refactor: Lint
This commit is contained in:
jstoobysmith
parent
006e29fd08
commit
fca3f02eca
16 changed files with 416 additions and 352 deletions
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@ -27,7 +27,8 @@ def join {φs : List 𝓕.States} (φsΛ : WickContraction φs.length)
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(φsucΛ : WickContraction [φsΛ]ᵘᶜ.length) : WickContraction φs.length :=
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⟨φsΛ.1 ∪ φsucΛ.1.map (Finset.mapEmbedding uncontractedListEmd).toEmbedding, by
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intro a ha
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simp at ha
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simp only [Finset.le_eq_subset, Finset.mem_union, Finset.mem_map,
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RelEmbedding.coe_toEmbedding] at ha
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rcases ha with ha | ha
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· exact φsΛ.2.1 a ha
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· obtain ⟨a, ha, rfl⟩ := ha
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@ -35,7 +36,8 @@ def join {φs : List 𝓕.States} (φsΛ : WickContraction φs.length)
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simp only [Finset.card_map]
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exact φsucΛ.2.1 a ha, by
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intro a ha b hb
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simp at ha hb
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simp only [Finset.le_eq_subset, Finset.mem_union, Finset.mem_map,
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RelEmbedding.coe_toEmbedding] at ha hb
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rcases ha with ha | ha <;> rcases hb with hb | hb
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· exact φsΛ.2.2 a ha b hb
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· obtain ⟨b, hb, rfl⟩ := hb
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@ -109,7 +111,7 @@ lemma jointLiftLeft_neq_joinLiftRight {φs : List 𝓕.States} {φsΛ : WickCont
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by_contra hn
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have h1 := jointLiftLeft_disjoint_joinLiftRight a b
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rw [hn] at h1
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simp at h1
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simp only [disjoint_self, Finset.bot_eq_empty] at h1
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have hj := (join φsΛ φsucΛ).2.1 (joinLiftRight b).1 (joinLiftRight b).2
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rw [h1] at hj
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simp at hj
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@ -125,17 +127,17 @@ lemma joinLift_injective {φs : List 𝓕.States} {φsΛ : WickContraction φs.l
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intro a b h
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match a, b with
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| Sum.inl a, Sum.inl b =>
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simp
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simp only [Sum.inl.injEq]
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exact jointLiftLeft_injective h
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| Sum.inr a, Sum.inr b =>
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simp
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simp only [Sum.inr.injEq]
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exact joinLiftRight_injective h
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| Sum.inl a, Sum.inr b =>
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simp [joinLift] at h
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simp only [joinLift] at h
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have h1 := jointLiftLeft_neq_joinLiftRight a b
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simp_all
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| Sum.inr a, Sum.inl b =>
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simp [joinLift] at h
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simp only [joinLift] at h
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have h1 := jointLiftLeft_neq_joinLiftRight b a
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simp_all
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@ -143,13 +145,14 @@ lemma joinLift_surjective {φs : List 𝓕.States} {φsΛ : WickContraction φs.
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{φsucΛ : WickContraction [φsΛ]ᵘᶜ.length} : Function.Surjective (@joinLift _ _ φsΛ φsucΛ) := by
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intro a
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have ha2 := a.2
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simp [- Finset.coe_mem, join] at ha2
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simp only [join, Finset.le_eq_subset, Finset.mem_union, Finset.mem_map,
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RelEmbedding.coe_toEmbedding] at ha2
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rcases ha2 with ha2 | ⟨a2, ha3⟩
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· use Sum.inl ⟨a, ha2⟩
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simp [joinLift, joinLiftLeft]
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· rw [Finset.mapEmbedding_apply] at ha3
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use Sum.inr ⟨a2, ha3.1⟩
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simp [joinLift, joinLiftRight]
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simp only [joinLift, joinLiftRight]
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refine Subtype.eq ?_
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exact ha3.2
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@ -171,7 +174,8 @@ lemma prod_join {φs : List 𝓕.States} (φsΛ : WickContraction φs.length)
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lemma joinLiftLeft_or_joinLiftRight_of_mem_join {φs : List 𝓕.States} (φsΛ : WickContraction φs.length)
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(φsucΛ : WickContraction [φsΛ]ᵘᶜ.length) {a : Finset (Fin φs.length)} (ha : a ∈ (join φsΛ φsucΛ).1) :
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(∃ b, a = (joinLiftLeft (φsucΛ := φsucΛ) b).1) ∨ (∃ b, a = (joinLiftRight (φsucΛ := φsucΛ) b).1):= by
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simp [join] at ha
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simp only [join, Finset.le_eq_subset, Finset.mem_union, Finset.mem_map,
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RelEmbedding.coe_toEmbedding] at ha
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rcases ha with ha | ⟨a, ha, rfl⟩
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· left
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use ⟨a, ha⟩
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@ -227,38 +231,39 @@ lemma join_sndFieldOfContract_joinLift {φs : List 𝓕.States} (φsΛ : WickCon
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lemma mem_join_right_iff {φs : List 𝓕.States} (φsΛ : WickContraction φs.length)
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(φsucΛ : WickContraction [φsΛ]ᵘᶜ.length) (a : Finset (Fin [φsΛ]ᵘᶜ.length)) :
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a ∈ φsucΛ.1 ↔ a.map uncontractedListEmd ∈ (join φsΛ φsucΛ).1 := by
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simp [join]
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simp only [join, Finset.le_eq_subset, Finset.mem_union, Finset.mem_map,
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RelEmbedding.coe_toEmbedding]
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have h1' : ¬ Finset.map uncontractedListEmd a ∈ φsΛ.1 :=
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uncontractedListEmd_finset_not_mem a
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simp [h1']
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simp only [h1', false_or]
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apply Iff.intro
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· intro h
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use a
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simp [h]
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simp only [h, true_and]
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rw [Finset.mapEmbedding_apply]
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· intro h
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obtain ⟨a, ha, h2⟩ := h
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rw [Finset.mapEmbedding_apply] at h2
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simp at h2
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simp only [Finset.map_inj] at h2
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subst h2
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exact ha
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lemma join_card {φs : List 𝓕.States} {φsΛ : WickContraction φs.length}
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{φsucΛ : WickContraction [φsΛ]ᵘᶜ.length} :
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(join φsΛ φsucΛ).1.card = φsΛ.1.card + φsucΛ.1.card := by
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simp [join]
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simp only [join, Finset.le_eq_subset]
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rw [Finset.card_union_of_disjoint]
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simp
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simp only [Finset.card_map]
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rw [@Finset.disjoint_left]
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intro a ha
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simp
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simp only [Finset.mem_map, RelEmbedding.coe_toEmbedding, not_exists, not_and]
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intro x hx
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by_contra hn
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have hdis : Disjoint (Finset.map uncontractedListEmd x) a := by
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exact uncontractedListEmd_finset_disjoint_left x a ha
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rw [Finset.mapEmbedding_apply] at hn
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rw [hn] at hdis
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simp at hdis
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simp only [disjoint_self, Finset.bot_eq_empty] at hdis
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have hcard := φsΛ.2.1 a ha
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simp_all
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@ -266,13 +271,13 @@ lemma join_card {φs : List 𝓕.States} {φsΛ : WickContraction φs.length}
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lemma empty_join {φs : List 𝓕.States} (φsΛ : WickContraction [empty (n := φs.length)]ᵘᶜ.length) :
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join empty φsΛ = congr (by simp) φsΛ := by
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apply Subtype.ext
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simp [join]
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simp only [join, Finset.le_eq_subset, uncontractedListEmd_empty]
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ext a
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conv_lhs =>
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left
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left
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rw [empty]
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simp
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simp only [Finset.empty_union, Finset.mem_map, RelEmbedding.coe_toEmbedding]
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rw [mem_congr_iff]
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apply Iff.intro
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· intro h
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@ -285,7 +290,7 @@ lemma empty_join {φs : List 𝓕.States} (φsΛ : WickContraction [empty (n :=
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simp
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· intro h
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use Finset.map (finCongr (by simp)).toEmbedding a
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simp [h]
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simp only [h, true_and]
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trans Finset.map (Equiv.refl _).toEmbedding a
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rw [Finset.mapEmbedding_apply, Finset.map_map]
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rfl
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@ -325,7 +330,8 @@ lemma mem_join_uncontracted_of_mem_right_uncontracted {φs : List 𝓕.States}
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uncontractedListEmd i ∈ (join φsΛ φsucΛ).uncontracted := by
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rw [mem_uncontracted_iff_not_contracted]
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intro p hp
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simp [join] at hp
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simp only [join, Finset.le_eq_subset, Finset.mem_union, Finset.mem_map,
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RelEmbedding.coe_toEmbedding] at hp
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rcases hp with hp | hp
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· have hi : uncontractedListEmd i ∈ φsΛ.uncontracted := by
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exact uncontractedListEmd_mem_uncontracted i
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@ -333,7 +339,7 @@ lemma mem_join_uncontracted_of_mem_right_uncontracted {φs : List 𝓕.States}
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exact hi p hp
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· obtain ⟨p, hp, rfl⟩ := hp
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rw [Finset.mapEmbedding_apply]
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simp
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simp only [Finset.mem_map']
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rw [mem_uncontracted_iff_not_contracted] at ha
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exact ha p hp
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@ -344,7 +350,8 @@ lemma exists_mem_left_uncontracted_of_mem_join_uncontracted {φs : List 𝓕.Sta
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i ∈ φsΛ.uncontracted := by
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rw [@mem_uncontracted_iff_not_contracted]
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rw [@mem_uncontracted_iff_not_contracted] at ha
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simp [join] at ha
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simp only [join, Finset.le_eq_subset, Finset.mem_union, Finset.mem_map,
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RelEmbedding.coe_toEmbedding] at ha
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intro p hp
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have hp' := ha p
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simp_all
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@ -357,15 +364,16 @@ lemma exists_mem_right_uncontracted_of_mem_join_uncontracted {φs : List 𝓕.St
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have hi' := exists_mem_left_uncontracted_of_mem_join_uncontracted _ _ i hi
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obtain ⟨j, rfl⟩ := uncontractedListEmd_surjective_mem_uncontracted i hi'
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use j
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simp
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simp only [true_and]
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rw [mem_uncontracted_iff_not_contracted] at hi
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rw [mem_uncontracted_iff_not_contracted]
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intro p hp
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have hip := hi (p.map uncontractedListEmd) (by
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simp [join]
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simp only [join, Finset.le_eq_subset, Finset.mem_union, Finset.mem_map,
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RelEmbedding.coe_toEmbedding]
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right
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use p
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simp [hp]
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simp only [hp, true_and]
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rw [Finset.mapEmbedding_apply])
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simpa using hip
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@ -377,7 +385,7 @@ lemma join_uncontractedList {φs : List 𝓕.States} (φsΛ : WickContraction φ
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rw [fin_finset_sort_map_monotone]
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congr
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ext a
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simp
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simp only [Finset.mem_map]
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apply Iff.intro
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· intro h
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obtain ⟨a, rfl, ha⟩ := exists_mem_right_uncontracted_of_mem_join_uncontracted _ _ a h
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@ -404,9 +412,12 @@ lemma join_uncontractedList_get {φs : List 𝓕.States} (φsΛ : WickContractio
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lemma join_uncontractedListGet {φs : List 𝓕.States} (φsΛ : WickContraction φs.length)
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(φsucΛ : WickContraction [φsΛ]ᵘᶜ.length) :
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(join φsΛ φsucΛ).uncontractedListGet = φsucΛ.uncontractedListGet := by
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simp [uncontractedListGet, join_uncontractedList]
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simp only [uncontractedListGet, join_uncontractedList, List.map_map, List.map_inj_left,
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Function.comp_apply, List.get_eq_getElem, List.getElem_map]
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intro a ha
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simp [uncontractedListEmd, uncontractedIndexEquiv]
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simp only [uncontractedListEmd, uncontractedIndexEquiv, List.get_eq_getElem,
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Equiv.trans_toEmbedding, Function.Embedding.trans_apply, Equiv.coe_toEmbedding, Equiv.coe_fn_mk,
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Function.Embedding.coe_subtype]
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rfl
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lemma join_uncontractedListEmb {φs : List 𝓕.States} (φsΛ : WickContraction φs.length)
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@ -427,7 +438,8 @@ lemma join_assoc {φs : List 𝓕.States} (φsΛ : WickContraction φs.length)
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ext a
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by_cases ha : a ∈ φsΛ.1
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· simp [ha, join]
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simp [ha, join]
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simp only [join, Finset.le_eq_subset, Finset.union_assoc, Finset.mem_union, ha, Finset.mem_map,
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RelEmbedding.coe_toEmbedding, false_or]
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apply Iff.intro
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· intro h
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rcases h with h | h
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@ -459,7 +471,7 @@ lemma join_assoc {φs : List 𝓕.States} (φsΛ : WickContraction φs.length)
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right
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let a' := congrLiftInv _ ⟨a, ha'⟩
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use a'
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simp
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simp only [Finset.coe_mem, true_and]
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simp only [a']
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rw [Finset.mapEmbedding_apply]
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rw [join_uncontractedListEmb]
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@ -478,7 +490,7 @@ lemma join_getDual?_apply_uncontractedListEmb_eq_none_iff {φs : List 𝓕.Stat
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apply Iff.intro
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· intro h
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obtain ⟨a, ha', ha⟩ := exists_mem_right_uncontracted_of_mem_join_uncontracted _ _ (uncontractedListEmd i) h
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simp at ha'
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simp only [EmbeddingLike.apply_eq_iff_eq] at ha'
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subst ha'
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exact ha
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· intro h
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@ -498,10 +510,11 @@ lemma join_getDual?_apply_uncontractedListEmb_some {φs : List 𝓕.States} (φs
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((join φsΛ φsucΛ).getDual? (uncontractedListEmd i)) =
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some (uncontractedListEmd ((φsucΛ.getDual? i).get (by simpa using hi))) := by
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rw [getDual?_eq_some_iff_mem]
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simp [join]
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simp only [join, Finset.le_eq_subset, Finset.mem_union, Finset.mem_map,
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RelEmbedding.coe_toEmbedding]
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right
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use {i, (φsucΛ.getDual? i).get (by simpa using hi)}
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simp
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simp only [self_getDual?_get_mem, true_and]
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rw [Finset.mapEmbedding_apply]
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simp
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@ -514,11 +527,11 @@ lemma join_getDual?_apply_uncontractedListEmb {φs : List 𝓕.States} (φsΛ :
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have h1 : (φsucΛ.getDual? i) =(φsucΛ.getDual? i).get (by simpa using h) :=
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Eq.symm (Option.some_get h)
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conv_rhs => rw [h1]
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simp [- Option.some_get]
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simp only [Option.map_some']
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exact (join_getDual?_apply_uncontractedListEmb_isSome_iff φsΛ φsucΛ i).mpr h
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· simp only [Bool.not_eq_true, Option.not_isSome, Option.isNone_iff_eq_none] at h
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rw [h]
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simp
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simp only [Option.map_none', join_getDual?_apply_uncontractedListEmb_eq_none_iff]
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exact h
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/-!
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@ -535,7 +548,8 @@ lemma join_sub_quot (S : Finset (Finset (Fin φs.length))) (ha : S ⊆ φsΛ.1)
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join (subContraction S ha) (quotContraction S ha) = φsΛ := by
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apply Subtype.ext
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ext a
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simp [join]
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simp only [join, Finset.le_eq_subset, Finset.mem_union, Finset.mem_map,
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RelEmbedding.coe_toEmbedding]
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apply Iff.intro
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· intro h
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rcases h with h | h
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@ -549,7 +563,7 @@ lemma join_sub_quot (S : Finset (Finset (Fin φs.length))) (ha : S ⊆ φsΛ.1)
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· right
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obtain ⟨a, rfl, ha⟩ := h1
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use a
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simp [ha]
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simp only [ha, true_and]
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rw [Finset.mapEmbedding_apply]
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lemma subContraction_card_plus_quotContraction_card_eq (S : Finset (Finset (Fin φs.length)))
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@ -565,10 +579,11 @@ lemma stat_signFinset_right {φs : List 𝓕.States} (φsΛ : WickContraction φ
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(φsucΛ : WickContraction [φsΛ]ᵘᶜ.length) (i j : Fin [φsΛ]ᵘᶜ.length) :
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(𝓕 |>ₛ ⟨[φsΛ]ᵘᶜ.get, φsucΛ.signFinset i j⟩) =
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(𝓕 |>ₛ ⟨φs.get, (φsucΛ.signFinset i j).map uncontractedListEmd⟩) := by
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simp [ofFinset]
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simp only [ofFinset]
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congr 1
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rw [← fin_finset_sort_map_monotone]
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simp
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simp only [List.map_map, List.map_inj_left, Finset.mem_sort, List.get_eq_getElem,
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Function.comp_apply, getElem_uncontractedListEmd, implies_true]
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intro i j h
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exact uncontractedListEmd_strictMono h
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@ -578,18 +593,19 @@ lemma signFinset_right_map_uncontractedListEmd_eq_filter {φs : List 𝓕.States
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((join φsΛ φsucΛ).signFinset (uncontractedListEmd i) (uncontractedListEmd j)).filter
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(fun c => c ∈ φsΛ.uncontracted) := by
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ext a
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simp
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simp only [Finset.mem_map, Finset.mem_filter]
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apply Iff.intro
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· intro h
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obtain ⟨a, ha, rfl⟩ := h
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apply And.intro
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· simp_all [signFinset]
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· simp_all only [signFinset, Finset.mem_filter, Finset.mem_univ, true_and,
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join_getDual?_apply_uncontractedListEmb, Option.map_eq_none', Option.isSome_map']
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apply And.intro
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· exact uncontractedListEmd_strictMono ha.1
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· apply And.intro
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· exact uncontractedListEmd_strictMono ha.2.1
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· have ha2 := ha.2.2
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simp_all
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simp_all only [and_true]
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||||
rcases ha2 with ha2 | ha2
|
||||
· simp [ha2]
|
||||
· right
|
||||
|
|
@ -605,7 +621,9 @@ lemma signFinset_right_map_uncontractedListEmd_eq_filter {φs : List 𝓕.States
|
|||
obtain ⟨a, rfl⟩ := h2'
|
||||
use a
|
||||
have h1 := h.1
|
||||
simp_all [signFinset]
|
||||
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]
|
||||
apply And.intro
|
||||
· have h1 := h.1
|
||||
rw [StrictMono.lt_iff_lt] at h1
|
||||
|
|
@ -617,7 +635,7 @@ lemma signFinset_right_map_uncontractedListEmd_eq_filter {φs : List 𝓕.States
|
|||
exact h1
|
||||
exact fun _ _ h => uncontractedListEmd_strictMono h
|
||||
· have h1 := h.2.2
|
||||
simp_all
|
||||
simp_all only [and_true]
|
||||
rcases h1 with h1 | h1
|
||||
· simp [h1]
|
||||
· right
|
||||
|
|
@ -655,16 +673,17 @@ lemma join_singleton_signFinset_eq_filter {φs : List 𝓕.States}
|
|||
((h1 : ((join (singleton h) φsucΛ).getDual? c).isSome) →
|
||||
(((join (singleton h) φsucΛ).getDual? c).get h1) < i))) := by
|
||||
ext a
|
||||
simp [signFinset, and_assoc]
|
||||
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
|
||||
rw [singleton_getDual?_eq_none_iff_neq]
|
||||
omega
|
||||
simp [h1]
|
||||
simp only [h1, Option.isSome_none, Bool.false_eq_true, IsEmpty.forall_iff, or_self, true_and]
|
||||
apply Iff.intro
|
||||
· intro h1 h2
|
||||
rcases h1 with h1 | h1
|
||||
· simp [h1]
|
||||
· simp only [h1, Option.isSome_none, Bool.false_eq_true, IsEmpty.exists_iff]
|
||||
have h2' : ¬ (((singleton h).join φsucΛ).getDual? a).isSome := by
|
||||
exact Option.not_isSome_iff_eq_none.mpr h1
|
||||
exact h2' h2
|
||||
|
|
@ -682,9 +701,9 @@ lemma join_singleton_signFinset_eq_filter {φs : List 𝓕.States}
|
|||
have hij : ((singleton h).join φsucΛ).getDual? i = j := by
|
||||
rw [@getDual?_eq_some_iff_mem]
|
||||
simp [join, singleton]
|
||||
simp [hn] at hij
|
||||
simp only [hn, getDual?_getDual?_get_get, Option.some.injEq] at hij
|
||||
omega
|
||||
· simp at h2'
|
||||
· simp only [Bool.not_eq_true, Option.not_isSome, Option.isNone_iff_eq_none] at h2'
|
||||
simp [h2']
|
||||
|
||||
|
||||
|
|
@ -756,7 +775,8 @@ lemma join_singleton_getDual?_right {φs : List 𝓕.States}
|
|||
(φsucΛ : WickContraction [singleton h]ᵘᶜ.length) :
|
||||
(join (singleton h) φsucΛ).getDual? j = some i := by
|
||||
rw [@getDual?_eq_some_iff_mem]
|
||||
simp [singleton, join]
|
||||
simp only [join, singleton, Finset.le_eq_subset, Finset.mem_union, Finset.mem_singleton,
|
||||
Finset.mem_map, RelEmbedding.coe_toEmbedding]
|
||||
left
|
||||
exact Finset.pair_comm j i
|
||||
|
||||
|
|
@ -779,36 +799,41 @@ lemma joinSignRightExtra_eq_i_j_finset_eq_if {φs : List 𝓕.States}
|
|||
rw [Finset.filter_comm]
|
||||
have h11 : (Finset.filter (fun c => c ∉ (singleton h).uncontracted) Finset.univ) = {i, j} := by
|
||||
ext x
|
||||
simp
|
||||
simp only [Finset.mem_filter, Finset.mem_univ, true_and, Finset.mem_insert,
|
||||
Finset.mem_singleton]
|
||||
rw [@mem_uncontracted_iff_not_contracted]
|
||||
simp [singleton]
|
||||
simp only [singleton, Finset.mem_singleton, forall_eq, Finset.mem_insert, not_or, not_and,
|
||||
Decidable.not_not]
|
||||
omega
|
||||
rw [h11]
|
||||
ext x
|
||||
simp
|
||||
simp only [Finset.mem_filter, Finset.mem_insert, Finset.mem_singleton, Finset.mem_union]
|
||||
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)
|
||||
by_cases hj1 : ¬ uncontractedListEmd (φsucΛ.fstFieldOfContract a) < j
|
||||
· simp [hj1]
|
||||
· simp only [hj1, false_and, ↓reduceIte, Finset.not_mem_empty, false_or]
|
||||
have hi1 : ¬ uncontractedListEmd (φsucΛ.fstFieldOfContract a) < i := by omega
|
||||
simp [hi1]
|
||||
simp only [hi1, false_and, ↓reduceIte, Finset.not_mem_empty, iff_false, not_and, not_or,
|
||||
not_forall, not_lt]
|
||||
intro hxij h1 h2
|
||||
omega
|
||||
· have hj1 : uncontractedListEmd (φsucΛ.fstFieldOfContract a) < j := by
|
||||
omega
|
||||
by_cases hi1 : ¬ i < uncontractedListEmd (φsucΛ.sndFieldOfContract a)
|
||||
· simp [hi1]
|
||||
· simp only [hi1, and_false, ↓reduceIte, Finset.not_mem_empty, or_false]
|
||||
have hj2 : ¬ j < uncontractedListEmd (φsucΛ.sndFieldOfContract a) := by omega
|
||||
simp [hj2]
|
||||
simp only [hj2, false_and, and_false, ↓reduceIte, Finset.not_mem_empty, iff_false, not_and,
|
||||
not_or, not_forall, not_lt]
|
||||
intro hxij h1 h2
|
||||
omega
|
||||
· have hi1 : i < uncontractedListEmd (φsucΛ.sndFieldOfContract a) := by
|
||||
omega
|
||||
simp [hi1, hj1]
|
||||
simp only [hj1, true_and, hi1, and_true]
|
||||
by_cases hi2 : ¬ uncontractedListEmd (φsucΛ.fstFieldOfContract a) < i
|
||||
· simp [hi2]
|
||||
· 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)
|
||||
· omega
|
||||
· have hj4 : j < uncontractedListEmd (φsucΛ.sndFieldOfContract a) := by omega
|
||||
|
|
@ -817,31 +842,35 @@ lemma joinSignRightExtra_eq_i_j_finset_eq_if {φs : List 𝓕.States}
|
|||
· subst h
|
||||
omega
|
||||
· subst h
|
||||
simp
|
||||
simp only [join_singleton_getDual?_right, reduceCtorEq, not_false_eq_true,
|
||||
Option.get_some, Option.isSome_some, exists_const, true_and]
|
||||
omega
|
||||
· have hi2 : uncontractedListEmd (φsucΛ.fstFieldOfContract a) < i := by omega
|
||||
simp [hi2]
|
||||
simp only [hi2, and_true, ↓reduceIte, Finset.mem_singleton]
|
||||
by_cases hj3 : ¬ j < uncontractedListEmd (φsucΛ.sndFieldOfContract a)
|
||||
· simp [hj3]
|
||||
· simp only [hj3, ↓reduceIte, Finset.not_mem_empty, false_or]
|
||||
apply Iff.intro
|
||||
· intro h
|
||||
omega
|
||||
· intro h
|
||||
subst h
|
||||
simp
|
||||
simp only [true_or, join_singleton_getDual?_left, reduceCtorEq, Option.isSome_some,
|
||||
Option.get_some, forall_const, false_or, true_and]
|
||||
omega
|
||||
· have hj3 : j < uncontractedListEmd (φsucΛ.sndFieldOfContract a) := by omega
|
||||
simp [hj3]
|
||||
simp only [hj3, ↓reduceIte, Finset.mem_singleton]
|
||||
apply Iff.intro
|
||||
· intro h
|
||||
omega
|
||||
· intro h
|
||||
rcases h with h1 | h1
|
||||
· subst h1
|
||||
simp
|
||||
simp only [or_true, join_singleton_getDual?_right, reduceCtorEq, Option.isSome_some,
|
||||
Option.get_some, forall_const, false_or, true_and]
|
||||
omega
|
||||
· subst h1
|
||||
simp
|
||||
simp only [true_or, join_singleton_getDual?_left, reduceCtorEq, Option.isSome_some,
|
||||
Option.get_some, forall_const, false_or, true_and]
|
||||
omega
|
||||
|
||||
|
||||
|
|
@ -860,13 +889,12 @@ lemma joinSignLeftExtra_eq_joinSignRightExtra {φs : List 𝓕.States}
|
|||
conv_lhs =>
|
||||
enter [2, 2, x]
|
||||
simp only [Equiv.symm_symm, Equiv.sumCompl_apply_inl, Equiv.sumCompl_apply_inr, e2]
|
||||
rw [if_neg (
|
||||
by
|
||||
simp
|
||||
rw [if_neg (by
|
||||
simp only [Finset.mem_filter, mem_signFinset, not_and, not_forall, not_lt, and_imp]
|
||||
intro h1 h2
|
||||
have hx := x.2
|
||||
simp_all)]
|
||||
simp
|
||||
simp only [Finset.mem_filter, mem_signFinset, map_one, Finset.prod_const_one, mul_one]
|
||||
rw [← ((singleton h).join φsucΛ).sigmaContractedEquiv.prod_comp]
|
||||
erw [Finset.prod_sigma]
|
||||
conv_lhs =>
|
||||
|
|
@ -896,38 +924,38 @@ lemma joinSignLeftExtra_eq_joinSignRightExtra {φs : List 𝓕.States}
|
|||
· have hi1 : ¬ uncontractedListEmd (φsucΛ.fstFieldOfContract a) < i := by omega
|
||||
simp [hj1, hi1]
|
||||
· have hj1 : uncontractedListEmd (φsucΛ.fstFieldOfContract a) < j := by omega
|
||||
simp [hj1]
|
||||
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 hj2 : ¬ j < uncontractedListEmd (φsucΛ.sndFieldOfContract a) := by omega
|
||||
simp [hi2, hj2, hi1]
|
||||
· have hi2 : i < uncontractedListEmd (φsucΛ.sndFieldOfContract a) := by omega
|
||||
have hi2n : ¬ uncontractedListEmd (φsucΛ.sndFieldOfContract a) < i := by omega
|
||||
simp [hi2, hi2n]
|
||||
simp only [hi2n, and_false, ↓reduceIte, map_one, hi2, true_and, one_mul, and_true]
|
||||
by_cases hj2 : ¬ j < uncontractedListEmd (φsucΛ.sndFieldOfContract a)
|
||||
· simp [hj2]
|
||||
· simp only [hj2, false_and, ↓reduceIte, Finset.empty_union]
|
||||
have hj2 : uncontractedListEmd (φsucΛ.sndFieldOfContract a) < j:= by omega
|
||||
simp [hj2]
|
||||
simp only [hj2, true_and]
|
||||
by_cases hi1 : ¬ uncontractedListEmd (φsucΛ.fstFieldOfContract a) < i
|
||||
· simp [hi1]
|
||||
· have hi1 : uncontractedListEmd (φsucΛ.fstFieldOfContract a) < i := by omega
|
||||
simp [hi1, ofFinset_singleton]
|
||||
simp only [hi1, ↓reduceIte, ofFinset_singleton, List.get_eq_getElem]
|
||||
erw [hs]
|
||||
exact exchangeSign_symm (𝓕|>ₛφs[↑j]) (𝓕|>ₛ[singleton h]ᵘᶜ[↑(φsucΛ.sndFieldOfContract a)])
|
||||
· simp at hj2
|
||||
simp [hj2]
|
||||
· simp only [not_lt, not_le] at hj2
|
||||
simp only [hj2, true_and]
|
||||
by_cases hi1 : ¬ uncontractedListEmd (φsucΛ.fstFieldOfContract a) < i
|
||||
· simp [hi1]
|
||||
· have hi1 : uncontractedListEmd (φsucΛ.fstFieldOfContract a) < i := by omega
|
||||
simp [hi1]
|
||||
simp only [hi1, and_true, ↓reduceIte]
|
||||
have hj2 : ¬ uncontractedListEmd (φsucΛ.sndFieldOfContract a) < j := by omega
|
||||
simp [hj2]
|
||||
simp only [hj2, ↓reduceIte, map_one]
|
||||
rw [← ofFinset_union_disjoint]
|
||||
simp only [instCommGroup, ofFinset_singleton, List.get_eq_getElem, hs]
|
||||
erw [hs]
|
||||
simp
|
||||
simp
|
||||
omega
|
||||
simp only [Fin.getElem_fin, mul_self, map_one]
|
||||
simp only [Finset.disjoint_singleton_right, Finset.mem_singleton]
|
||||
exact Fin.ne_of_lt h
|
||||
|
||||
lemma join_sign_singleton {φs : List 𝓕.States}
|
||||
{i j : Fin φs.length} (h : i < j) (hs : (𝓕 |>ₛ φs[i]) = (𝓕 |>ₛ φs[j]))
|
||||
|
|
@ -968,21 +996,21 @@ lemma exists_join_singleton_of_card_ge_zero {φs : List 𝓕.States} (φsΛ : Wi
|
|||
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 [← subContraction_singleton_eq_singleton]
|
||||
simp only [Fin.getElem_fin]
|
||||
apply And.intro
|
||||
· have h1 := join_congr (subContraction_singleton_eq_singleton _ ⟨a, ha⟩).symm (φsucΛ := φsucΛ)
|
||||
simp [h1, 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
|
||||
· simp [φsucΛ]
|
||||
· simp only [id_eq, eq_mpr_eq_cast, φsucΛ]
|
||||
rw [gradingCompliant_congr (φs' := [(φsΛ.subContraction {a} (by simpa using ha))]ᵘᶜ)]
|
||||
simp
|
||||
simp only [id_eq, eq_mpr_eq_cast, congr_trans_apply, congr_refl]
|
||||
exact quotContraction_gradingCompliant hc
|
||||
rw [← subContraction_singleton_eq_singleton]
|
||||
· simp [φsucΛ]
|
||||
· simp only [id_eq, eq_mpr_eq_cast, card_congr, φsucΛ]
|
||||
have h1 := subContraction_card_plus_quotContraction_card_eq _ {a} (by simpa using ha)
|
||||
simp [subContraction] at h1
|
||||
simp only [subContraction, Finset.card_singleton, id_eq, eq_mpr_eq_cast] at h1
|
||||
omega
|
||||
|
||||
lemma join_sign_induction {φs : List 𝓕.States} (φsΛ : WickContraction φs.length)
|
||||
|
|
@ -992,7 +1020,7 @@ lemma join_sign_induction {φs : List 𝓕.States} (φsΛ : WickContraction φs.
|
|||
| 0, hn => by
|
||||
rw [@card_zero_iff_empty] at hn
|
||||
subst hn
|
||||
simp [sign_empty]
|
||||
simp only [empty_join, sign_empty, one_mul]
|
||||
apply sign_congr
|
||||
simp
|
||||
| Nat.succ n, hn => by
|
||||
|
|
@ -1005,7 +1033,7 @@ lemma join_sign_induction {φs : List 𝓕.States} (φsΛ : WickContraction φs.
|
|||
rw [join_sign_induction φsucΛ' (congr (by simp [join_uncontractedListGet]) φsucΛ) h2
|
||||
n hn]
|
||||
rw [mul_assoc]
|
||||
simp [sign_congr]
|
||||
simp only [mul_eq_mul_left_iff]
|
||||
left
|
||||
left
|
||||
apply sign_congr
|
||||
|
|
@ -1019,11 +1047,12 @@ lemma join_sign {φs : List 𝓕.States} (φsΛ : WickContraction φs.length)
|
|||
lemma join_not_gradingCompliant_of_left_not_gradingCompliant {φs : List 𝓕.States}
|
||||
(φsΛ : WickContraction φs.length) (φsucΛ : WickContraction [φsΛ]ᵘᶜ.length)
|
||||
(hc : ¬ φsΛ.GradingCompliant) : ¬ (join φsΛ φsucΛ).GradingCompliant := by
|
||||
simp_all [GradingCompliant]
|
||||
simp_all only [GradingCompliant, Fin.getElem_fin, Subtype.forall, not_forall]
|
||||
obtain ⟨a, ha, ha2⟩ := hc
|
||||
use (joinLiftLeft (φsucΛ := φsucΛ) ⟨a, ha⟩).1
|
||||
use (joinLiftLeft (φsucΛ := φsucΛ) ⟨a, ha⟩).2
|
||||
simp
|
||||
simp only [Subtype.coe_eta, join_fstFieldOfContract_joinLiftLeft,
|
||||
join_sndFieldOfContract_joinLift]
|
||||
exact ha2
|
||||
|
||||
lemma join_sign_timeContract {φs : List 𝓕.States} (φsΛ : WickContraction φs.length)
|
||||
|
|
|
|||
Loading…
Add table
Add a link
Reference in a new issue