFieldOfContract_some_incl (φ : 𝓕.States) (φs : List 𝓕.States)
(c : WickContraction φs.length) (i : Fin φs.length.succ) (j : c.uncontracted) :
(insertList φ φs c i (some j)).fstFieldOfContract
(insertList φ c i (some j)).fstFieldOfContract
(congrLift (insertIdx_length_fin φ φs i).symm ⟨{i, i.succAbove j}, by simp [insert]⟩) =
if i < i.succAbove j.1 then
finCongr (insertIdx_length_fin φ φs i).symm i else
finCongr (insertIdx_length_fin φ φs i).symm (i.succAbove j.1) := by
split
· rename_i h
refine (insertList φ φs c i (some j)).eq_fstFieldOfContract_of_mem
refine (insertList φ c i (some j)).eq_fstFieldOfContract_of_mem
(a := congrLift (insertIdx_length_fin φ φs i).symm ⟨{i, i.succAbove j}, by simp [insert]⟩)
(i := finCongr (insertIdx_length_fin φ φs i).symm i) (j :=
finCongr (insertIdx_length_fin φ φs i).symm (i.succAbove j)) ?_ ?_ ?_
@ -69,7 +69,7 @@ lemma insertList_fstFieldOfContract_some_incl (φ : 𝓕.States) (φs : List
· rw [Fin.lt_def] at h ⊢
simp_all
· rename_i h
refine (insertList φ φs c i (some j)).eq_fstFieldOfContract_of_mem
refine (insertList φ c i (some j)).eq_fstFieldOfContract_of_mem
(a := congrLift (insertIdx_length_fin φ φs i).symm ⟨{i, i.succAbove j}, by simp [insert]⟩)
(i := finCongr (insertIdx_length_fin φ φs i).symm (i.succAbove j))
(j := finCongr (insertIdx_length_fin φ φs i).symm i) ?_ ?_ ?_
@ -88,7 +88,7 @@ lemma insertList_fstFieldOfContract_some_incl (φ : 𝓕.States) (φs : List
@[simp]
lemma insertList_none_getDual?_self (φ : 𝓕.States) (φs : List 𝓕.States)
(c : WickContraction φs.length) (i : Fin φs.length.succ) :
(insertList φ φs c i none).getDual? (Fin.cast (insertIdx_length_fin φ φs i).symm i) = none := by
(insertList φ c i none).getDual? (Fin.cast (insertIdx_length_fin φ φs i).symm i) = none := by
simp only [Nat.succ_eq_add_one, insertList, getDual?_congr, finCongr_apply, Fin.cast_trans,
Fin.cast_eq_self, Option.map_eq_none']
have h1 := c.insert_none_getDual?_isNone i
@ -96,27 +96,27 @@ lemma insertList_none_getDual?_self (φ : 𝓕.States) (φs : List 𝓕.States)
lemma insertList_isSome_getDual?_self (φ : 𝓕.States) (φs : List 𝓕.States)
(c : WickContraction φs.length) (i : Fin φs.length.succ) (j : c.uncontracted) :
((insertList φ φs c i (some j)).getDual?
((insertList φ c i (some j)).getDual?
(Fin.cast (insertIdx_length_fin φ φs i).symm i)).isSome := by
simp [insertList, getDual?_congr]
lemma insertList_some_getDual?_self_not_none (φ : 𝓕.States) (φs : List 𝓕.States)
(c : WickContraction φs.length) (i : Fin φs.length.succ) (j : c.uncontracted) :
¬ ((insertList φ φs c i (some j)).getDual?
¬ ((insertList φ c i (some j)).getDual?
(Fin.cast (insertIdx_length_fin φ φs i).symm i)) = none := by
simp [insertList, getDual?_congr]
@[simp]
lemma insertList_some_getDual?_self_eq (φ : 𝓕.States) (φs : List 𝓕.States)
(c : WickContraction φs.length) (i : Fin φs.length.succ) (j : c.uncontracted) :
((insertList φ φs c i (some j)).getDual? (Fin.cast (insertIdx_length_fin φ φs i).symm i))
((insertList φ c i (some j)).getDual? (Fin.cast (insertIdx_length_fin φ φs i).symm i))
= some (Fin.cast (insertIdx_length_fin φ φs i).symm (i.succAbove j)) := by
simp [insertList, getDual?_congr]
@[simp]
lemma insertList_some_getDual?_some_eq (φ : 𝓕.States) (φs : List 𝓕.States)
(c : WickContraction φs.length) (i : Fin φs.length.succ) (j : c.uncontracted) :
((insertList φ φs c i (some j)).getDual?
((insertList φ c i (some j)).getDual?
(Fin.cast (insertIdx_length_fin φ φs i).symm (i.succAbove j)))
= some (Fin.cast (insertIdx_length_fin φ φs i).symm i) := by
rw [getDual?_eq_some_iff_mem]
@ -127,7 +127,7 @@ lemma insertList_some_getDual?_some_eq (φ : 𝓕.States) (φs : List 𝓕.State
@[simp]
lemma insertList_none_succAbove_getDual?_eq_none_iff (φ : 𝓕.States) (φs : List 𝓕.States)
(c : WickContraction φs.length) (i : Fin φs.length.succ) (j : Fin φs.length) :
(insertList φ φs c i none).getDual? (Fin.cast (insertIdx_length_fin φ φs i).symm
(insertList φ c i none).getDual? (Fin.cast (insertIdx_length_fin φ φs i).symm
(i.succAbove j)) = none ↔ c.getDual? j = none := by
simp [insertList, getDual?_congr]
@ -135,7 +135,7 @@ lemma insertList_none_succAbove_getDual?_eq_none_iff (φ : 𝓕.States) (φs : L
lemma insertList_some_succAbove_getDual?_eq_option (φ : 𝓕.States) (φs : List 𝓕.States)
(c : WickContraction φs.length) (i : Fin φs.length.succ) (j : Fin φs.length)
(k : c.uncontracted) (hkj : j ≠ k.1) :
(insertList φ φs c i (some k)).getDual? (Fin.cast (insertIdx_length_fin φ φs i).symm
(insertList φ c i (some k)).getDual? (Fin.cast (insertIdx_length_fin φ φs i).symm
(i.succAbove j)) = Option.map (Fin.cast (insertIdx_length_fin φ φs i).symm ∘ i.succAbove)
(c.getDual? j) := by
simp only [Nat.succ_eq_add_one, insertList, getDual?_congr, finCongr_apply, Fin.cast_trans,
@ -145,7 +145,7 @@ lemma insertList_some_succAbove_getDual?_eq_option (φ : 𝓕.States) (φs : Lis
@[simp]
lemma insertList_none_succAbove_getDual?_isSome_iff (φ : 𝓕.States) (φs : List 𝓕.States)
(c : WickContraction φs.length) (i : Fin φs.length.succ) (j : Fin φs.length) :
((insertList φ φs c i none).getDual? (Fin.cast (insertIdx_length_fin φ φs i).symm
((insertList φ c i none).getDual? (Fin.cast (insertIdx_length_fin φ φs i).symm
(i.succAbove j))).isSome ↔ (c.getDual? j).isSome := by
rw [← not_iff_not]
simp
@ -153,9 +153,9 @@ lemma insertList_none_succAbove_getDual?_isSome_iff (φ : 𝓕.States) (φs : Li
@[simp]
lemma insertList_none_getDual?_get_eq (φ : 𝓕.States) (φs : List 𝓕.States)
(c : WickContraction φs.length) (i : Fin φs.length.succ) (j : Fin φs.length)
(h : ((insertList φ φs c i none).getDual? (Fin.cast (insertIdx_length_fin φ φs i).symm
(h : ((insertList φ c i none).getDual? (Fin.cast (insertIdx_length_fin φ φs i).symm
(i.succAbove j))).isSome) :
((insertList φ φs c i none).getDual? (Fin.cast (insertIdx_length_fin φ φs i).symm
((insertList φ c i none).getDual? (Fin.cast (insertIdx_length_fin φ φs i).symm
(i.succAbove j))).get h = Fin.cast (insertIdx_length_fin φ φs i).symm
(i.succAbove ((c.getDual? j).get (by simpa using h))) := by
simp [insertList, getDual?_congr_get]
@ -164,14 +164,14 @@ lemma insertList_none_getDual?_get_eq (φ : 𝓕.States) (φs : List 𝓕.States
@[simp]
lemma insertList_sndFieldOfContract_some_incl (φ : 𝓕.States) (φs : List 𝓕.States)
(c : WickContraction φs.length) (i : Fin φs.length.succ) (j : c.uncontracted) :
(insertList φ φs c i (some j)).sndFieldOfContract
(insertList φ c i (some j)).sndFieldOfContract
(congrLift (insertIdx_length_fin φ φs i).symm ⟨{i, i.succAbove j}, by simp [insert]⟩) =
if i < i.succAbove j.1 then
finCongr (insertIdx_length_fin φ φs i).symm (i.succAbove j.1) else
finCongr (insertIdx_length_fin φ φs i).symm i := by
split
· rename_i h
refine (insertList φ φs c i (some j)).eq_sndFieldOfContract_of_mem
refine (insertList φ c i (some j)).eq_sndFieldOfContract_of_mem
(a := congrLift (insertIdx_length_fin φ φs i).symm ⟨{i, i.succAbove j}, by simp [insert]⟩)
(i := finCongr (insertIdx_length_fin φ φs i).symm i) (j :=
finCongr (insertIdx_length_fin φ φs i).symm (i.succAbove j)) ?_ ?_ ?_
@ -180,7 +180,7 @@ lemma insertList_sndFieldOfContract_some_incl (φ : 𝓕.States) (φs : List
· rw [Fin.lt_def] at h ⊢
simp_all
· rename_i h
refine (insertList φ φs c i (some j)).eq_sndFieldOfContract_of_mem
refine (insertList φ c i (some j)).eq_sndFieldOfContract_of_mem
(a := congrLift (insertIdx_length_fin φ φs i).symm ⟨{i, i.succAbove j}, by simp [insert]⟩)
(i := finCongr (insertIdx_length_fin φ φs i).symm (i.succAbove j))
(j := finCongr (insertIdx_length_fin φ φs i).symm i) ?_ ?_ ?_
@ -193,7 +193,7 @@ lemma insertList_sndFieldOfContract_some_incl (φ : 𝓕.States) (φs : List
lemma insertList_none_prod_contractions (φ : 𝓕.States) (φs : List 𝓕.States)
(c : WickContraction φs.length) (i : Fin φs.length.succ)
(f : (c.insertList φ φs i none).1 → M) [CommMonoid M] :
(f : (c.insertList φ i none).1 → M) [CommMonoid M] :
∏ a, f a = ∏ (a : c.1), f (congrLift (insertIdx_length_fin φ φs i).symm
(insertLift i none a)) := by
let e1 := Equiv.ofBijective (c.insertLift i none) (insertLift_none_bijective i)
@ -205,7 +205,7 @@ lemma insertList_none_prod_contractions (φ : 𝓕.States) (φs : List 𝓕.Stat
lemma insertList_some_prod_contractions (φ : 𝓕.States) (φs : List 𝓕.States)
(c : WickContraction φs.length) (i : Fin φs.length.succ) (j : c.uncontracted)
(f : (c.insertList φ φs i (some j)).1 → M) [CommMonoid M] :
(f : (c.insertList φ i (some j)).1 → M) [CommMonoid M] :
∏ a, f a = f (congrLift (insertIdx_length_fin φ φs i).symm
⟨{i, i.succAbove j}, by simp [insert]⟩) *
∏ (a : c.1), f (congrLift (insertIdx_length_fin φ φs i).symm (insertLift i (some j) a)) := by
@ -268,7 +268,7 @@ lemma insert_fin_eq_self (φ : 𝓕.States) {φs : List 𝓕.States}
lemma insertList_uncontractedList_none_map (φ : 𝓕.States) {φs : List 𝓕.States}
(c : WickContraction φs.length) (i : Fin φs.length.succ) :
List.map (List.insertIdx (↑i) φ φs).get (insertList φ φs c i none).uncontractedList =
List.map (List.insertIdx (↑i) φ φs).get (insertList φ c i none).uncontractedList =
List.insertIdx (c.uncontractedListOrderPos i) φ (List.map φs.get c.uncontractedList) := by
simp only [Nat.succ_eq_add_one, insertList]
rw [congr_uncontractedList]
@ -285,7 +285,7 @@ lemma insertList_uncontractedList_none_map (φ : 𝓕.States) {φs : List 𝓕.S
lemma insertLift_sum (φ : 𝓕.States) {φs : List 𝓕.States}
(i : Fin φs.length.succ) [AddCommMonoid M] (f : WickContraction (φs.insertIdx i φ).length → M) :
∑ c, f c = ∑ (c : WickContraction φs.length), ∑ (k : Option (c.uncontracted)),
f (insertList φ φs c i k) := by
f (insertList φ c i k) := by
rw [sum_extractEquiv_congr (finCongr (insertIdx_length_fin φ φs i).symm i) f
(insertIdx_length_fin φ φs i)]
rfl

24
HepLean/PerturbationTheory/WickContraction/Sign.lean
View file

@ -30,7 +30,7 @@ def signFinset (c : WickContraction n) (i1 i2 : Fin n) : Finset (Fin n) :=
lemma signFinset_insertList_none (φ : 𝓕.States) (φs : List 𝓕.States)
(c : WickContraction φs.length)
(i : Fin φs.length.succ) (i1 i2 : Fin φs.length) :
(c.insertList φ φs i none).signFinset (finCongr (insertIdx_length_fin φ φs i).symm
(c.insertList φ i none).signFinset (finCongr (insertIdx_length_fin φ φs i).symm
(i.succAbove i1)) (finCongr (insertIdx_length_fin φ φs i).symm (i.succAbove i2)) =
if i.succAbove i1 < i ∧ i < i.succAbove i2 then
Insert.insert (finCongr (insertIdx_length_fin φ φs i).symm i)
@ -180,7 +180,7 @@ lemma stat_ofFinset_eq_one_of_gradingCompliant (φs : List 𝓕.States)
lemma signFinset_insertList_some (φ : 𝓕.States) (φs : List 𝓕.States)
(c : WickContraction φs.length) (i : Fin φs.length.succ) (i1 i2 : Fin φs.length)
(j : c.uncontracted) :
(c.insertList φ φs i (some j)).signFinset (finCongr (insertIdx_length_fin φ φs i).symm
(c.insertList φ i (some j)).signFinset (finCongr (insertIdx_length_fin φ φs i).symm
(i.succAbove i1)) (finCongr (insertIdx_length_fin φ φs i).symm (i.succAbove i2)) =
if i.succAbove i1 < i ∧ i < i.succAbove i2 ∧ (i1 < j) then
Insert.insert (finCongr (insertIdx_length_fin φ φs i).symm i)
@ -341,7 +341,7 @@ def signInsertNone (φ : 𝓕.States) (φs : List 𝓕.States) (c : WickContract
lemma sign_insert_none (φ : 𝓕.States) (φs : List 𝓕.States) (c : WickContraction φs.length)
(i : Fin φs.length.succ) :
(c.insertList φ φs i none).sign = (c.signInsertNone φ φs i) * c.sign := by
(c.insertList φ i none).sign = (c.signInsertNone φ φs i) * c.sign := by
rw [sign]
rw [signInsertNone, sign, ← Finset.prod_mul_distrib]
rw [insertList_none_prod_contractions]
@ -508,13 +508,13 @@ def signInsertSomeProd (φ : 𝓕.States) (φs : List 𝓕.States) (c : WickCont
coming from putting `i` next to `j`. -/
def signInsertSomeCoef (φ : 𝓕.States) (φs : List 𝓕.States) (c : WickContraction φs.length)
(i : Fin φs.length.succ) (j : c.uncontracted) : ℂ :=
let a : (c.insertList φ φs i (some j)).1 :=
let a : (c.insertList φ i (some j)).1 :=
congrLift (insertIdx_length_fin φ φs i).symm
⟨{i, i.succAbove j}, by simp [insert]⟩;
𝓢(𝓕 |>ₛ (φs.insertIdx i φ)[(c.insertList φ φs i (some j)).sndFieldOfContract a],
𝓢(𝓕 |>ₛ (φs.insertIdx i φ)[(c.insertList φ i (some j)).sndFieldOfContract a],
𝓕 |>ₛ ⟨(φs.insertIdx i φ).get, signFinset
(c.insertList φ φs i (some j)) ((c.insertList φ φs i (some j)).fstFieldOfContract a)
((c.insertList φ φs i (some j)).sndFieldOfContract a)⟩)
(c.insertList φ i (some j)) ((c.insertList φ i (some j)).fstFieldOfContract a)
((c.insertList φ i (some j)).sndFieldOfContract a)⟩)
/-- Given a Wick contraction `c` associated with a list of states `φs`
and an `i : Fin φs.length.succ`, the change in sign of the contraction associated with
@ -525,7 +525,7 @@ def signInsertSome (φ : 𝓕.States) (φs : List 𝓕.States) (c : WickContract
lemma sign_insert_some (φ : 𝓕.States) (φs : List 𝓕.States) (c : WickContraction φs.length)
(i : Fin φs.length.succ) (j : c.uncontracted) :
(c.insertList φ φs i (some j)).sign = (c.signInsertSome φ φs i j) * c.sign := by
(c.insertList φ i (some j)).sign = (c.signInsertSome φ φs i j) * c.sign := by
rw [sign]
rw [signInsertSome, signInsertSomeProd, sign, mul_assoc, ← Finset.prod_mul_distrib]
rw [insertList_some_prod_contractions]
@ -730,11 +730,11 @@ lemma signInsertSomeCoef_if (φ : 𝓕.States) (φs : List 𝓕.States) (c : Wic
c.signInsertSomeCoef φ φs i j =
if i < i.succAbove j then
𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ ⟨(φs.insertIdx i φ).get,
(signFinset (c.insertList φ φs i (some j)) (finCongr (insertIdx_length_fin φ φs i).symm i)
(signFinset (c.insertList φ i (some j)) (finCongr (insertIdx_length_fin φ φs i).symm i)
(finCongr (insertIdx_length_fin φ φs i).symm (i.succAbove j)))⟩)
else
𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ ⟨(φs.insertIdx i φ).get,
signFinset (c.insertList φ φs i (some j))
signFinset (c.insertList φ i (some j))
(finCongr (insertIdx_length_fin φ φs i).symm (i.succAbove j))
(finCongr (insertIdx_length_fin φ φs i).symm i)⟩) := by
simp only [signInsertSomeCoef, instCommGroup.eq_1, Nat.succ_eq_add_one,
@ -748,7 +748,7 @@ lemma stat_signFinset_insert_some_self_fst
(φ : 𝓕.States) (φs : List 𝓕.States) (c : WickContraction φs.length)
(i : Fin φs.length.succ) (j : c.uncontracted) :
(𝓕 |>ₛ ⟨(φs.insertIdx i φ).get,
(signFinset (c.insertList φ φs i (some j)) (finCongr (insertIdx_length_fin φ φs i).symm i)
(signFinset (c.insertList φ i (some j)) (finCongr (insertIdx_length_fin φ φs i).symm i)
(finCongr (insertIdx_length_fin φ φs i).symm (i.succAbove j)))⟩) =
𝓕 |>ₛ ⟨φs.get,
(Finset.univ.filter (fun x => i < i.succAbove x ∧ x < j ∧ ((c.getDual? x = none) ∨
@ -824,7 +824,7 @@ lemma stat_signFinset_insert_some_self_fst
lemma stat_signFinset_insert_some_self_snd (φ : 𝓕.States) (φs : List 𝓕.States)
(c : WickContraction φs.length) (i : Fin φs.length.succ) (j : c.uncontracted) :
(𝓕 |>ₛ ⟨(φs.insertIdx i φ).get,
(signFinset (c.insertList φ φs i (some j))
(signFinset (c.insertList φ i (some j))
(finCongr (insertIdx_length_fin φ φs i).symm (i.succAbove j))
(finCongr (insertIdx_length_fin φ φs i).symm i))⟩) =
𝓕 |>ₛ ⟨φs.get,

8
HepLean/PerturbationTheory/WickContraction/TimeContract.lean
View file

@ -30,7 +30,7 @@ noncomputable def timeContract (𝓞 : 𝓕.ProtoOperatorAlgebra) {φs : List
@[simp]
lemma timeContract_insertList_none (𝓞 : 𝓕.ProtoOperatorAlgebra) (φ : 𝓕.States) (φs : List 𝓕.States)
(c : WickContraction φs.length) (i : Fin φs.length.succ) :
(c.insertList φ φs i none).timeContract 𝓞 = c.timeContract 𝓞 := by
(c.insertList φ i none).timeContract 𝓞 = c.timeContract 𝓞 := by
rw [timeContract, insertList_none_prod_contractions]
congr
ext a
@ -38,7 +38,7 @@ lemma timeContract_insertList_none (𝓞 : 𝓕.ProtoOperatorAlgebra) (φ : 𝓕
lemma timeConract_insertList_some (𝓞 : 𝓕.ProtoOperatorAlgebra) (φ : 𝓕.States) (φs : List 𝓕.States)
(c : WickContraction φs.length) (i : Fin φs.length.succ) (j : c.uncontracted) :
(c.insertList φ φs i (some j)).timeContract 𝓞 =
(c.insertList φ i (some j)).timeContract 𝓞 =
(if i < i.succAbove j then
⟨𝓞.timeContract φ φs[j.1], 𝓞.timeContract_mem_center _ _⟩
else ⟨𝓞.timeContract φs[j.1] φ, 𝓞.timeContract_mem_center _ _⟩) * c.timeContract 𝓞 := by
@ -59,7 +59,7 @@ lemma timeConract_insertList_some_eq_mul_contractStateAtIndex_lt
(𝓞 : 𝓕.ProtoOperatorAlgebra) (φ : 𝓕.States) (φs : List 𝓕.States)
(c : WickContraction φs.length) (i : Fin φs.length.succ) (k : c.uncontracted)
(ht : 𝓕.timeOrderRel φ φs[k.1]) (hik : i < i.succAbove k) :
(c.insertList φ φs i (some k)).timeContract 𝓞 =
(c.insertList φ i (some k)).timeContract 𝓞 =
𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ ⟨φs.get, (c.uncontracted.filter (fun x => x < k))⟩)
• (𝓞.contractStateAtIndex φ (List.map φs.get c.uncontractedList)
((uncontractedStatesEquiv φs c) (some k)) * c.timeContract 𝓞) := by
@ -94,7 +94,7 @@ lemma timeConract_insertList_some_eq_mul_contractStateAtIndex_not_lt
(𝓞 : 𝓕.ProtoOperatorAlgebra) (φ : 𝓕.States) (φs : List 𝓕.States)
(c : WickContraction φs.length) (i : Fin φs.length.succ) (k : c.uncontracted)
(ht : ¬ 𝓕.timeOrderRel φs[k.1] φ) (hik : ¬ i < i.succAbove k) :
(c.insertList φ φs i (some k)).timeContract 𝓞 =
(c.insertList φ i (some k)).timeContract 𝓞 =
𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ ⟨φs.get, (c.uncontracted.filter (fun x => x ≤ k))⟩)
• (𝓞.contractStateAtIndex φ (List.map φs.get c.uncontractedList)
((uncontractedStatesEquiv φs c) (some k)) * c.timeContract 𝓞) := by