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refactor: Rename States to FieldOps

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jstoobysmith 2025-02-03 11:28:14 +00:00
parent 171e80fc04
commit 8f41de5785
36 changed files with 946 additions and 946 deletions
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6
HepLean/PerturbationTheory/WickContraction/Sign/Basic.lean
View file

@ -32,16 +32,16 @@ def signFinset (c : WickContraction n) (i1 i2 : Fin n) : Finset (Fin n) :=
the sign associated with `φsΛ` is sign corresponding to the number
of fermionic-fermionic exchanges one must do to put elements in contracted pairs
of `φsΛ` next to each other. -/
def sign (φs : List 𝓕.States) (φsΛ : WickContraction φs.length) : ℂ :=
def sign (φs : List 𝓕.FieldOp) (φsΛ : WickContraction φs.length) : ℂ :=
∏ (a : φsΛ.1), 𝓢(𝓕 |>ₛ φs[φsΛ.sndFieldOfContract a],
𝓕 |>ₛ ⟨φs.get, φsΛ.signFinset (φsΛ.fstFieldOfContract a) (φsΛ.sndFieldOfContract a)⟩)
lemma sign_empty (φs : List 𝓕.States) :
lemma sign_empty (φs : List 𝓕.FieldOp) :
sign φs empty = 1 := by
rw [sign]
simp [empty]
lemma sign_congr {φs φs' : List 𝓕.States} (h : φs = φs') (φsΛ : WickContraction φs.length) :
lemma sign_congr {φs φs' : List 𝓕.FieldOp} (h : φs = φs') (φsΛ : WickContraction φs.length) :
sign φs' (congr (by simp [h]) φsΛ) = sign φs φsΛ := by
subst h
rfl

18
HepLean/PerturbationTheory/WickContraction/Sign/InsertNone.lean
View file

@ -19,7 +19,7 @@ variable {n : ℕ} (c : WickContraction n)
open HepLean.List
open FieldStatistic
lemma signFinset_insertAndContract_none (φ : 𝓕.States) (φs : List 𝓕.States)
lemma signFinset_insertAndContract_none (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp)
(φsΛ : WickContraction φs.length)
(i : Fin φs.length.succ) (i1 i2 : Fin φs.length) :
(φsΛ ↩Λ φ i none).signFinset (finCongr (insertIdx_length_fin φ φs i).symm
@ -81,14 +81,14 @@ lemma signFinset_insertAndContract_none (φ : 𝓕.States) (φs : List 𝓕.Stat
For each contracted pair `{a1, a2}` in `φsΛ` if `a1 < a2` such that `i` is within the range
`a1 < i < a2` we pick up a sign equal to `𝓢(φ, φs[a2])`. -/
def signInsertNone (φ : 𝓕.States) (φs : List 𝓕.States) (φsΛ : WickContraction φs.length)
def signInsertNone (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp) (φsΛ : WickContraction φs.length)
(i : Fin φs.length.succ) : ℂ :=
∏ (a : φsΛ.1),
if i.succAbove (φsΛ.fstFieldOfContract a) < i ∧ i < i.succAbove (φsΛ.sndFieldOfContract a) then
𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φs[φsΛ.sndFieldOfContract a])
else 1
lemma sign_insert_none (φ : 𝓕.States) (φs : List 𝓕.States) (φsΛ : WickContraction φs.length)
lemma sign_insert_none (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp) (φsΛ : WickContraction φs.length)
(i : Fin φs.length.succ) :
(φsΛ ↩Λ φ i none).sign = (φsΛ.signInsertNone φ φs i) * φsΛ.sign := by
rw [sign]
@ -109,7 +109,7 @@ lemma sign_insert_none (φ : 𝓕.States) (φs : List 𝓕.States) (φsΛ : Wick
simp
· rw [stat_ofFinset_of_insertAndContractLiftFinset]
lemma signInsertNone_eq_mul_fst_snd (φ : 𝓕.States) (φs : List 𝓕.States)
lemma signInsertNone_eq_mul_fst_snd (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp)
(φsΛ : WickContraction φs.length) (i : Fin φs.length.succ) :
φsΛ.signInsertNone φ φs i = ∏ (a : φsΛ.1),
(if i.succAbove (φsΛ.fstFieldOfContract a) < i then
@ -143,7 +143,7 @@ lemma signInsertNone_eq_mul_fst_snd (φ : 𝓕.States) (φs : List 𝓕.States)
have hx := (Fin.succAbove_lt_succAbove_iff (p := i)).mpr hn
omega
lemma signInsertNone_eq_prod_prod (φ : 𝓕.States) (φs : List 𝓕.States)
lemma signInsertNone_eq_prod_prod (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp)
(φsΛ : WickContraction φs.length) (i : Fin φs.length.succ) (hG : GradingCompliant φs φsΛ) :
φsΛ.signInsertNone φ φs i = ∏ (a : φsΛ.1), ∏ (x : a),
(if i.succAbove x < i then 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φs[x.1]) else 1) := by
@ -158,12 +158,12 @@ lemma signInsertNone_eq_prod_prod (φ : 𝓕.States) (φs : List 𝓕.States)
erw [hG a]
rfl
lemma sign_insert_none_zero (φ : 𝓕.States) (φs : List 𝓕.States) (φsΛ : WickContraction φs.length) :
lemma sign_insert_none_zero (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp) (φsΛ : WickContraction φs.length) :
(φsΛ ↩Λ φ 0 none).sign = φsΛ.sign := by
rw [sign_insert_none]
simp [signInsertNone]
lemma signInsertNone_eq_prod_getDual?_Some (φ : 𝓕.States) (φs : List 𝓕.States)
lemma signInsertNone_eq_prod_getDual?_Some (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp)
(φsΛ : WickContraction φs.length) (i : Fin φs.length.succ) (hG : GradingCompliant φs φsΛ) :
φsΛ.signInsertNone φ φs i = ∏ (x : Fin φs.length),
if (φsΛ.getDual? x).isSome then
@ -190,7 +190,7 @@ lemma signInsertNone_eq_prod_getDual?_Some (φ : 𝓕.States) (φs : List 𝓕.S
rfl
exact hG
lemma signInsertNone_eq_filter_map (φ : 𝓕.States) (φs : List 𝓕.States)
lemma signInsertNone_eq_filter_map (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp)
(φsΛ : WickContraction φs.length) (i : Fin φs.length.succ) (hG : GradingCompliant φs φsΛ) :
φsΛ.signInsertNone φ φs i =
𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ ((List.filter (fun x => (φsΛ.getDual? x).isSome ∧ i.succAbove x < i)
@ -216,7 +216,7 @@ lemma signInsertNone_eq_filter_map (φ : 𝓕.States) (φs : List 𝓕.States)
/-- The change in sign when inserting a field `φ` at `i` into `φsΛ` is equal
to the sign got by moving `φ` through each field `φ₀…φᵢ₋₁` which has a dual. -/
lemma signInsertNone_eq_filterset (φ : 𝓕.States) (φs : List 𝓕.States)
lemma signInsertNone_eq_filterset (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp)
(φsΛ : WickContraction φs.length) (i : Fin φs.length.succ) (hG : GradingCompliant φs φsΛ) :
φsΛ.signInsertNone φ φs i = 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ ⟨φs.get, Finset.univ.filter
(fun x => (φsΛ.getDual? x).isSome ∧ i.succAbove x < i)⟩) := by

34
HepLean/PerturbationTheory/WickContraction/Sign/InsertSome.lean
View file

@ -27,7 +27,7 @@ open FieldStatistic
-/
lemma stat_ofFinset_eq_one_of_gradingCompliant (φs : List 𝓕.States)
lemma stat_ofFinset_eq_one_of_gradingCompliant (φs : List 𝓕.FieldOp)
(a : Finset (Fin φs.length)) (φsΛ : WickContraction φs.length) (hg : GradingCompliant φs φsΛ)
(hnon : ∀ i, φsΛ.getDual? i = none → i ∉ a)
(hsom : ∀ i, (h : (φsΛ.getDual? i).isSome) → i ∈ a → (φsΛ.getDual? i).get h ∈ a) :
@ -64,7 +64,7 @@ lemma stat_ofFinset_eq_one_of_gradingCompliant (φs : List 𝓕.States)
rfl
lemma signFinset_insertAndContract_some (φ : 𝓕.States) (φs : List 𝓕.States)
lemma signFinset_insertAndContract_some (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp)
(φsΛ : WickContraction φs.length) (i : Fin φs.length.succ) (i1 i2 : Fin φs.length)
(j : φsΛ.uncontracted) :
(φsΛ ↩Λ φ i (some j)).signFinset (finCongr (insertIdx_length_fin φ φs i).symm
@ -206,7 +206,7 @@ lemma signFinset_insertAndContract_some (φ : 𝓕.States) (φs : List 𝓕.Stat
inserting `φ` into `φs` at position `i` and contracting it with `j : c.uncontracted`
coming from contractions other then the `i` and `j` contraction but which
are effected by this new contraction. -/
def signInsertSomeProd (φ : 𝓕.States) (φs : List 𝓕.States) (φsΛ : WickContraction φs.length)
def signInsertSomeProd (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp) (φsΛ : WickContraction φs.length)
(i : Fin φs.length.succ) (j : φsΛ.uncontracted) : ℂ :=
∏ (a : φsΛ.1),
if i.succAbove (φsΛ.fstFieldOfContract a) < i ∧ i < i.succAbove (φsΛ.sndFieldOfContract a) ∧
@ -223,7 +223,7 @@ def signInsertSomeProd (φ : 𝓕.States) (φs : List 𝓕.States) (φsΛ : Wick
and an `i : Fin φs.length.succ`, the change in sign of the contraction associated with
inserting `φ` into `φs` at position `i` and contracting it with `j : c.uncontracted`
coming from putting `i` next to `j`. -/
def signInsertSomeCoef (φ : 𝓕.States) (φs : List 𝓕.States) (φsΛ : WickContraction φs.length)
def signInsertSomeCoef (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp) (φsΛ : WickContraction φs.length)
(i : Fin φs.length.succ) (j : φsΛ.uncontracted) : ℂ :=
let a : (φsΛ ↩Λ φ i (some j)).1 := congrLift (insertIdx_length_fin φ φs i).symm
⟨{i, i.succAbove j}, by simp [insertAndContractNat]⟩;
@ -235,11 +235,11 @@ def signInsertSomeCoef (φ : 𝓕.States) (φs : List 𝓕.States) (φsΛ : Wick
/-- Given a Wick contraction `φsΛ` associated with a list of states `φs`
and an `i : Fin φs.length.succ`, the change in sign of the contraction associated with
inserting `φ` into `φs` at position `i` and contracting it with `j : c.uncontracted`. -/
def signInsertSome (φ : 𝓕.States) (φs : List 𝓕.States) (φsΛ : WickContraction φs.length)
def signInsertSome (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp) (φsΛ : WickContraction φs.length)
(i : Fin φs.length.succ) (j : φsΛ.uncontracted) : ℂ :=
signInsertSomeCoef φ φs φsΛ i j * signInsertSomeProd φ φs φsΛ i j
lemma sign_insert_some (φ : 𝓕.States) (φs : List 𝓕.States) (φsΛ : WickContraction φs.length)
lemma sign_insert_some (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp) (φsΛ : WickContraction φs.length)
(i : Fin φs.length.succ) (j : φsΛ.uncontracted) :
(φsΛ ↩Λ φ i (some j)).sign = (φsΛ.signInsertSome φ φs i j) * φsΛ.sign := by
rw [sign]
@ -280,7 +280,7 @@ lemma sign_insert_some (φ : 𝓕.States) (φs : List 𝓕.States) (φsΛ : Wick
rw [stat_ofFinset_of_insertAndContractLiftFinset]
simp_all
lemma signInsertSomeProd_eq_one_if (φ : 𝓕.States) (φs : List 𝓕.States)
lemma signInsertSomeProd_eq_one_if (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp)
(φsΛ : WickContraction φs.length) (i : Fin φs.length.succ) (j : φsΛ.uncontracted)
(hφj : (𝓕 |>ₛ φ) = (𝓕 |>ₛ φs[j.1])) :
φsΛ.signInsertSomeProd φ φs i j =
@ -313,7 +313,7 @@ lemma signInsertSomeProd_eq_one_if (φ : 𝓕.States) (φs : List 𝓕.States)
implies_true, and_true]
omega
lemma signInsertSomeProd_eq_prod_prod (φ : 𝓕.States) (φs : List 𝓕.States)
lemma signInsertSomeProd_eq_prod_prod (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp)
(φsΛ : WickContraction φs.length)
(i : Fin φs.length.succ) (j : φsΛ.uncontracted) (hφj : (𝓕 |>ₛ φ) = (𝓕 |>ₛ φs[j.1]))
(hg : GradingCompliant φs φsΛ) :
@ -347,7 +347,7 @@ lemma signInsertSomeProd_eq_prod_prod (φ : 𝓕.States) (φs : List 𝓕.States
· omega
simp [hφj]
lemma signInsertSomeProd_eq_prod_fin (φ : 𝓕.States) (φs : List 𝓕.States)
lemma signInsertSomeProd_eq_prod_fin (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp)
(φsΛ : WickContraction φs.length)
(i : Fin φs.length.succ) (j : φsΛ.uncontracted) (hφj : (𝓕 |>ₛ φ) = (𝓕 |>ₛ φs[j.1]))
(hg : GradingCompliant φs φsΛ) :
@ -380,7 +380,7 @@ lemma signInsertSomeProd_eq_prod_fin (φ : 𝓕.States) (φs : List 𝓕.States)
simp only [hφj, Fin.getElem_fin]
exact hg
lemma signInsertSomeProd_eq_list (φ : 𝓕.States) (φs : List 𝓕.States)
lemma signInsertSomeProd_eq_list (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp)
(φsΛ : WickContraction φs.length)
(i : Fin φs.length.succ) (j : φsΛ.uncontracted) (hφj : (𝓕 |>ₛ φ) = (𝓕 |>ₛ φs[j.1]))
(hg : GradingCompliant φs φsΛ) :
@ -414,7 +414,7 @@ lemma signInsertSomeProd_eq_list (φ : 𝓕.States) (φs : List 𝓕.States)
simp only [hφj, Fin.getElem_fin]
exact hg
lemma signInsertSomeProd_eq_finset (φ : 𝓕.States) (φs : List 𝓕.States)
lemma signInsertSomeProd_eq_finset (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp)
(φsΛ : WickContraction φs.length)
(i : Fin φs.length.succ) (j : φsΛ.uncontracted) (hφj : (𝓕 |>ₛ φ) = (𝓕 |>ₛ φs[j.1]))
(hg : GradingCompliant φs φsΛ) :
@ -442,7 +442,7 @@ lemma signInsertSomeProd_eq_finset (φ : 𝓕.States) (φs : List 𝓕.States)
simp only [hφj, Fin.getElem_fin]
exact hg
lemma signInsertSomeCoef_if (φ : 𝓕.States) (φs : List 𝓕.States) (φsΛ : WickContraction φs.length)
lemma signInsertSomeCoef_if (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp) (φsΛ : WickContraction φs.length)
(i : Fin φs.length.succ) (j : φsΛ.uncontracted) (hφj : (𝓕 |>ₛ φ) = (𝓕 |>ₛ φs[j.1])) :
φsΛ.signInsertSomeCoef φ φs i j =
if i < i.succAbove j then
@ -462,7 +462,7 @@ lemma signInsertSomeCoef_if (φ : 𝓕.States) (φs : List 𝓕.States) (φsΛ :
· simp [hφj]
lemma stat_signFinset_insert_some_self_fst
(φ : 𝓕.States) (φs : List 𝓕.States) (φsΛ : WickContraction φs.length)
(φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp) (φsΛ : WickContraction φs.length)
(i : Fin φs.length.succ) (j : φsΛ.uncontracted) :
(𝓕 |>ₛ ⟨(φs.insertIdx i φ).get,
(signFinset (φsΛ ↩Λ φ i (some j)) (finCongr (insertIdx_length_fin φ φs i).symm i)
@ -538,7 +538,7 @@ lemma stat_signFinset_insert_some_self_fst
have hy2 := hy2 h
simpa [Option.get_map] using hy2
lemma stat_signFinset_insert_some_self_snd (φ : 𝓕.States) (φs : List 𝓕.States)
lemma stat_signFinset_insert_some_self_snd (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp)
(φsΛ : WickContraction φs.length) (i : Fin φs.length.succ) (j : φsΛ.uncontracted) :
(𝓕 |>ₛ ⟨(φs.insertIdx i φ).get,
(signFinset (φsΛ ↩Λ φ i (some j))
@ -618,7 +618,7 @@ lemma stat_signFinset_insert_some_self_snd (φ : 𝓕.States) (φs : List 𝓕.S
simp only [Option.get_map, Function.comp_apply, Fin.coe_cast, Fin.val_fin_lt]
exact Fin.succAbove_lt_succAbove_iff.mpr hy2
lemma signInsertSomeCoef_eq_finset (φ : 𝓕.States) (φs : List 𝓕.States)
lemma signInsertSomeCoef_eq_finset (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp)
(φsΛ : WickContraction φs.length) (i : Fin φs.length.succ) (j : φsΛ.uncontracted)
(hφj : (𝓕 |>ₛ φ) = (𝓕 |>ₛ φs[j.1])) : φsΛ.signInsertSomeCoef φ φs i j =
if i < i.succAbove j then
@ -637,7 +637,7 @@ lemma signInsertSomeCoef_eq_finset (φ : 𝓕.States) (φs : List 𝓕.States)
contracting it with `k` (`k < i`) is equal
to the sign got by moving `φ` through each field `φ₀…φᵢ₋₁`
multiplied by the sign got moving `φ` through each uncontracted field `φ₀…φₖ`. -/
lemma signInsertSome_mul_filter_contracted_of_lt (φ : 𝓕.States) (φs : List 𝓕.States)
lemma signInsertSome_mul_filter_contracted_of_lt (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp)
(φsΛ : WickContraction φs.length) (i : Fin φs.length.succ) (k : φsΛ.uncontracted)
(hk : i.succAbove k < i) (hg : GradingCompliant φs φsΛ ∧ (𝓕 |>ₛ φ) = 𝓕 |>ₛ φs[k.1]) :
signInsertSome φ φs φsΛ i k *
@ -744,7 +744,7 @@ lemma signInsertSome_mul_filter_contracted_of_lt (φ : 𝓕.States) (φs : List
contracting it with `k` (`i < k`) is equal
to the sign got by moving `φ` through each field `φ₀…φᵢ₋₁`
multiplied by the sign got moving `φ` through each uncontracted field `φ₀…φₖ₋₁`. -/
lemma signInsertSome_mul_filter_contracted_of_not_lt (φ : 𝓕.States) (φs : List 𝓕.States)
lemma signInsertSome_mul_filter_contracted_of_not_lt (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp)
(φsΛ : WickContraction φs.length) (i : Fin φs.length.succ) (k : φsΛ.uncontracted)
(hk : ¬ i.succAbove k < i) (hg : GradingCompliant φs φsΛ ∧ (𝓕 |>ₛ φ) = 𝓕 |>ₛ φs[k.1]) :
signInsertSome φ φs φsΛ i k *

30
HepLean/PerturbationTheory/WickContraction/Sign/Join.lean
View file

@ -20,7 +20,7 @@ open FieldOpAlgebra
open FieldStatistic
lemma stat_signFinset_right {φs : List 𝓕.States} (φsΛ : WickContraction φs.length)
lemma stat_signFinset_right {φs : List 𝓕.FieldOp} (φsΛ : WickContraction φs.length)
(φsucΛ : WickContraction [φsΛ]ᵘᶜ.length) (i j : Fin [φsΛ]ᵘᶜ.length) :
(𝓕 |>ₛ ⟨[φsΛ]ᵘᶜ.get, φsucΛ.signFinset i j⟩) =
(𝓕 |>ₛ ⟨φs.get, (φsucΛ.signFinset i j).map uncontractedListEmd⟩) := by
@ -32,7 +32,7 @@ lemma stat_signFinset_right {φs : List 𝓕.States} (φsΛ : WickContraction φ
intro i j h
exact uncontractedListEmd_strictMono h
lemma signFinset_right_map_uncontractedListEmd_eq_filter {φs : List 𝓕.States}
lemma signFinset_right_map_uncontractedListEmd_eq_filter {φs : List 𝓕.FieldOp}
(φsΛ : WickContraction φs.length) (φsucΛ : WickContraction [φsΛ]ᵘᶜ.length)
(i j : Fin [φsΛ]ᵘᶜ.length) : (φsucΛ.signFinset i j).map uncontractedListEmd =
((join φsΛ φsucΛ).signFinset (uncontractedListEmd i) (uncontractedListEmd j)).filter
@ -90,7 +90,7 @@ lemma signFinset_right_map_uncontractedListEmd_eq_filter {φs : List 𝓕.States
exact hl
exact fun _ _ h => uncontractedListEmd_strictMono h
lemma sign_right_eq_prod_mul_prod {φs : List 𝓕.States} (φsΛ : WickContraction φs.length)
lemma sign_right_eq_prod_mul_prod {φs : List 𝓕.FieldOp} (φsΛ : WickContraction φs.length)
(φsucΛ : WickContraction [φsΛ]ᵘᶜ.length) :
φsucΛ.sign = (∏ a, 𝓢(𝓕|>ₛ [φsΛ]ᵘᶜ[φsucΛ.sndFieldOfContract a], 𝓕|>ₛ ⟨φs.get,
((join φsΛ φsucΛ).signFinset (uncontractedListEmd (φsucΛ.fstFieldOfContract a))
@ -107,7 +107,7 @@ lemma sign_right_eq_prod_mul_prod {φs : List 𝓕.States} (φsΛ : WickContract
rw [stat_signFinset_right, signFinset_right_map_uncontractedListEmd_eq_filter]
rw [ofFinset_filter]
lemma join_singleton_signFinset_eq_filter {φs : List 𝓕.States}
lemma join_singleton_signFinset_eq_filter {φs : List 𝓕.FieldOp}
{i j : Fin φs.length} (h : i < j)
(φsucΛ : WickContraction [singleton h]ᵘᶜ.length) :
(join (singleton h) φsucΛ).signFinset i j =
@ -149,7 +149,7 @@ lemma join_singleton_signFinset_eq_filter {φs : List 𝓕.States}
· simp only [Bool.not_eq_true, Option.not_isSome, Option.isNone_iff_eq_none] at h2'
simp [h2']
lemma join_singleton_left_signFinset_eq_filter {φs : List 𝓕.States}
lemma join_singleton_left_signFinset_eq_filter {φs : List 𝓕.FieldOp}
{i j : Fin φs.length} (h : i < j)
(φsucΛ : WickContraction [singleton h]ᵘᶜ.length) :
(𝓕 |>ₛ ⟨φs.get, (singleton h).signFinset i j⟩)
@ -166,7 +166,7 @@ lemma join_singleton_left_signFinset_eq_filter {φs : List 𝓕.States}
/-- The difference in sign between `φsucΛ.sign` and the direct contribution of `φsucΛ` to
`(join (singleton h) φsucΛ)`. -/
def joinSignRightExtra {φs : List 𝓕.States}
def joinSignRightExtra {φs : List 𝓕.FieldOp}
{i j : Fin φs.length} (h : i < j)
(φsucΛ : WickContraction [singleton h]ᵘᶜ.length) : ℂ :=
∏ a, 𝓢(𝓕|>ₛ [singleton h]ᵘᶜ[φsucΛ.sndFieldOfContract a], 𝓕|>ₛ ⟨φs.get,
@ -176,7 +176,7 @@ def joinSignRightExtra {φs : List 𝓕.States}
/-- The difference in sign between `(singleton h).sign` and the direct contribution of
`(singleton h)` to `(join (singleton h) φsucΛ)`. -/
def joinSignLeftExtra {φs : List 𝓕.States}
def joinSignLeftExtra {φs : List 𝓕.FieldOp}
{i j : Fin φs.length} (h : i < j)
(φsucΛ : WickContraction [singleton h]ᵘᶜ.length) : ℂ :=
𝓢(𝓕 |>ₛ φs[j], (𝓕 |>ₛ ⟨φs.get, ((singleton h).signFinset i j).filter (fun c =>
@ -184,7 +184,7 @@ def joinSignLeftExtra {φs : List 𝓕.States}
((h1 : ((join (singleton h) φsucΛ).getDual? c).isSome) →
(((join (singleton h) φsucΛ).getDual? c).get h1) < i)))⟩))
lemma join_singleton_sign_left {φs : List 𝓕.States}
lemma join_singleton_sign_left {φs : List 𝓕.FieldOp}
{i j : Fin φs.length} (h : i < j)
(φsucΛ : WickContraction [singleton h]ᵘᶜ.length) :
(singleton h).sign = 𝓢(𝓕 |>ₛ φs[j],
@ -194,7 +194,7 @@ lemma join_singleton_sign_left {φs : List 𝓕.States}
rw [map_mul]
rfl
lemma join_singleton_sign_right {φs : List 𝓕.States}
lemma join_singleton_sign_right {φs : List 𝓕.FieldOp}
{i j : Fin φs.length} (h : i < j)
(φsucΛ : WickContraction [singleton h]ᵘᶜ.length) :
φsucΛ.sign =
@ -206,7 +206,7 @@ lemma join_singleton_sign_right {φs : List 𝓕.States}
rfl
lemma joinSignRightExtra_eq_i_j_finset_eq_if {φs : List 𝓕.States}
lemma joinSignRightExtra_eq_i_j_finset_eq_if {φs : List 𝓕.FieldOp}
{i j : Fin φs.length} (h : i < j)
(φsucΛ : WickContraction [singleton h]ᵘᶜ.length) :
joinSignRightExtra h φsucΛ = ∏ a,
@ -298,7 +298,7 @@ lemma joinSignRightExtra_eq_i_j_finset_eq_if {φs : List 𝓕.States}
Option.get_some, forall_const, false_or, true_and]
omega
lemma joinSignLeftExtra_eq_joinSignRightExtra {φs : List 𝓕.States}
lemma joinSignLeftExtra_eq_joinSignRightExtra {φs : List 𝓕.FieldOp}
{i j : Fin φs.length} (h : i < j) (hs : (𝓕 |>ₛ φs[i]) = (𝓕 |>ₛ φs[j]))
(φsucΛ : WickContraction [singleton h]ᵘᶜ.length) :
joinSignLeftExtra h φsucΛ = joinSignRightExtra h φsucΛ := by
@ -380,7 +380,7 @@ lemma joinSignLeftExtra_eq_joinSignRightExtra {φs : List 𝓕.States}
simp only [Finset.disjoint_singleton_right, Finset.mem_singleton]
exact Fin.ne_of_lt h
lemma join_sign_singleton {φs : List 𝓕.States}
lemma join_sign_singleton {φs : List 𝓕.FieldOp}
{i j : Fin φs.length} (h : i < j) (hs : (𝓕 |>ₛ φs[i]) = (𝓕 |>ₛ φs[j]))
(φsucΛ : WickContraction [singleton h]ᵘᶜ.length) :
(join (singleton h) φsucΛ).sign = (singleton h).sign * φsucΛ.sign := by
@ -401,7 +401,7 @@ lemma join_sign_singleton {φs : List 𝓕.States}
· funext a
simp
lemma join_sign_induction {φs : List 𝓕.States} (φsΛ : WickContraction φs.length)
lemma join_sign_induction {φs : List 𝓕.FieldOp} (φsΛ : WickContraction φs.length)
(φsucΛ : WickContraction [φsΛ]ᵘᶜ.length) (hc : φsΛ.GradingCompliant) :
(n : ℕ) → (hn : φsΛ.1.card = n) →
(join φsΛ φsucΛ).sign = φsΛ.sign * φsucΛ.sign
@ -428,12 +428,12 @@ lemma join_sign_induction {φs : List 𝓕.States} (φsΛ : WickContraction φs.
apply sign_congr
exact join_uncontractedListGet (singleton hij) φsucΛ'
lemma join_sign {φs : List 𝓕.States} (φsΛ : WickContraction φs.length)
lemma join_sign {φs : List 𝓕.FieldOp} (φsΛ : WickContraction φs.length)
(φsucΛ : WickContraction [φsΛ]ᵘᶜ.length) (hc : φsΛ.GradingCompliant) :
(join φsΛ φsucΛ).sign = φsΛ.sign * φsucΛ.sign := by
exact join_sign_induction φsΛ φsucΛ hc (φsΛ).1.card rfl
lemma join_sign_timeContract {φs : List 𝓕.States} (φsΛ : WickContraction φs.length)
lemma join_sign_timeContract {φs : List 𝓕.FieldOp} (φsΛ : WickContraction φs.length)
(φsucΛ : WickContraction [φsΛ]ᵘᶜ.length) :
(join φsΛ φsucΛ).sign • (join φsΛ φsucΛ).timeContract.1 =
(φsΛ.sign • φsΛ.timeContract.1) * (φsucΛ.sign • φsucΛ.timeContract.1) := by