refactor: Lint
This commit is contained in:
jstoobysmith
parent
22636db606
commit
32aefb7eb7
9 changed files with 209 additions and 166 deletions
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@ -25,7 +25,7 @@ def statisticSubmodule (f : FieldStatistic) : Submodule ℂ 𝓕.CrAnAlgebra :=
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Submodule.span ℂ {a | ∃ φs, a = ofCrAnList φs ∧ (𝓕 |>ₛ φs) = f}
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lemma ofCrAnList_mem_statisticSubmodule_of (φs : List 𝓕.CrAnStates) (f : FieldStatistic)
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(h : (𝓕 |>ₛ φs) = f) :
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(h : (𝓕 |>ₛ φs) = f) :
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ofCrAnList φs ∈ statisticSubmodule f := by
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refine Submodule.mem_span.mpr fun _ a => a ⟨φs, ⟨rfl, h⟩⟩
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@ -234,6 +234,7 @@ lemma directSum_eq_bosonic_plus_fermionic
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conv_lhs => rw [hx, hy]
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abel
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/-- The instance of a graded algebra on `CrAnAlgebra`. -/
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instance crAnAlgebraGrade : GradedAlgebra (A := 𝓕.CrAnAlgebra) statisticSubmodule where
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one_mem := by
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simp only [statisticSubmodule]
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@ -134,7 +134,6 @@ lemma ι_superCommute_ofCrAnState_superCommute_ofCrAnState_ofCrAnState (φ1 φ2
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left
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use φ1, φ2, φ3
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@[simp]
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lemma ι_superCommute_superCommute_ofCrAnState_ofCrAnState_ofCrAnState (φ1 φ2 φ3 : 𝓕.CrAnStates) :
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ι [[ofCrAnState φ1, ofCrAnState φ2]ₛca, ofCrAnState φ3]ₛca = 0 := by
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rw [← ofCrAnList_singleton, ← ofCrAnList_singleton, ← ofCrAnList_singleton]
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@ -147,16 +146,15 @@ lemma ι_superCommute_superCommute_ofCrAnState_ofCrAnState_ofCrAnState (φ1 φ2
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· rw [superCommute_fermionic_fermionic_symm h h']
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simp [ofCrAnList_singleton]
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@[simp]
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lemma ι_superCommute_superCommute_ofCrAnState_ofCrAnState_ofCrAnList (φ1 φ2 : 𝓕.CrAnStates)
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(φs : List 𝓕.CrAnStates) :
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ι [[ofCrAnState φ1, ofCrAnState φ2]ₛca, ofCrAnList φs]ₛca = 0 := by
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rw [← ofCrAnList_singleton, ← ofCrAnList_singleton]
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rcases superCommute_ofCrAnList_ofCrAnList_bosonic_or_fermionic [φ1] [φ2] with h | h
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· rw [superCommute_bosonic_ofCrAnList_eq_sum _ _ h]
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simp [ofCrAnList_singleton]
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simp [ofCrAnList_singleton, ι_superCommute_superCommute_ofCrAnState_ofCrAnState_ofCrAnState]
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· rw [superCommute_fermionic_ofCrAnList_eq_sum _ _ h]
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simp [ofCrAnList_singleton]
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simp [ofCrAnList_singleton, ι_superCommute_superCommute_ofCrAnState_ofCrAnState_ofCrAnState]
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@[simp]
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lemma ι_superCommute_superCommute_ofCrAnState_ofCrAnState_crAnAlgebra (φ1 φ2 : 𝓕.CrAnStates)
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@ -164,7 +162,7 @@ lemma ι_superCommute_superCommute_ofCrAnState_ofCrAnState_crAnAlgebra (φ1 φ2
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change (ι.toLinearMap ∘ₗ superCommute [ofCrAnState φ1, ofCrAnState φ2]ₛca) a = _
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have h1 : (ι.toLinearMap ∘ₗ superCommute [ofCrAnState φ1, ofCrAnState φ2]ₛca) = 0 := by
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apply (ofCrAnListBasis.ext fun l ↦ ?_)
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simp
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simp [ι_superCommute_superCommute_ofCrAnState_ofCrAnState_ofCrAnList]
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rw [h1]
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simp
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@ -239,7 +237,7 @@ lemma bosonicProj_mem_fieldOpIdealSet_or_zero (x : CrAnAlgebra 𝓕) (hx : x ∈
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lemma fermionicProj_mem_fieldOpIdealSet_or_zero (x : CrAnAlgebra 𝓕) (hx : x ∈ 𝓕.fieldOpIdealSet) :
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x.fermionicProj.1 ∈ 𝓕.fieldOpIdealSet ∨ x.fermionicProj = 0 := by
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have hx' := hx
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simp only [fieldOpIdealSet, exists_prop, Set.mem_setOf_eq] at hx
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simp only [fieldOpIdealSet, exists_prop, Set.mem_setOf_eq] at hx
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rcases hx with ⟨φ1, φ2, φ3, rfl⟩ | ⟨φc, φc', hφc, hφc', rfl⟩ | ⟨φa, φa', hφa, hφa', rfl⟩ |
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⟨φ, φ', hdiff, rfl⟩
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· rcases superCommute_superCommute_ofCrAnState_bosonic_or_fermionic φ1 φ2 φ3 with h | h
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@ -49,8 +49,11 @@ lemma ι_superCommute_eq_of_equiv_right (a b1 b2 : 𝓕.CrAnAlgebra) (h : b1 ≈
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simp only [LinearMap.mem_ker, ← map_sub]
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exact ι_superCommute_right_zero_of_mem_ideal a (b1 - b2) h
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noncomputable def superCommuteRight (a : 𝓕.CrAnAlgebra) : FieldOpAlgebra 𝓕 →ₗ[ℂ] FieldOpAlgebra 𝓕 where
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toFun := Quotient.lift (ι.toLinearMap ∘ₗ CrAnAlgebra.superCommute a) (ι_superCommute_eq_of_equiv_right a)
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/-- The super commutor on the `FieldOpAlgebra` defined as a linear map `[a,_]ₛ`. -/
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noncomputable def superCommuteRight (a : 𝓕.CrAnAlgebra) :
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FieldOpAlgebra 𝓕 →ₗ[ℂ] FieldOpAlgebra 𝓕 where
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toFun := Quotient.lift (ι.toLinearMap ∘ₗ CrAnAlgebra.superCommute a)
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(ι_superCommute_eq_of_equiv_right a)
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map_add' x y := by
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obtain ⟨x, hx⟩ := ι_surjective x
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obtain ⟨y, hy⟩ := ι_surjective y
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@ -64,11 +67,11 @@ noncomputable def superCommuteRight (a : 𝓕.CrAnAlgebra) : FieldOpAlgebra 𝓕
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rw [← map_smul, ι_apply, ι_apply]
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simp
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lemma superCommuteRight_apply_ι (a b : 𝓕.CrAnAlgebra) : superCommuteRight a (ι b) = ι [a, b]ₛca := by
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rfl
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lemma superCommuteRight_apply_ι (a b : 𝓕.CrAnAlgebra) :
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superCommuteRight a (ι b) = ι [a, b]ₛca := by rfl
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lemma superCommuteRight_apply_quot (a b : 𝓕.CrAnAlgebra) : superCommuteRight a ⟦b⟧= ι [a, b]ₛca := by
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rfl
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lemma superCommuteRight_apply_quot (a b : 𝓕.CrAnAlgebra) :
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superCommuteRight a ⟦b⟧= ι [a, b]ₛca := by rfl
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lemma superCommuteRight_eq_of_equiv (a1 a2 : 𝓕.CrAnAlgebra) (h : a1 ≈ a2) :
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superCommuteRight a1 = superCommuteRight a2 := by
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@ -85,6 +88,7 @@ lemma superCommuteRight_eq_of_equiv (a1 a2 : 𝓕.CrAnAlgebra) (h : a1 ≈ a2) :
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simp only [add_sub_cancel]
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simp
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/-- The super commutor on the `FieldOpAlgebra`. -/
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noncomputable def superCommute : FieldOpAlgebra 𝓕 →ₗ[ℂ]
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FieldOpAlgebra 𝓕 →ₗ[ℂ] FieldOpAlgebra 𝓕 where
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toFun := Quotient.lift superCommuteRight superCommuteRight_eq_of_equiv
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@ -20,33 +20,35 @@ namespace FieldOpAlgebra
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variable {𝓕 : FieldSpecification}
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lemma ι_timeOrder_superCommute_superCommute_eq_time_ofCrAnList {φ1 φ2 φ3 : 𝓕.CrAnStates}
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(φs1 φs2 : List 𝓕.CrAnStates) (h :
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(φs1 φs2 : List 𝓕.CrAnStates) (h :
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crAnTimeOrderRel φ1 φ2 ∧ crAnTimeOrderRel φ1 φ3 ∧
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crAnTimeOrderRel φ2 φ1 ∧ crAnTimeOrderRel φ2 φ3 ∧
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crAnTimeOrderRel φ3 φ1 ∧ crAnTimeOrderRel φ3 φ2):
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ι 𝓣ᶠ(ofCrAnList φs1 * [ofCrAnState φ1, [ofCrAnState φ2, ofCrAnState φ3]ₛca]ₛca * ofCrAnList φs2)
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= 0 := by
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crAnTimeOrderRel φ3 φ1 ∧ crAnTimeOrderRel φ3 φ2) :
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ι 𝓣ᶠ(ofCrAnList φs1 * [ofCrAnState φ1, [ofCrAnState φ2, ofCrAnState φ3]ₛca]ₛca *
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ofCrAnList φs2) = 0 := by
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let l1 :=
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(List.takeWhile (fun c => ¬ crAnTimeOrderRel φ1 c) ((φs1 ++ φs2).insertionSort crAnTimeOrderRel))
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(List.takeWhile (fun c => ¬ crAnTimeOrderRel φ1 c)
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((φs1 ++ φs2).insertionSort crAnTimeOrderRel))
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++ (List.filter (fun c => crAnTimeOrderRel φ1 c ∧ crAnTimeOrderRel c φ1) φs1)
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let l2 := (List.filter (fun c => crAnTimeOrderRel φ1 c ∧ crAnTimeOrderRel c φ1) φs2)
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++ (List.filter (fun c => crAnTimeOrderRel φ1 c ∧ ¬ crAnTimeOrderRel c φ1) ((φs1 ++ φs2).insertionSort crAnTimeOrderRel))
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++ (List.filter (fun c => crAnTimeOrderRel φ1 c ∧ ¬ crAnTimeOrderRel c φ1)
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((φs1 ++ φs2).insertionSort crAnTimeOrderRel))
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have h123 : ι 𝓣ᶠ(ofCrAnList (φs1 ++ φ1 :: φ2 :: φ3 :: φs2)) =
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crAnTimeOrderSign (φs1 ++ φ1 :: φ2 :: φ3 :: φs2)
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• (ι (ofCrAnList l1) * ι (ofCrAnList [φ1, φ2, φ3]) * ι (ofCrAnList l2)):= by
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• (ι (ofCrAnList l1) * ι (ofCrAnList [φ1, φ2, φ3]) * ι (ofCrAnList l2)) := by
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have h1 := insertionSort_of_eq_list 𝓕.crAnTimeOrderRel φ1 φs1 [φ1, φ2, φ3] φs2
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(by simp_all)
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rw [timeOrder_ofCrAnList, show φs1 ++ φ1 :: φ2 :: φ3 :: φs2 = φs1 ++ [φ1, φ2, φ3] ++ φs2 by simp,
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crAnTimeOrderList, h1]
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(by simp_all)
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rw [timeOrder_ofCrAnList, show φs1 ++ φ1 :: φ2 :: φ3 :: φs2 = φs1 ++ [φ1, φ2, φ3] ++ φs2
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by simp, crAnTimeOrderList, h1]
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simp only [List.append_assoc, List.singleton_append, decide_not,
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Bool.decide_and, ofCrAnList_append, map_smul, map_mul, l1, l2, mul_assoc]
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have h132 : ι 𝓣ᶠ(ofCrAnList (φs1 ++ φ1 :: φ3 :: φ2 :: φs2)) =
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crAnTimeOrderSign (φs1 ++ φ1 :: φ2 :: φ3 :: φs2)
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• (ι (ofCrAnList l1) * ι (ofCrAnList [φ1, φ3, φ2]) * ι (ofCrAnList l2)):= by
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• (ι (ofCrAnList l1) * ι (ofCrAnList [φ1, φ3, φ2]) * ι (ofCrAnList l2)) := by
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have h1 := insertionSort_of_eq_list 𝓕.crAnTimeOrderRel φ1 φs1 [φ1, φ3, φ2] φs2
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(by simp_all)
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rw [timeOrder_ofCrAnList, show φs1 ++ φ1 :: φ3 :: φ2 :: φs2 = φs1 ++ [φ1, φ3, φ2] ++ φs2 by simp,
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crAnTimeOrderList, h1]
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(by simp_all)
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rw [timeOrder_ofCrAnList, show φs1 ++ φ1 :: φ3 :: φ2 :: φs2 = φs1 ++ [φ1, φ3, φ2] ++ φs2
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by simp, crAnTimeOrderList, h1]
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simp only [List.singleton_append, decide_not,
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Bool.decide_and, ofCrAnList_append, map_smul, map_mul, l1, l2, mul_assoc]
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congr 1
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@ -67,11 +69,11 @@ lemma ι_timeOrder_superCommute_superCommute_eq_time_ofCrAnList {φ1 φ2 φ3 :
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exact List.Perm.swap φ1 φ2 [φ3]
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have h231 : ι 𝓣ᶠ(ofCrAnList (φs1 ++ φ2 :: φ3 :: φ1 :: φs2)) =
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crAnTimeOrderSign (φs1 ++ φ1 :: φ2 :: φ3 :: φs2)
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• (ι (ofCrAnList l1) * ι (ofCrAnList [φ2, φ3, φ1]) * ι (ofCrAnList l2)):= by
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• (ι (ofCrAnList l1) * ι (ofCrAnList [φ2, φ3, φ1]) * ι (ofCrAnList l2)) := by
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have h1 := insertionSort_of_eq_list 𝓕.crAnTimeOrderRel φ1 φs1 [φ2, φ3, φ1] φs2
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(by simp_all)
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rw [timeOrder_ofCrAnList, show φs1 ++ φ2 :: φ3 :: φ1 :: φs2 = φs1 ++ [φ2, φ3, φ1] ++ φs2 by simp,
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crAnTimeOrderList, h1]
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(by simp_all)
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rw [timeOrder_ofCrAnList, show φs1 ++ φ2 :: φ3 :: φ1 :: φs2 = φs1 ++ [φ2, φ3, φ1] ++ φs2
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by simp, crAnTimeOrderList, h1]
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simp only [List.singleton_append, decide_not,
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Bool.decide_and, ofCrAnList_append, map_smul, map_mul, l1, l2, mul_assoc]
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congr 1
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@ -85,11 +87,11 @@ lemma ι_timeOrder_superCommute_superCommute_eq_time_ofCrAnList {φ1 φ2 φ3 :
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simp_all
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have h321 : ι 𝓣ᶠ(ofCrAnList (φs1 ++ φ3 :: φ2 :: φ1 :: φs2)) =
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crAnTimeOrderSign (φs1 ++ φ1 :: φ2 :: φ3 :: φs2)
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• (ι (ofCrAnList l1) * ι (ofCrAnList [φ3, φ2, φ1]) * ι (ofCrAnList l2)):= by
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• (ι (ofCrAnList l1) * ι (ofCrAnList [φ3, φ2, φ1]) * ι (ofCrAnList l2)) := by
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have h1 := insertionSort_of_eq_list 𝓕.crAnTimeOrderRel φ1 φs1 [φ3, φ2, φ1] φs2
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(by simp_all)
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rw [timeOrder_ofCrAnList, show φs1 ++ φ3 :: φ2 :: φ1 :: φs2 = φs1 ++ [φ3, φ2, φ1] ++ φs2 by simp,
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crAnTimeOrderList, h1]
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(by simp_all)
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rw [timeOrder_ofCrAnList, show φs1 ++ φ3 :: φ2 :: φ1 :: φs2 = φs1 ++ [φ3, φ2, φ1] ++ φs2
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by simp, crAnTimeOrderList, h1]
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simp only [List.singleton_append, decide_not,
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Bool.decide_and, ofCrAnList_append, map_smul, map_mul, l1, l2, mul_assoc]
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congr 1
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@ -125,7 +127,8 @@ lemma ι_timeOrder_superCommute_superCommute_eq_time_ofCrAnList {φ1 φ2 φ3 :
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repeat rw [mul_assoc]
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rw [← mul_sub, ← mul_sub, ← mul_sub]
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rw [← sub_mul, ← sub_mul, ← sub_mul]
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trans ι (ofCrAnList l1) * ι [ofCrAnState φ1, [ofCrAnState φ2, ofCrAnState φ3]ₛca]ₛca * ι (ofCrAnList l2)
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trans ι (ofCrAnList l1) * ι [ofCrAnState φ1, [ofCrAnState φ2, ofCrAnState φ3]ₛca]ₛca *
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ι (ofCrAnList l2)
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rw [mul_assoc]
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congr
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rw [← ofCrAnList_singleton, ← ofCrAnList_singleton, ← ofCrAnList_singleton]
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@ -137,7 +140,7 @@ lemma ι_timeOrder_superCommute_superCommute_eq_time_ofCrAnList {φ1 φ2 φ3 :
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simp_all
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lemma ι_timeOrder_superCommute_superCommute_ofCrAnList {φ1 φ2 φ3 : 𝓕.CrAnStates}
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(φs1 φs2 : List 𝓕.CrAnStates):
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(φs1 φs2 : List 𝓕.CrAnStates) :
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ι 𝓣ᶠ(ofCrAnList φs1 * [ofCrAnState φ1, [ofCrAnState φ2, ofCrAnState φ3]ₛca]ₛca * ofCrAnList φs2)
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= 0 := by
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by_cases h :
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@ -150,7 +153,7 @@ lemma ι_timeOrder_superCommute_superCommute_ofCrAnList {φ1 φ2 φ3 : 𝓕.CrAn
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simp
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@[simp]
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lemma ι_timeOrder_superCommute_superCommute {φ1 φ2 φ3 : 𝓕.CrAnStates} (a b : 𝓕.CrAnAlgebra):
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lemma ι_timeOrder_superCommute_superCommute {φ1 φ2 φ3 : 𝓕.CrAnStates} (a b : 𝓕.CrAnAlgebra) :
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ι 𝓣ᶠ(a * [ofCrAnState φ1, [ofCrAnState φ2, ofCrAnState φ3]ₛca]ₛca * b) = 0 := by
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let pb (b : 𝓕.CrAnAlgebra) (hc : b ∈ Submodule.span ℂ (Set.range ofCrAnListBasis)) :
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Prop := ι 𝓣ᶠ(a * [ofCrAnState φ1, [ofCrAnState φ2, ofCrAnState φ3]ₛca]ₛca * b) = 0
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@ -215,20 +218,21 @@ lemma ι_timeOrder_superCommute_eq_time {φ ψ : 𝓕.CrAnStates}
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rw [h1]
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simp only [map_smul]
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have h1 := insertionSort_of_eq_list 𝓕.crAnTimeOrderRel φ φs' [φ, ψ] φs
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(by simp_all)
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(by simp_all)
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rw [crAnTimeOrderList, show φs' ++ φ :: ψ :: φs = φs' ++ [φ, ψ] ++ φs by simp, h1]
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have h2 := insertionSort_of_eq_list 𝓕.crAnTimeOrderRel φ φs' [ψ, φ] φs
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(by simp_all)
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(by simp_all)
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rw [crAnTimeOrderList, show φs' ++ ψ :: φ :: φs = φs' ++ [ψ, φ] ++ φs by simp, h2]
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repeat rw [ofCrAnList_append]
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rw [smul_smul, mul_comm, ← smul_smul, ← smul_sub]
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rw [map_mul, map_mul, map_mul, map_mul, map_mul, map_mul, map_mul, map_mul]
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rw [← mul_smul_comm]
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rw [mul_assoc, mul_assoc, mul_assoc ,mul_assoc ,mul_assoc ,mul_assoc]
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rw [mul_assoc, mul_assoc, mul_assoc, mul_assoc, mul_assoc, mul_assoc]
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rw [← mul_sub, ← mul_sub, mul_smul_comm, mul_smul_comm, ← smul_mul_assoc,
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← smul_mul_assoc]
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rw [← sub_mul]
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have h1 : (ι (ofCrAnList [φ, ψ]) - (exchangeSign (𝓕.crAnStatistics φ)) (𝓕.crAnStatistics ψ) • ι (ofCrAnList [ψ, φ])) =
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have h1 : (ι (ofCrAnList [φ, ψ]) -
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(exchangeSign (𝓕.crAnStatistics φ)) (𝓕.crAnStatistics ψ) • ι (ofCrAnList [ψ, φ])) =
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ι [ofCrAnState φ, ofCrAnState ψ]ₛca := by
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rw [superCommute_ofCrAnState_ofCrAnState]
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rw [← ofCrAnList_singleton, ← ofCrAnList_singleton, ← ofCrAnList_append]
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@ -237,7 +241,8 @@ lemma ι_timeOrder_superCommute_eq_time {φ ψ : 𝓕.CrAnStates}
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rw [← ofCrAnList_append]
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simp
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rw [h1]
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have hc : ι ((superCommute (ofCrAnState φ)) (ofCrAnState ψ)) ∈ Subalgebra.center ℂ 𝓕.FieldOpAlgebra := by
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have hc : ι ((superCommute (ofCrAnState φ)) (ofCrAnState ψ)) ∈
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Subalgebra.center ℂ 𝓕.FieldOpAlgebra := by
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apply ι_superCommute_ofCrAnState_ofCrAnState_mem_center
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rw [Subalgebra.mem_center_iff] at hc
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repeat rw [← mul_assoc]
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@ -272,7 +277,6 @@ lemma ι_timeOrder_superCommute_eq_time {φ ψ : 𝓕.CrAnStates}
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· intro x hx hpx
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simp_all [pb, hpx]
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lemma ι_timeOrder_superCommute_neq_time {φ ψ : 𝓕.CrAnStates}
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(hφψ : ¬ (crAnTimeOrderRel φ ψ ∧ crAnTimeOrderRel ψ φ)) (a b : 𝓕.CrAnAlgebra) :
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ι 𝓣ᶠ(a * [ofCrAnState φ, ofCrAnState ψ]ₛca * b) = 0 := by
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@ -280,18 +284,16 @@ lemma ι_timeOrder_superCommute_neq_time {φ ψ : 𝓕.CrAnStates}
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have hφψ : ¬ (crAnTimeOrderRel φ ψ) ∨ ¬ (crAnTimeOrderRel ψ φ) := by
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exact Decidable.not_and_iff_or_not.mp hφψ
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rcases hφψ with hφψ | hφψ
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· rw [timeOrder_superCommute_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel ]
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have ht := IsTotal.total (r := crAnTimeOrderRel) φ ψ
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· rw [timeOrder_superCommute_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel]
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simp_all only [false_and, not_false_eq_true, false_or, mul_zero, zero_mul, map_zero]
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simp_all
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· rw [superCommute_ofCrAnState_ofCrAnState_symm]
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simp only [instCommGroup.eq_1, neg_smul, map_neg, map_smul, mul_neg, Algebra.mul_smul_comm,
|
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neg_mul, Algebra.smul_mul_assoc, neg_eq_zero, smul_eq_zero]
|
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rw [timeOrder_superCommute_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel ]
|
||||
rw [timeOrder_superCommute_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel]
|
||||
simp only [mul_zero, zero_mul, map_zero, or_true]
|
||||
simp_all
|
||||
|
||||
|
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/-!
|
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|
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## Defining normal order for `FiedOpAlgebra`.
|
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|
|
|
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|
|
@ -280,7 +280,7 @@ lemma koszulSign_swap_eq_rel {ψ φ : 𝓕} (h1 : le φ ψ) (h2 : le ψ φ) : (
|
|||
apply Wick.koszulSignInsert_eq_perm
|
||||
exact List.Perm.append_left φs (List.Perm.swap ψ φ φs')
|
||||
|
||||
lemma koszulSign_eq_rel_eq_stat_append {ψ φ : 𝓕} [IsTrans 𝓕 le] [IsTotal 𝓕 le]
|
||||
lemma koszulSign_eq_rel_eq_stat_append {ψ φ : 𝓕} [IsTrans 𝓕 le]
|
||||
(h1 : le φ ψ) (h2 : le ψ φ) (hq : q ψ = q φ) : (φs : List 𝓕) →
|
||||
koszulSign q le (φ :: ψ :: φs) = koszulSign q le φs := by
|
||||
intro φs
|
||||
|
|
@ -294,7 +294,7 @@ lemma koszulSign_eq_rel_eq_stat_append {ψ φ : 𝓕} [IsTrans 𝓕 le] [IsTotal
|
|||
rw [koszulSignInsert_eq_rel_eq_stat q le h1 h2 hq]
|
||||
simp
|
||||
|
||||
lemma koszulSign_eq_rel_eq_stat {ψ φ : 𝓕} [IsTrans 𝓕 le] [IsTotal 𝓕 le]
|
||||
lemma koszulSign_eq_rel_eq_stat {ψ φ : 𝓕} [IsTrans 𝓕 le]
|
||||
(h1 : le φ ψ) (h2 : le ψ φ) (hq : q ψ = q φ) : (φs' φs : List 𝓕) →
|
||||
koszulSign q le (φs' ++ φ :: ψ :: φs) = koszulSign q le (φs' ++ φs)
|
||||
| [], φs => by
|
||||
|
|
@ -395,7 +395,7 @@ lemma koszulSign_of_append_eq_insertionSort [IsTotal 𝓕 le] [IsTrans 𝓕 le]
|
|||
|
||||
-/
|
||||
|
||||
lemma koszulSign_perm_eq_append [IsTotal 𝓕 le] [IsTrans 𝓕 le] (φ : 𝓕) (φs φs' φs2 : List 𝓕)
|
||||
lemma koszulSign_perm_eq_append [IsTrans 𝓕 le] (φ : 𝓕) (φs φs' φs2 : List 𝓕)
|
||||
(hp : φs.Perm φs') : (h : ∀ φ' ∈ φs, le φ φ' ∧ le φ' φ) →
|
||||
koszulSign q le (φs ++ φs2) = koszulSign q le (φs' ++ φs2) := by
|
||||
let motive (φs φs' : List 𝓕) (hp : φs.Perm φs') : Prop :=
|
||||
|
|
@ -423,7 +423,7 @@ lemma koszulSign_perm_eq_append [IsTotal 𝓕 le] [IsTrans 𝓕 le] (φ : 𝓕)
|
|||
refine h φ' ?_
|
||||
exact (List.Perm.mem_iff (id (List.Perm.symm h1))).mp hφ
|
||||
|
||||
lemma koszulSign_perm_eq [IsTotal 𝓕 le] [IsTrans 𝓕 le] (φ : 𝓕) : (φs1 φs φs' φs2 : List 𝓕) →
|
||||
lemma koszulSign_perm_eq [IsTrans 𝓕 le] (φ : 𝓕) : (φs1 φs φs' φs2 : List 𝓕) →
|
||||
(h : ∀ φ' ∈ φs, le φ φ' ∧ le φ' φ) → (hp : φs.Perm φs') →
|
||||
koszulSign q le (φs1 ++ φs ++ φs2) = koszulSign q le (φs1 ++ φs' ++ φs2)
|
||||
| [], φs, φs', φs2, h, hp => by
|
||||
|
|
|
|||
|
|
@ -247,7 +247,7 @@ lemma koszulSignInsert_of_le_mem (φ0 : 𝓕) : (φs : List 𝓕) → (h : ∀ b
|
|||
exact h b (List.mem_cons_of_mem _ hb)
|
||||
· exact h φ1 (List.mem_cons_self _ _)
|
||||
|
||||
lemma koszulSignInsert_eq_rel_eq_stat {ψ φ : 𝓕} [IsTotal 𝓕 le] [IsTrans 𝓕 le]
|
||||
lemma koszulSignInsert_eq_rel_eq_stat {ψ φ : 𝓕} [IsTrans 𝓕 le]
|
||||
(h1 : le φ ψ) (h2 : le ψ φ) (hq : q ψ = q φ) : (φs : List 𝓕) →
|
||||
koszulSignInsert q le φ φs = koszulSignInsert q le ψ φs
|
||||
| [] => by
|
||||
|
|
@ -268,7 +268,7 @@ lemma koszulSignInsert_eq_rel_eq_stat {ψ φ : 𝓕} [IsTotal 𝓕 le] [IsTrans
|
|||
simp only [hr, ↓reduceIte, hψφ']
|
||||
rw [koszulSignInsert_eq_rel_eq_stat h1 h2 hq φs]
|
||||
|
||||
lemma koszulSignInsert_eq_remove_same_stat_append {ψ φ φ' : 𝓕} [IsTotal 𝓕 le] [IsTrans 𝓕 le]
|
||||
lemma koszulSignInsert_eq_remove_same_stat_append {ψ φ φ' : 𝓕} [IsTrans 𝓕 le]
|
||||
(h1 : le φ ψ) (h2 : le ψ φ) (hq : q ψ = q φ) : (φs : List 𝓕) →
|
||||
koszulSignInsert q le φ' (φ :: ψ :: φs) = koszulSignInsert q le φ' φs := by
|
||||
intro φs
|
||||
|
|
|
|||
Loading…
Add table
Add a link
Reference in a new issue