refactor: Lint
This commit is contained in:
jstoobysmith
parent
fca3f02eca
commit
c8e9c285a3
8 changed files with 229 additions and 194 deletions
|
|
@ -23,6 +23,9 @@ variable {n : ℕ} (c : WickContraction n)
|
|||
open HepLean.List
|
||||
open FieldOpAlgebra
|
||||
|
||||
/-- Given a Wick contraction `φsΛ` on `φs` and a Wick contraction `φsucΛ` on the uncontracted fields
|
||||
within `φsΛ`, a Wick contraction on `φs`consisting of the contractins in `φsΛ` and
|
||||
the contractions in `φsucΛ`. -/
|
||||
def join {φs : List 𝓕.States} (φsΛ : WickContraction φs.length)
|
||||
(φsucΛ : WickContraction [φsΛ]ᵘᶜ.length) : WickContraction φs.length :=
|
||||
⟨φsΛ.1 ∪ φsucΛ.1.map (Finset.mapEmbedding uncontractedListEmd).toEmbedding, by
|
||||
|
|
@ -60,12 +63,13 @@ def join {φs : List 𝓕.States} (φsΛ : WickContraction φs.length)
|
|||
|
||||
lemma join_congr {φs : List 𝓕.States} {φsΛ : WickContraction φs.length}
|
||||
{φsucΛ : WickContraction [φsΛ]ᵘᶜ.length} {φsΛ' : WickContraction φs.length}
|
||||
(h1 : φsΛ = φsΛ') :
|
||||
join φsΛ φsucΛ = join φsΛ' (congr (by simp [h1]) φsucΛ):= by
|
||||
(h1 : φsΛ = φsΛ') :
|
||||
join φsΛ φsucΛ = join φsΛ' (congr (by simp [h1]) φsucΛ) := by
|
||||
subst h1
|
||||
rfl
|
||||
|
||||
|
||||
/-- Given a contracting pair within `φsΛ` the corresponding contracting pair within
|
||||
`(join φsΛ φsucΛ)`. -/
|
||||
def joinLiftLeft {φs : List 𝓕.States} {φsΛ : WickContraction φs.length}
|
||||
{φsucΛ : WickContraction [φsΛ]ᵘᶜ.length} : φsΛ.1 → (join φsΛ φsucΛ).1 :=
|
||||
fun a => ⟨a, by simp [join]⟩
|
||||
|
|
@ -78,6 +82,8 @@ lemma jointLiftLeft_injective {φs : List 𝓕.States} {φsΛ : WickContraction
|
|||
rw [Subtype.mk_eq_mk] at h
|
||||
refine Subtype.eq h
|
||||
|
||||
/-- Given a contracting pair within `φsucΛ` the corresponding contracting pair within
|
||||
`(join φsΛ φsucΛ)`. -/
|
||||
def joinLiftRight {φs : List 𝓕.States} {φsΛ : WickContraction φs.length}
|
||||
{φsucΛ : WickContraction [φsΛ]ᵘᶜ.length} : φsucΛ.1 → (join φsΛ φsucΛ).1 :=
|
||||
fun a => ⟨a.1.map uncontractedListEmd, by
|
||||
|
|
@ -116,6 +122,8 @@ lemma jointLiftLeft_neq_joinLiftRight {φs : List 𝓕.States} {φsΛ : WickCont
|
|||
rw [h1] at hj
|
||||
simp at hj
|
||||
|
||||
/-- The map from contracted pairs of `φsΛ` and `φsucΛ` to contracted pairs in
|
||||
`(join φsΛ φsucΛ)`. -/
|
||||
def joinLift {φs : List 𝓕.States} {φsΛ : WickContraction φs.length}
|
||||
{φsucΛ : WickContraction [φsΛ]ᵘᶜ.length} : φsΛ.1 ⊕ φsucΛ.1 → (join φsΛ φsucΛ).1 := fun a =>
|
||||
match a with
|
||||
|
|
@ -163,7 +171,7 @@ lemma joinLift_bijective {φs : List 𝓕.States} {φsΛ : WickContraction φs.l
|
|||
· exact joinLift_surjective
|
||||
|
||||
lemma prod_join {φs : List 𝓕.States} (φsΛ : WickContraction φs.length)
|
||||
(φsucΛ : WickContraction [φsΛ]ᵘᶜ.length) (f : (join φsΛ φsucΛ).1 → M) [ CommMonoid M]:
|
||||
(φsucΛ : WickContraction [φsΛ]ᵘᶜ.length) (f : (join φsΛ φsucΛ).1 → M) [CommMonoid M]:
|
||||
∏ (a : (join φsΛ φsucΛ).1), f a = (∏ (a : φsΛ.1), f (joinLiftLeft a)) *
|
||||
∏ (a : φsucΛ.1), f (joinLiftRight a) := by
|
||||
let e1 := Equiv.ofBijective (@joinLift _ _ φsΛ φsucΛ) joinLift_bijective
|
||||
|
|
@ -171,9 +179,12 @@ lemma prod_join {φs : List 𝓕.States} (φsΛ : WickContraction φs.length)
|
|||
simp only [Fintype.prod_sum_type, Finset.univ_eq_attach]
|
||||
rfl
|
||||
|
||||
lemma joinLiftLeft_or_joinLiftRight_of_mem_join {φs : List 𝓕.States} (φsΛ : WickContraction φs.length)
|
||||
(φsucΛ : WickContraction [φsΛ]ᵘᶜ.length) {a : Finset (Fin φs.length)} (ha : a ∈ (join φsΛ φsucΛ).1) :
|
||||
(∃ b, a = (joinLiftLeft (φsucΛ := φsucΛ) b).1) ∨ (∃ b, a = (joinLiftRight (φsucΛ := φsucΛ) b).1):= by
|
||||
lemma joinLiftLeft_or_joinLiftRight_of_mem_join {φs : List 𝓕.States}
|
||||
(φsΛ : WickContraction φs.length)
|
||||
(φsucΛ : WickContraction [φsΛ]ᵘᶜ.length) {a : Finset (Fin φs.length)}
|
||||
(ha : a ∈ (join φsΛ φsucΛ).1) :
|
||||
(∃ b, a = (joinLiftLeft (φsucΛ := φsucΛ) b).1) ∨
|
||||
(∃ b, a = (joinLiftRight (φsucΛ := φsucΛ) b).1) := by
|
||||
simp only [join, Finset.le_eq_subset, Finset.mem_union, Finset.mem_map,
|
||||
RelEmbedding.coe_toEmbedding] at ha
|
||||
rcases ha with ha | ⟨a, ha, rfl⟩
|
||||
|
|
@ -184,7 +195,6 @@ lemma joinLiftLeft_or_joinLiftRight_of_mem_join {φs : List 𝓕.States} (φsΛ
|
|||
use ⟨a, ha⟩
|
||||
rfl
|
||||
|
||||
|
||||
@[simp]
|
||||
lemma join_fstFieldOfContract_joinLiftRight {φs : List 𝓕.States} (φsΛ : WickContraction φs.length)
|
||||
(φsucΛ : WickContraction [φsΛ]ᵘᶜ.length) (a : φsucΛ.1) :
|
||||
|
|
@ -227,13 +237,12 @@ lemma join_sndFieldOfContract_joinLift {φs : List 𝓕.States} (φsΛ : WickCon
|
|||
· simp [joinLiftLeft]
|
||||
· exact fstFieldOfContract_lt_sndFieldOfContract φsΛ a
|
||||
|
||||
|
||||
lemma mem_join_right_iff {φs : List 𝓕.States} (φsΛ : WickContraction φs.length)
|
||||
(φsucΛ : WickContraction [φsΛ]ᵘᶜ.length) (a : Finset (Fin [φsΛ]ᵘᶜ.length)) :
|
||||
a ∈ φsucΛ.1 ↔ a.map uncontractedListEmd ∈ (join φsΛ φsucΛ).1 := by
|
||||
a ∈ φsucΛ.1 ↔ a.map uncontractedListEmd ∈ (join φsΛ φsucΛ).1 := by
|
||||
simp only [join, Finset.le_eq_subset, Finset.mem_union, Finset.mem_map,
|
||||
RelEmbedding.coe_toEmbedding]
|
||||
have h1' : ¬ Finset.map uncontractedListEmd a ∈ φsΛ.1 :=
|
||||
have h1' : ¬ Finset.map uncontractedListEmd a ∈ φsΛ.1 :=
|
||||
uncontractedListEmd_finset_not_mem a
|
||||
simp only [h1', false_or]
|
||||
apply Iff.intro
|
||||
|
|
@ -259,7 +268,7 @@ lemma join_card {φs : List 𝓕.States} {φsΛ : WickContraction φs.length}
|
|||
simp only [Finset.mem_map, RelEmbedding.coe_toEmbedding, not_exists, not_and]
|
||||
intro x hx
|
||||
by_contra hn
|
||||
have hdis : Disjoint (Finset.map uncontractedListEmd x) a := by
|
||||
have hdis : Disjoint (Finset.map uncontractedListEmd x) a := by
|
||||
exact uncontractedListEmd_finset_disjoint_left x a ha
|
||||
rw [Finset.mapEmbedding_apply] at hn
|
||||
rw [hn] at hdis
|
||||
|
|
@ -333,7 +342,7 @@ lemma mem_join_uncontracted_of_mem_right_uncontracted {φs : List 𝓕.States}
|
|||
simp only [join, Finset.le_eq_subset, Finset.mem_union, Finset.mem_map,
|
||||
RelEmbedding.coe_toEmbedding] at hp
|
||||
rcases hp with hp | hp
|
||||
· have hi : uncontractedListEmd i ∈ φsΛ.uncontracted := by
|
||||
· have hi : uncontractedListEmd i ∈ φsΛ.uncontracted := by
|
||||
exact uncontractedListEmd_mem_uncontracted i
|
||||
rw [mem_uncontracted_iff_not_contracted] at hi
|
||||
exact hi p hp
|
||||
|
|
@ -353,7 +362,6 @@ lemma exists_mem_left_uncontracted_of_mem_join_uncontracted {φs : List 𝓕.Sta
|
|||
simp only [join, Finset.le_eq_subset, Finset.mem_union, Finset.mem_map,
|
||||
RelEmbedding.coe_toEmbedding] at ha
|
||||
intro p hp
|
||||
have hp' := ha p
|
||||
simp_all
|
||||
|
||||
lemma exists_mem_right_uncontracted_of_mem_join_uncontracted {φs : List 𝓕.States}
|
||||
|
|
@ -399,12 +407,12 @@ lemma join_uncontractedList {φs : List 𝓕.States} (φsΛ : WickContraction φ
|
|||
lemma join_uncontractedList_get {φs : List 𝓕.States} (φsΛ : WickContraction φs.length)
|
||||
(φsucΛ : WickContraction [φsΛ]ᵘᶜ.length) :
|
||||
((join φsΛ φsucΛ).uncontractedList).get =
|
||||
φsΛ.uncontractedListEmd ∘ (φsucΛ.uncontractedList).get ∘ (
|
||||
Fin.cast (by rw [join_uncontractedList]; simp) ):= by
|
||||
have h1 {n : ℕ} (l1 l2 : List (Fin n)) (h : l1 = l2) : l1.get = l2.get ∘ Fin.cast (by rw [h]):= by
|
||||
φsΛ.uncontractedListEmd ∘ (φsucΛ.uncontractedList).get ∘
|
||||
(Fin.cast (by rw [join_uncontractedList]; simp)) := by
|
||||
have h1 {n : ℕ} (l1 l2 : List (Fin n)) (h : l1 = l2) :
|
||||
l1.get = l2.get ∘ Fin.cast (by rw [h]) := by
|
||||
subst h
|
||||
rfl
|
||||
have hl := h1 _ _ (join_uncontractedList φsΛ φsucΛ)
|
||||
conv_lhs => rw [h1 _ _ (join_uncontractedList φsΛ φsucΛ)]
|
||||
ext i
|
||||
simp
|
||||
|
|
@ -420,10 +428,11 @@ lemma join_uncontractedListGet {φs : List 𝓕.States} (φsΛ : WickContraction
|
|||
Function.Embedding.coe_subtype]
|
||||
rfl
|
||||
|
||||
lemma join_uncontractedListEmb {φs : List 𝓕.States} (φsΛ : WickContraction φs.length)
|
||||
lemma join_uncontractedListEmb {φs : List 𝓕.States} (φsΛ : WickContraction φs.length)
|
||||
(φsucΛ : WickContraction [φsΛ]ᵘᶜ.length) :
|
||||
(join φsΛ φsucΛ).uncontractedListEmd =
|
||||
((finCongr (congrArg List.length (join_uncontractedListGet _ _))).toEmbedding.trans φsucΛ.uncontractedListEmd).trans φsΛ.uncontractedListEmd := by
|
||||
((finCongr (congrArg List.length (join_uncontractedListGet _ _))).toEmbedding.trans
|
||||
φsucΛ.uncontractedListEmd).trans φsΛ.uncontractedListEmd := by
|
||||
refine Function.Embedding.ext_iff.mpr (congrFun ?_)
|
||||
change uncontractedListEmd.toFun = _
|
||||
rw [uncontractedListEmd_toFun_eq_get]
|
||||
|
|
@ -447,8 +456,8 @@ lemma join_assoc {φs : List 𝓕.States} (φsΛ : WickContraction φs.length)
|
|||
use a
|
||||
simp [ha']
|
||||
· obtain ⟨a, ha', rfl⟩ := h
|
||||
let a' := congrLift (congrArg List.length (join_uncontractedListGet _ _)) ⟨a, ha'⟩
|
||||
let a'' := joinLiftRight a'
|
||||
let a' := congrLift (congrArg List.length (join_uncontractedListGet _ _)) ⟨a, ha'⟩
|
||||
let a'' := joinLiftRight a'
|
||||
use a''
|
||||
apply And.intro
|
||||
· right
|
||||
|
|
@ -481,39 +490,43 @@ lemma join_assoc {φs : List 𝓕.States} (φsΛ : WickContraction φs.length)
|
|||
change Finset.map (Equiv.refl _).toEmbedding a = _
|
||||
simp only [Equiv.refl_toEmbedding, Finset.map_refl]
|
||||
|
||||
@[simp]
|
||||
lemma join_getDual?_apply_uncontractedListEmb_eq_none_iff {φs : List 𝓕.States} (φsΛ : WickContraction φs.length)
|
||||
lemma join_getDual?_apply_uncontractedListEmb_eq_none_iff {φs : List 𝓕.States}
|
||||
(φsΛ : WickContraction φs.length)
|
||||
(φsucΛ : WickContraction [φsΛ]ᵘᶜ.length) (i : Fin [φsΛ]ᵘᶜ.length) :
|
||||
(join φsΛ φsucΛ).getDual? (uncontractedListEmd i) = none
|
||||
(join φsΛ φsucΛ).getDual? (uncontractedListEmd i) = none
|
||||
↔ φsucΛ.getDual? i = none := by
|
||||
rw [getDual?_eq_none_iff_mem_uncontracted, getDual?_eq_none_iff_mem_uncontracted]
|
||||
apply Iff.intro
|
||||
· intro h
|
||||
obtain ⟨a, ha', ha⟩ := exists_mem_right_uncontracted_of_mem_join_uncontracted _ _ (uncontractedListEmd i) h
|
||||
obtain ⟨a, ha', ha⟩ := exists_mem_right_uncontracted_of_mem_join_uncontracted _ _
|
||||
(uncontractedListEmd i) h
|
||||
simp only [EmbeddingLike.apply_eq_iff_eq] at ha'
|
||||
subst ha'
|
||||
exact ha
|
||||
· intro h
|
||||
exact mem_join_uncontracted_of_mem_right_uncontracted φsΛ φsucΛ i h
|
||||
|
||||
@[simp]
|
||||
lemma join_getDual?_apply_uncontractedListEmb_isSome_iff {φs : List 𝓕.States} (φsΛ : WickContraction φs.length)
|
||||
lemma join_getDual?_apply_uncontractedListEmb_isSome_iff {φs : List 𝓕.States}
|
||||
(φsΛ : WickContraction φs.length)
|
||||
(φsucΛ : WickContraction [φsΛ]ᵘᶜ.length) (i : Fin [φsΛ]ᵘᶜ.length) :
|
||||
((join φsΛ φsucΛ).getDual? (uncontractedListEmd i)).isSome
|
||||
((join φsΛ φsucΛ).getDual? (uncontractedListEmd i)).isSome
|
||||
↔ (φsucΛ.getDual? i).isSome := by
|
||||
rw [← Decidable.not_iff_not]
|
||||
simp
|
||||
simp [join_getDual?_apply_uncontractedListEmb_eq_none_iff]
|
||||
|
||||
lemma join_getDual?_apply_uncontractedListEmb_some {φs : List 𝓕.States} (φsΛ : WickContraction φs.length)
|
||||
lemma join_getDual?_apply_uncontractedListEmb_some {φs : List 𝓕.States}
|
||||
(φsΛ : WickContraction φs.length)
|
||||
(φsucΛ : WickContraction [φsΛ]ᵘᶜ.length) (i : Fin [φsΛ]ᵘᶜ.length)
|
||||
(hi :((join φsΛ φsucΛ).getDual? (uncontractedListEmd i)).isSome) :
|
||||
((join φsΛ φsucΛ).getDual? (uncontractedListEmd i)) =
|
||||
some (uncontractedListEmd ((φsucΛ.getDual? i).get (by simpa using hi))) := by
|
||||
(hi :((join φsΛ φsucΛ).getDual? (uncontractedListEmd i)).isSome) :
|
||||
((join φsΛ φsucΛ).getDual? (uncontractedListEmd i)) =
|
||||
some (uncontractedListEmd ((φsucΛ.getDual? i).get (by
|
||||
simpa [join_getDual?_apply_uncontractedListEmb_isSome_iff]using hi))) := by
|
||||
rw [getDual?_eq_some_iff_mem]
|
||||
simp only [join, Finset.le_eq_subset, Finset.mem_union, Finset.mem_map,
|
||||
RelEmbedding.coe_toEmbedding]
|
||||
right
|
||||
use {i, (φsucΛ.getDual? i).get (by simpa using hi)}
|
||||
use {i, (φsucΛ.getDual? i).get (by
|
||||
simpa [join_getDual?_apply_uncontractedListEmb_isSome_iff] using hi)}
|
||||
simp only [self_getDual?_get_mem, true_and]
|
||||
rw [Finset.mapEmbedding_apply]
|
||||
simp
|
||||
|
|
@ -521,10 +534,11 @@ lemma join_getDual?_apply_uncontractedListEmb_some {φs : List 𝓕.States} (φs
|
|||
@[simp]
|
||||
lemma join_getDual?_apply_uncontractedListEmb {φs : List 𝓕.States} (φsΛ : WickContraction φs.length)
|
||||
(φsucΛ : WickContraction [φsΛ]ᵘᶜ.length) (i : Fin [φsΛ]ᵘᶜ.length) :
|
||||
((join φsΛ φsucΛ).getDual? (uncontractedListEmd i)) = Option.map uncontractedListEmd (φsucΛ.getDual? i) := by
|
||||
((join φsΛ φsucΛ).getDual? (uncontractedListEmd i)) =
|
||||
Option.map uncontractedListEmd (φsucΛ.getDual? i) := by
|
||||
by_cases h : (φsucΛ.getDual? i).isSome
|
||||
· rw [join_getDual?_apply_uncontractedListEmb_some]
|
||||
have h1 : (φsucΛ.getDual? i) =(φsucΛ.getDual? i).get (by simpa using h) :=
|
||||
have h1 : (φsucΛ.getDual? i) = (φsucΛ.getDual? i).get (by simpa using h) :=
|
||||
Eq.symm (Option.some_get h)
|
||||
conv_rhs => rw [h1]
|
||||
simp only [Option.map_some']
|
||||
|
|
@ -610,8 +624,6 @@ lemma signFinset_right_map_uncontractedListEmd_eq_filter {φs : List 𝓕.States
|
|||
· simp [ha2]
|
||||
· right
|
||||
intro h
|
||||
have h1 : (φsucΛ.getDual? a) = some ((φsucΛ.getDual? a).get h) :=
|
||||
Eq.symm (Option.some_get h)
|
||||
apply lt_of_lt_of_eq (uncontractedListEmd_strictMono (ha2 h))
|
||||
rw [Option.get_map]
|
||||
· exact uncontractedListEmd_mem_uncontracted a
|
||||
|
|
@ -620,7 +632,6 @@ lemma signFinset_right_map_uncontractedListEmd_eq_filter {φs : List 𝓕.States
|
|||
have h2' := uncontractedListEmd_surjective_mem_uncontracted a h.2
|
||||
obtain ⟨a, rfl⟩ := h2'
|
||||
use a
|
||||
have h1 := h.1
|
||||
simp_all only [signFinset, Finset.mem_filter, Finset.mem_univ,
|
||||
join_getDual?_apply_uncontractedListEmb, Option.map_eq_none', Option.isSome_map', true_and,
|
||||
and_true, and_self]
|
||||
|
|
@ -650,10 +661,11 @@ lemma signFinset_right_map_uncontractedListEmd_eq_filter {φs : List 𝓕.States
|
|||
|
||||
lemma sign_right_eq_prod_mul_prod {φs : List 𝓕.States} (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)) (uncontractedListEmd (φsucΛ.sndFieldOfContract a))).filter
|
||||
(fun c => ¬ c ∈ φsΛ.uncontracted)⟩)) *
|
||||
(∏ a, 𝓢(𝓕|>ₛ [φsΛ]ᵘᶜ[φsucΛ.sndFieldOfContract a], 𝓕|>ₛ ⟨φs.get ,
|
||||
φsucΛ.sign = (∏ a, 𝓢(𝓕|>ₛ [φsΛ]ᵘᶜ[φsucΛ.sndFieldOfContract a], 𝓕|>ₛ ⟨φs.get,
|
||||
((join φsΛ φsucΛ).signFinset (uncontractedListEmd (φsucΛ.fstFieldOfContract a))
|
||||
(uncontractedListEmd (φsucΛ.sndFieldOfContract a))).filter
|
||||
(fun c => ¬ c ∈ φsΛ.uncontracted)⟩)) *
|
||||
(∏ a, 𝓢(𝓕|>ₛ [φsΛ]ᵘᶜ[φsucΛ.sndFieldOfContract a], 𝓕|>ₛ ⟨φs.get,
|
||||
((join φsΛ φsucΛ).signFinset (uncontractedListEmd (φsucΛ.fstFieldOfContract a))
|
||||
(uncontractedListEmd (φsucΛ.sndFieldOfContract a)))⟩)) := by
|
||||
rw [← Finset.prod_mul_distrib, sign]
|
||||
|
|
@ -666,7 +678,7 @@ lemma sign_right_eq_prod_mul_prod {φs : List 𝓕.States} (φsΛ : WickContract
|
|||
|
||||
lemma join_singleton_signFinset_eq_filter {φs : List 𝓕.States}
|
||||
{i j : Fin φs.length} (h : i < j)
|
||||
(φsucΛ : WickContraction [singleton h]ᵘᶜ.length) :
|
||||
(φsucΛ : WickContraction [singleton h]ᵘᶜ.length) :
|
||||
(join (singleton h) φsucΛ).signFinset i j =
|
||||
((singleton h).signFinset i j).filter (fun c => ¬
|
||||
(((join (singleton h) φsucΛ).getDual? c).isSome ∧
|
||||
|
|
@ -676,7 +688,7 @@ lemma join_singleton_signFinset_eq_filter {φs : List 𝓕.States}
|
|||
simp only [signFinset, Finset.mem_filter, Finset.mem_univ, true_and, not_and, not_forall, not_lt,
|
||||
and_assoc, and_congr_right_iff]
|
||||
intro h1 h2
|
||||
have h1 : (singleton h).getDual? a = none := by
|
||||
have h1 : (singleton h).getDual? a = none := by
|
||||
rw [singleton_getDual?_eq_none_iff_neq]
|
||||
omega
|
||||
simp only [h1, Option.isSome_none, Bool.false_eq_true, IsEmpty.forall_iff, or_self, true_and]
|
||||
|
|
@ -684,14 +696,14 @@ lemma join_singleton_signFinset_eq_filter {φs : List 𝓕.States}
|
|||
· intro h1 h2
|
||||
rcases h1 with h1 | h1
|
||||
· simp only [h1, Option.isSome_none, Bool.false_eq_true, IsEmpty.exists_iff]
|
||||
have h2' : ¬ (((singleton h).join φsucΛ).getDual? a).isSome := by
|
||||
have h2' : ¬ (((singleton h).join φsucΛ).getDual? a).isSome := by
|
||||
exact Option.not_isSome_iff_eq_none.mpr h1
|
||||
exact h2' h2
|
||||
use h2
|
||||
have h1 := h1 h2
|
||||
omega
|
||||
· intro h2
|
||||
by_cases h2' : (((singleton h).join φsucΛ).getDual? a).isSome = true
|
||||
by_cases h2' : (((singleton h).join φsucΛ).getDual? a).isSome = true
|
||||
· have h2 := h2 h2'
|
||||
obtain ⟨hb, h2⟩ := h2
|
||||
right
|
||||
|
|
@ -706,10 +718,9 @@ 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}
|
||||
{i j : Fin φs.length} (h : i < j)
|
||||
(φsucΛ : WickContraction [singleton h]ᵘᶜ.length) :
|
||||
(φsucΛ : WickContraction [singleton h]ᵘᶜ.length) :
|
||||
(𝓕 |>ₛ ⟨φs.get, (singleton h).signFinset i j⟩)
|
||||
= (𝓕 |>ₛ ⟨φs.get, (join (singleton h) φsucΛ).signFinset i j⟩) *
|
||||
(𝓕 |>ₛ ⟨φs.get, ((singleton h).signFinset i j).filter (fun c =>
|
||||
|
|
@ -722,18 +733,21 @@ lemma join_singleton_left_signFinset_eq_filter {φs : List 𝓕.States}
|
|||
rw [mul_comm]
|
||||
exact (ofFinset_filter_mul_neg 𝓕.statesStatistic _ _ _).symm
|
||||
|
||||
/-- The difference in sign between `φsucΛ.sign` and the direct contribution of `φsucΛ` to
|
||||
`(join (singleton h) φsucΛ)`. -/
|
||||
def joinSignRightExtra {φs : List 𝓕.States}
|
||||
{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 ,
|
||||
(φsucΛ : WickContraction [singleton h]ᵘᶜ.length) : ℂ :=
|
||||
∏ a, 𝓢(𝓕|>ₛ [singleton h]ᵘᶜ[φsucΛ.sndFieldOfContract a], 𝓕|>ₛ ⟨φs.get,
|
||||
((join (singleton h) φsucΛ).signFinset (uncontractedListEmd (φsucΛ.fstFieldOfContract a))
|
||||
(uncontractedListEmd (φsucΛ.sndFieldOfContract a))).filter
|
||||
(fun c => ¬ c ∈ (singleton h).uncontracted)⟩)
|
||||
|
||||
(fun c => ¬ c ∈ (singleton h).uncontracted)⟩)
|
||||
|
||||
/-- 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}
|
||||
{i j : Fin φs.length} (h : i < j)
|
||||
(φsucΛ : WickContraction [singleton h]ᵘᶜ.length) : ℂ :=
|
||||
(φsucΛ : WickContraction [singleton h]ᵘᶜ.length) : ℂ :=
|
||||
𝓢(𝓕 |>ₛ φs[j], (𝓕 |>ₛ ⟨φs.get, ((singleton h).signFinset i j).filter (fun c =>
|
||||
(((join (singleton h) φsucΛ).getDual? c).isSome ∧
|
||||
((h1 : ((join (singleton h) φsucΛ).getDual? c).isSome) →
|
||||
|
|
@ -741,30 +755,29 @@ def joinSignLeftExtra {φs : List 𝓕.States}
|
|||
|
||||
lemma join_singleton_sign_left {φs : List 𝓕.States}
|
||||
{i j : Fin φs.length} (h : i < j)
|
||||
(φsucΛ : WickContraction [singleton h]ᵘᶜ.length) :
|
||||
(singleton h).sign = 𝓢(𝓕 |>ₛ φs[j], (𝓕 |>ₛ ⟨φs.get, (join (singleton h) φsucΛ).signFinset i j⟩)) *
|
||||
(joinSignLeftExtra h φsucΛ) := by
|
||||
(φsucΛ : WickContraction [singleton h]ᵘᶜ.length) :
|
||||
(singleton h).sign = 𝓢(𝓕 |>ₛ φs[j],
|
||||
(𝓕 |>ₛ ⟨φs.get, (join (singleton h) φsucΛ).signFinset i j⟩)) * (joinSignLeftExtra h φsucΛ) := by
|
||||
rw [singleton_sign_expand]
|
||||
rw [join_singleton_left_signFinset_eq_filter h φsucΛ]
|
||||
rw [map_mul]
|
||||
rfl
|
||||
|
||||
|
||||
lemma join_singleton_sign_right {φs : List 𝓕.States}
|
||||
{i j : Fin φs.length} (h : i < j)
|
||||
(φsucΛ : WickContraction [singleton h]ᵘᶜ.length) :
|
||||
(φsucΛ : WickContraction [singleton h]ᵘᶜ.length) :
|
||||
φsucΛ.sign =
|
||||
(joinSignRightExtra h φsucΛ) *
|
||||
(∏ a, 𝓢(𝓕|>ₛ [singleton h]ᵘᶜ[φsucΛ.sndFieldOfContract a], 𝓕|>ₛ ⟨φs.get ,
|
||||
(∏ a, 𝓢(𝓕|>ₛ [singleton h]ᵘᶜ[φsucΛ.sndFieldOfContract a], 𝓕|>ₛ ⟨φs.get,
|
||||
((join (singleton h) φsucΛ).signFinset (uncontractedListEmd (φsucΛ.fstFieldOfContract a))
|
||||
(uncontractedListEmd (φsucΛ.sndFieldOfContract a)))⟩)) := by
|
||||
(uncontractedListEmd (φsucΛ.sndFieldOfContract a)))⟩)) := by
|
||||
rw [sign_right_eq_prod_mul_prod]
|
||||
rfl
|
||||
|
||||
@[simp]
|
||||
lemma join_singleton_getDual?_left {φs : List 𝓕.States}
|
||||
{i j : Fin φs.length} (h : i < j)
|
||||
(φsucΛ : WickContraction [singleton h]ᵘᶜ.length) :
|
||||
(φsucΛ : WickContraction [singleton h]ᵘᶜ.length) :
|
||||
(join (singleton h) φsucΛ).getDual? i = some j := by
|
||||
rw [@getDual?_eq_some_iff_mem]
|
||||
simp [singleton, join]
|
||||
|
|
@ -772,7 +785,7 @@ lemma join_singleton_getDual?_left {φs : List 𝓕.States}
|
|||
@[simp]
|
||||
lemma join_singleton_getDual?_right {φs : List 𝓕.States}
|
||||
{i j : Fin φs.length} (h : i < j)
|
||||
(φsucΛ : WickContraction [singleton h]ᵘᶜ.length) :
|
||||
(φsucΛ : WickContraction [singleton h]ᵘᶜ.length) :
|
||||
(join (singleton h) φsucΛ).getDual? j = some i := by
|
||||
rw [@getDual?_eq_some_iff_mem]
|
||||
simp only [join, singleton, Finset.le_eq_subset, Finset.mem_union, Finset.mem_singleton,
|
||||
|
|
@ -780,17 +793,16 @@ lemma join_singleton_getDual?_right {φs : List 𝓕.States}
|
|||
left
|
||||
exact Finset.pair_comm j i
|
||||
|
||||
|
||||
lemma joinSignRightExtra_eq_i_j_finset_eq_if {φs : List 𝓕.States}
|
||||
{i j : Fin φs.length} (h : i < j)
|
||||
(φsucΛ : WickContraction [singleton h]ᵘᶜ.length) :
|
||||
(φsucΛ : WickContraction [singleton h]ᵘᶜ.length) :
|
||||
joinSignRightExtra h φsucΛ = ∏ a,
|
||||
𝓢((𝓕|>ₛ [singleton h]ᵘᶜ[φsucΛ.sndFieldOfContract a]),
|
||||
𝓕 |>ₛ ⟨φs.get, (if uncontractedListEmd (φsucΛ.fstFieldOfContract a) < j ∧
|
||||
j < uncontractedListEmd (φsucΛ.sndFieldOfContract a) ∧
|
||||
uncontractedListEmd (φsucΛ.fstFieldOfContract a) < i then {j} else ∅)
|
||||
uncontractedListEmd (φsucΛ.fstFieldOfContract a) < i then {j} else ∅)
|
||||
∪ (if uncontractedListEmd (φsucΛ.fstFieldOfContract a) < i ∧
|
||||
i < uncontractedListEmd (φsucΛ.sndFieldOfContract a) then {i} else ∅)⟩) := by
|
||||
i < uncontractedListEmd (φsucΛ.sndFieldOfContract a) then {i} else ∅)⟩) := by
|
||||
rw [joinSignRightExtra]
|
||||
congr
|
||||
funext a
|
||||
|
|
@ -831,7 +843,7 @@ lemma joinSignRightExtra_eq_i_j_finset_eq_if {φs : List 𝓕.States}
|
|||
· have hi1 : i < uncontractedListEmd (φsucΛ.sndFieldOfContract a) := by
|
||||
omega
|
||||
simp only [hj1, true_and, hi1, and_true]
|
||||
by_cases hi2 : ¬ uncontractedListEmd (φsucΛ.fstFieldOfContract a) < i
|
||||
by_cases hi2 : ¬ uncontractedListEmd (φsucΛ.fstFieldOfContract a) < i
|
||||
· simp only [hi2, and_false, ↓reduceIte, Finset.not_mem_empty, or_self, iff_false, not_and,
|
||||
not_or, not_forall, not_lt]
|
||||
by_cases hj3 : ¬ j < uncontractedListEmd (φsucΛ.sndFieldOfContract a)
|
||||
|
|
@ -873,7 +885,6 @@ 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}
|
||||
{i j : Fin φs.length} (h : i < j) (hs : (𝓕 |>ₛ φs[i]) = (𝓕 |>ₛ φs[j]))
|
||||
(φsucΛ : WickContraction [singleton h]ᵘᶜ.length) :
|
||||
|
|
@ -882,7 +893,8 @@ lemma joinSignLeftExtra_eq_joinSignRightExtra {φs : List 𝓕.States}
|
|||
rw [joinSignLeftExtra]
|
||||
rw [ofFinset_eq_prod]
|
||||
rw [map_prod]
|
||||
let e2 : Fin φs.length ≃ {x // (((singleton h).join φsucΛ).getDual? x).isSome} ⊕ {x // ¬ (((singleton h).join φsucΛ).getDual? x).isSome} := by
|
||||
let e2 : Fin φs.length ≃ {x // (((singleton h).join φsucΛ).getDual? x).isSome} ⊕
|
||||
{x // ¬ (((singleton h).join φsucΛ).getDual? x).isSome} := by
|
||||
exact (Equiv.sumCompl fun a => (((singleton h).join φsucΛ).getDual? a).isSome = true).symm
|
||||
rw [← e2.symm.prod_comp]
|
||||
simp only [Fin.getElem_fin, Fintype.prod_sum_type, instCommGroup]
|
||||
|
|
@ -913,11 +925,9 @@ lemma joinSignLeftExtra_eq_joinSignRightExtra {φs : List 𝓕.States}
|
|||
rw [joinSignRightExtra_eq_i_j_finset_eq_if]
|
||||
congr
|
||||
funext a
|
||||
have hjneqfst := singleton_uncontractedEmd_neq_right h (φsucΛ.fstFieldOfContract a)
|
||||
have hjneqsnd := singleton_uncontractedEmd_neq_right h (φsucΛ.sndFieldOfContract a)
|
||||
have hineqfst := singleton_uncontractedEmd_neq_left h (φsucΛ.fstFieldOfContract a)
|
||||
have hineqsnd := singleton_uncontractedEmd_neq_left h (φsucΛ.sndFieldOfContract a)
|
||||
have hl : uncontractedListEmd (φsucΛ.fstFieldOfContract a) < uncontractedListEmd (φsucΛ.sndFieldOfContract a) := by
|
||||
have hl : uncontractedListEmd (φsucΛ.fstFieldOfContract a) <
|
||||
uncontractedListEmd (φsucΛ.sndFieldOfContract a) := by
|
||||
apply uncontractedListEmd_strictMono
|
||||
exact fstFieldOfContract_lt_sndFieldOfContract φsucΛ a
|
||||
by_cases hj1 : ¬ uncontractedListEmd (φsucΛ.fstFieldOfContract a) < j
|
||||
|
|
@ -926,7 +936,7 @@ lemma joinSignLeftExtra_eq_joinSignRightExtra {φs : List 𝓕.States}
|
|||
· have hj1 : uncontractedListEmd (φsucΛ.fstFieldOfContract a) < j := by omega
|
||||
simp only [hj1, and_true, instCommGroup, Fin.getElem_fin, true_and]
|
||||
by_cases hi2 : ¬ i < uncontractedListEmd (φsucΛ.sndFieldOfContract a)
|
||||
· have hi1 : ¬ i < uncontractedListEmd (φsucΛ.fstFieldOfContract a) := by omega
|
||||
· have hi1 : ¬ i < uncontractedListEmd (φsucΛ.fstFieldOfContract a) := by omega
|
||||
have hj2 : ¬ j < uncontractedListEmd (φsucΛ.sndFieldOfContract a) := by omega
|
||||
simp [hi2, hj2, hi1]
|
||||
· have hi2 : i < uncontractedListEmd (φsucΛ.sndFieldOfContract a) := by omega
|
||||
|
|
@ -934,7 +944,7 @@ lemma joinSignLeftExtra_eq_joinSignRightExtra {φs : List 𝓕.States}
|
|||
simp only [hi2n, and_false, ↓reduceIte, map_one, hi2, true_and, one_mul, and_true]
|
||||
by_cases hj2 : ¬ j < uncontractedListEmd (φsucΛ.sndFieldOfContract a)
|
||||
· simp only [hj2, false_and, ↓reduceIte, Finset.empty_union]
|
||||
have hj2 : uncontractedListEmd (φsucΛ.sndFieldOfContract a) < j:= by omega
|
||||
have hj2 : uncontractedListEmd (φsucΛ.sndFieldOfContract a) < j:= by omega
|
||||
simp only [hj2, true_and]
|
||||
by_cases hi1 : ¬ uncontractedListEmd (φsucΛ.fstFieldOfContract a) < i
|
||||
· simp [hi1]
|
||||
|
|
@ -948,7 +958,7 @@ lemma joinSignLeftExtra_eq_joinSignRightExtra {φs : List 𝓕.States}
|
|||
· simp [hi1]
|
||||
· have hi1 : uncontractedListEmd (φsucΛ.fstFieldOfContract a) < i := by omega
|
||||
simp only [hi1, and_true, ↓reduceIte]
|
||||
have hj2 : ¬ uncontractedListEmd (φsucΛ.sndFieldOfContract a) < j := by omega
|
||||
have hj2 : ¬ uncontractedListEmd (φsucΛ.sndFieldOfContract a) < j := by omega
|
||||
simp only [hj2, ↓reduceIte, map_one]
|
||||
rw [← ofFinset_union_disjoint]
|
||||
simp only [instCommGroup, ofFinset_singleton, List.get_eq_getElem, hs]
|
||||
|
|
@ -993,15 +1003,16 @@ lemma exists_join_singleton_of_card_ge_zero {φs : List 𝓕.States} (φsΛ : Wi
|
|||
use φsΛ.sndFieldOfContract ⟨a, ha⟩
|
||||
use φsΛ.fstFieldOfContract_lt_sndFieldOfContract ⟨a, ha⟩
|
||||
let φsucΛ :
|
||||
WickContraction [singleton (φsΛ.fstFieldOfContract_lt_sndFieldOfContract ⟨a, ha⟩)]ᵘᶜ.length :=
|
||||
congr (by simp [← subContraction_singleton_eq_singleton]) (φsΛ.quotContraction {a} (by simpa using ha))
|
||||
WickContraction [singleton (φsΛ.fstFieldOfContract_lt_sndFieldOfContract ⟨a, ha⟩)]ᵘᶜ.length :=
|
||||
congr (by simp [← subContraction_singleton_eq_singleton])
|
||||
(φsΛ.quotContraction {a} (by simpa using ha))
|
||||
use φsucΛ
|
||||
simp only [Fin.getElem_fin]
|
||||
apply And.intro
|
||||
· have h1 := join_congr (subContraction_singleton_eq_singleton _ ⟨a, ha⟩).symm (φsucΛ := φsucΛ)
|
||||
simp only [id_eq, eq_mpr_eq_cast, h1, congr_trans_apply, congr_refl, φsucΛ]
|
||||
rw [join_sub_quot]
|
||||
· apply And.intro (hc ⟨a, ha⟩)
|
||||
· apply And.intro (hc ⟨a, ha⟩)
|
||||
apply And.intro
|
||||
· simp only [id_eq, eq_mpr_eq_cast, φsucΛ]
|
||||
rw [gradingCompliant_congr (φs' := [(φsΛ.subContraction {a} (by simpa using ha))]ᵘᶜ)]
|
||||
|
|
@ -1024,11 +1035,12 @@ lemma join_sign_induction {φs : List 𝓕.States} (φsΛ : WickContraction φs.
|
|||
apply sign_congr
|
||||
simp
|
||||
| Nat.succ n, hn => by
|
||||
obtain ⟨i, j, hij, φsucΛ', rfl, h1, h2, h3⟩ := exists_join_singleton_of_card_ge_zero φsΛ (by simp [hn]) hc
|
||||
obtain ⟨i, j, hij, φsucΛ', rfl, h1, h2, h3⟩ :=
|
||||
exists_join_singleton_of_card_ge_zero φsΛ (by simp [hn]) hc
|
||||
rw [join_assoc]
|
||||
rw [join_sign_singleton hij h1 ]
|
||||
rw [join_sign_singleton hij h1 ]
|
||||
have hn : φsucΛ'.1.card = n := by
|
||||
rw [join_sign_singleton hij h1]
|
||||
rw [join_sign_singleton hij h1]
|
||||
have hn : φsucΛ'.1.card = n := by
|
||||
omega
|
||||
rw [join_sign_induction φsucΛ' (congr (by simp [join_uncontractedListGet]) φsucΛ) h2
|
||||
n hn]
|
||||
|
|
@ -1058,7 +1070,7 @@ lemma join_not_gradingCompliant_of_left_not_gradingCompliant {φs : List 𝓕.St
|
|||
lemma join_sign_timeContract {φs : List 𝓕.States} (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
|
||||
(φsΛ.sign • φsΛ.timeContract.1) * (φsucΛ.sign • φsucΛ.timeContract.1) := by
|
||||
rw [join_timeContract]
|
||||
by_cases h : φsΛ.GradingCompliant
|
||||
· rw [join_sign _ _ h]
|
||||
|
|
|
|||
Loading…
Add table
Add a link
Reference in a new issue