feat: Fill in sorries
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jstoobysmith
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5 changed files with 1432 additions and 454 deletions
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@ -27,15 +27,15 @@ namespace Wick
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noncomputable section
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class SuperCommuteCenterMap {A : Type} [Semiring A] [Algebra ℂ A]
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(f : FreeAlgebra ℂ I →ₐ[ℂ] A) : Prop where
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prop : ∀ i j, f (superCommute q (FreeAlgebra.ι ℂ i) (FreeAlgebra.ι ℂ j)) ∈ Subalgebra.center ℂ A
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(q : I → Fin 2) (F : FreeAlgebra ℂ I →ₐ[ℂ] A) : Prop where
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prop : ∀ i j, F (superCommute q (FreeAlgebra.ι ℂ i) (FreeAlgebra.ι ℂ j)) ∈ Subalgebra.center ℂ A
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dif_grade : ∀ i j, q i ≠ q j → F (superCommute q (FreeAlgebra.ι ℂ i) (FreeAlgebra.ι ℂ j)) = 0
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namespace SuperCommuteCenterMap
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variable {I : Type} {A : Type} [Semiring A] [Algebra ℂ A]
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(f : FreeAlgebra ℂ I →ₐ[ℂ] A) [SuperCommuteCenterMap f]
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(f : FreeAlgebra ℂ I →ₐ[ℂ] A) (q : I → Fin 2) [SuperCommuteCenterMap q f]
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lemma ofList_fst (q : I → Fin 2) (i j : I) :
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lemma ofList_fst (i j : I) :
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f (superCommute q (ofList [i] xa) (FreeAlgebra.ι ℂ j)) ∈ Subalgebra.center ℂ A := by
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have h1 : f (superCommute q (ofList [i] xa) (FreeAlgebra.ι ℂ j)) =
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xa • f (superCommute q (FreeAlgebra.ι ℂ i) (FreeAlgebra.ι ℂ j)) := by
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@ -51,7 +51,7 @@ lemma ofList_fst (q : I → Fin 2) (i j : I) :
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lemma ofList_freeAlgebraMap {I : Type} {f : I → Type} [∀ i, Fintype (f i)]
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(q : I → Fin 2) (c : (Σ i, f i)) (x : ℂ)
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{A : Type} [Semiring A] [Algebra ℂ A] (F : FreeAlgebra ℂ (Σ i, f i) →ₐ[ℂ] A)
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[SuperCommuteCenterMap F] (b : I) :
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[SuperCommuteCenterMap (fun i => q i.1) F] (b : I) :
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F ((superCommute fun i => q i.fst) (ofList [c] x) ((freeAlgebraMap f) (FreeAlgebra.ι ℂ b)))
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∈ Subalgebra.center ℂ A := by
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rw [freeAlgebraMap_ι]
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@ -64,7 +64,7 @@ end SuperCommuteCenterMap
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lemma superCommuteTake_superCommuteCenterMap {I : Type} (q : I → Fin 2) (lb : List I) (xa xb : ℂ) (n : ℕ)
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(hn : n < lb.length) {A : Type} [Semiring A] [Algebra ℂ A] (f : FreeAlgebra ℂ I →ₐ[ℂ] A)
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[SuperCommuteCenterMap f] (i : I) :
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[SuperCommuteCenterMap q f] (i : I) :
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f (superCommuteTake q [i] lb xa xb n hn) =
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f (superCommute q (ofList [i] xa) (FreeAlgebra.ι ℂ (lb.get ⟨n, hn⟩)))
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* (superCommuteCoef q [i] (List.take n lb) •
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@ -83,7 +83,7 @@ lemma superCommuteTakeM_F {I : Type} {f : I → Type} [∀ i, Fintype (f i)]
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(q : I → Fin 2) (c : (Σ i, f i)) (r : List I) (x y : ℂ) (n : ℕ)
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(hn : n < r.length)
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{A : Type} [Semiring A] [Algebra ℂ A] (F : FreeAlgebra ℂ (Σ i, f i) →ₐ[ℂ] A)
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[SuperCommuteCenterMap F] :
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[SuperCommuteCenterMap (fun i => q i.1) F] :
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F (superCommuteTakeM q [c] r x y n hn) = superCommuteCoefM q [c] (List.take n r) •
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(F (superCommute (fun i => q i.1) (ofList [c] x) (freeAlgebraMap f (FreeAlgebra.ι ℂ (r.get ⟨n, hn⟩))))
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* F (ofListM f (List.eraseIdx r n) y)) := by
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@ -105,104 +105,102 @@ lemma superCommuteTakeM_F {I : Type} {f : I → Type} [∀ i, Fintype (f i)]
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lemma superCommute_koszulOrder_le_ofList {I : Type}
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(q : I → Fin 2) (r : List I) (x : ℂ)
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(le1 :I → I → Prop) [DecidableRel le1]
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(le1 :I → I → Prop) [DecidableRel le1] [IsTotal I le1] [IsTrans I le1]
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(i : I)
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{A : Type} [Semiring A] [Algebra ℂ A]
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(F : FreeAlgebra ℂ I →ₐ A) [SuperCommuteCenterMap F] :
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(F : FreeAlgebra ℂ I →ₐ A) [SuperCommuteCenterMap q F] :
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F ((superCommute q (FreeAlgebra.ι ℂ i) (koszulOrder le1 q (ofList r x)))) =
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∑ n : Fin r.length, (superCommuteCoef q [r.get n] (r.take n)) •
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(F (((superCommute q) (ofList [i] 1)) (FreeAlgebra.ι ℂ (r.get n))) *
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F ((koszulOrder le1 q) (ofList (r.eraseIdx ↑n) x))) := by
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rw [koszulOrder_ofList]
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rw [map_smul, map_smul, ← ofList_singleton]
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rw [superCommute_ofList_sum]
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rw [map_sum]
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rw [← (HepLean.List.insertionSortEquiv le1 r).sum_comp]
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rw [koszulOrder_ofList, map_smul, map_smul, ← ofList_singleton, superCommute_ofList_sum]
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rw [map_sum, ← (HepLean.List.insertionSortEquiv le1 r).sum_comp]
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conv_lhs =>
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rhs
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rhs
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enter [2, 2]
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intro n
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rw [superCommuteTake_superCommuteCenterMap]
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lhs
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rhs
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rhs
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rhs
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enter [1, 2, 2, 2]
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change ((List.insertionSort le1 r).get ∘ (HepLean.List.insertionSortEquiv le1 r)) n
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rw [HepLean.List.insertionSort_get_comp_insertionSortEquiv]
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have hListErase (n : Fin r.length ) : (List.insertionSort le1 r).eraseIdx ↑((HepLean.List.insertionSortEquiv le1 r) n)
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= List.insertionSort le1 (r.eraseIdx n) := by sorry
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conv_lhs =>
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rhs
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rhs
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enter [2, 2]
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intro n
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rw [hListErase]
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rw [HepLean.List.eraseIdx_insertionSort_fin le1 r n]
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rw [ofList_insertionSort_eq_koszulOrder le1 q]
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rw [Finset.smul_sum]
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conv_lhs =>
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rhs
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intro n
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rw [map_smul]
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rw [smul_smul]
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rw [Algebra.mul_smul_comm]
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rw [smul_smul]
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rw [map_smul, smul_smul, Algebra.mul_smul_comm, smul_smul]
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congr
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funext n
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congr 1
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trans superCommuteCoefLE q le1 r i n
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· rw [superCommuteCoefLE]
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rw [mul_assoc]
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exact superCommuteCoefLE_eq_get q le1 r i n
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by_cases hq : q i ≠ q (r.get n)
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· have hn := SuperCommuteCenterMap.dif_grade (q := q) (F := F) i (r.get n) hq
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conv_lhs =>
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enter [2, 1]
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rw [ofList_singleton, hn]
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conv_rhs =>
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enter [2, 1]
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rw [ofList_singleton, hn]
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simp
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· congr 1
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trans superCommuteCoefLE q le1 r i n
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· rw [superCommuteCoefLE, mul_assoc]
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refine superCommuteCoefLE_eq_get q le1 r i n ?_
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simpa using hq
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lemma koszulOrder_of_le_all_ofList {I : Type}
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(q : I → Fin 2) (r : List I) (x : ℂ) (le1 : I → I → Prop) [DecidableRel le1]
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(i : I) (hi : ∀ j, le1 j i)
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{A : Type} [Semiring A] [Algebra ℂ A]
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(F : FreeAlgebra ℂ I →ₐ A) [SuperCommuteCenterMap F] :
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(F : FreeAlgebra ℂ I →ₐ A) [SuperCommuteCenterMap q F] :
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F (koszulOrder le1 q (ofList r x * FreeAlgebra.ι ℂ i))
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= superCommuteCoef q [i] r • F (koszulOrder le1 q (FreeAlgebra.ι ℂ i * ofList r x)) := by
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conv_lhs =>
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rhs
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rhs
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enter [2, 2]
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rw [← ofList_singleton]
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rw [ofListM_ofList_superCommute' q]
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rw [map_sub]
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rw [sub_eq_add_neg]
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rw [map_add]
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conv_lhs =>
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rhs
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rhs
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enter [2, 2]
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rw [map_smul]
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rw [← neg_smul]
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rw [map_smul, map_smul, map_smul]
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sorry
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lemma le_all_mul_koszulOrder_ofList {I : Type}
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(q : I → Fin 2) (r : List I) (x : ℂ) (le1 : I → I→ Prop) [DecidableRel le1]
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(i : I) (hi : ∀ (j : I), le1 j i)
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{A : Type} [Semiring A] [Algebra ℂ A]
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(F : FreeAlgebra ℂ I →ₐ A) [SuperCommuteCenterMap F] :
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(F : FreeAlgebra ℂ I →ₐ A) [SuperCommuteCenterMap q F] :
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F (FreeAlgebra.ι ℂ i * koszulOrder le1 q (ofList r x)) =
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F ((koszulOrder le1 q) (FreeAlgebra.ι ℂ i * ofList r x)) +
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F (((superCommute q) (ofList [i] 1)) ((koszulOrder le1 q) (ofList r x))) := by
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rw [koszulOrder_ofList, Algebra.mul_smul_comm, map_smul, ← ofList_singleton,
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ofList_ofList_superCommute q, map_add, smul_add, ← map_smul]
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conv_lhs =>
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lhs
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rhs
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enter [1, 2]
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rw [← Algebra.smul_mul_assoc, smul_smul, mul_comm, ← smul_smul, ← koszulOrder_ofList,
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Algebra.smul_mul_assoc, ofList_singleton]
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rw [koszulOrder_mul_ge, map_smul]
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congr
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· rw [koszulOrder_of_le_all_ofList]
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sorry
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sorry
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rw [superCommuteCoef_perm_snd q [i] (List.insertionSort le1 r) r
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(List.perm_insertionSort le1 r)]
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rw [smul_smul]
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rw [superCommuteCoef_mul_self]
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simp [ofList_singleton]
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exact fun j => hi j
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· rw [map_smul, map_smul]
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· exact fun j => hi j
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def superCommuteCenterOrder {I : Type}
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(q : I → Fin 2) (r : List I) (i : I)
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{A : Type} [Semiring A] [Algebra ℂ A]
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(F : FreeAlgebra ℂ I →ₐ A) [SuperCommuteCenterMap F]
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(F : FreeAlgebra ℂ I →ₐ A) [SuperCommuteCenterMap q F]
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(n : Option (Fin r.length)) : A :=
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match n with
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| none => 1
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@ -212,7 +210,7 @@ def superCommuteCenterOrder {I : Type}
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lemma superCommuteCenterOrder_none {I : Type}
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(q : I → Fin 2) (r : List I) (i : I)
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{A : Type} [Semiring A] [Algebra ℂ A]
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(F : FreeAlgebra ℂ I →ₐ A) [SuperCommuteCenterMap F] :
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(F : FreeAlgebra ℂ I →ₐ A) [SuperCommuteCenterMap q F] :
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superCommuteCenterOrder q r i F none = 1 := by
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simp [superCommuteCenterOrder]
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@ -220,9 +218,10 @@ open HepLean.List
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lemma le_all_mul_koszulOrder_ofList_expand {I : Type}
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(q : I → Fin 2) (r : List I) (x : ℂ) (le1 : I → I→ Prop) [DecidableRel le1]
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[IsTotal I le1] [IsTrans I le1]
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(i : I) (hi : ∀ (j : I), le1 j i)
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{A : Type} [Semiring A] [Algebra ℂ A]
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(F : FreeAlgebra ℂ I →ₐ A) [SuperCommuteCenterMap F] :
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(F : FreeAlgebra ℂ I →ₐ A) [SuperCommuteCenterMap q F] :
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F (FreeAlgebra.ι ℂ i * koszulOrder le1 q (ofList r x)) =
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∑ n, superCommuteCenterOrder q r i F n * F ((koszulOrder le1 q) (ofList (optionEraseZ r i n) x)) := by
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rw [le_all_mul_koszulOrder_ofList]
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@ -242,9 +241,10 @@ lemma le_all_mul_koszulOrder_ofList_expand {I : Type}
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lemma le_all_mul_koszulOrder_ofListM_expand {I : Type} {f : I → Type} [∀ i, Fintype (f i)]
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(q : I → Fin 2) (r : List I) (x : ℂ) (le1 : (Σ i, f i) → (Σ i, f i) → Prop) [DecidableRel le1]
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[IsTotal (Σ i, f i) le1] [IsTrans (Σ i, f i) le1]
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(i : (Σ i, f i)) (hi : ∀ (j : (Σ i, f i)), le1 j i)
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{A : Type} [Semiring A] [Algebra ℂ A]
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(F : FreeAlgebra ℂ (Σ i, f i) →ₐ A) [SuperCommuteCenterMap F] :
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(F : FreeAlgebra ℂ (Σ i, f i) →ₐ A) [SuperCommuteCenterMap (fun i => q i.1) F] :
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F (ofList [i] 1 * koszulOrder le1 (fun i => q i.1) (ofListM f r x)) =
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F ((koszulOrder le1 fun i => q i.fst) (ofList [i] 1 * ofListM f r x)) +
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∑ n : (Fin r.length), superCommuteCoef q [r.get n] (List.take (↑n) r) •
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@ -318,19 +318,19 @@ lemma le_all_mul_koszulOrder_ofListM_expand {I : Type} {f : I → Type} [∀ i,
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rhs
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intro a0
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rw [← Finset.mul_sum]
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have hl (a0 : f (r0 :: r)[↑n]) (a : (i : Fin r.length) → f (r0 :: r)[↑(n.succAbove i)]):
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have hl (n : Fin (r0 :: r).length) (a0 : f (r0 :: r)[↑n]) (a : (i : Fin r.length) → f (r0 :: r)[↑(n.succAbove i)]):
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(ofList (optionEraseZ (liftM f (r0 :: r) ((liftMCongrEquiv f r0 r n).symm (a0, a))) i (some (Fin.cast (by simp ) n))) x)
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= ofList ((liftM f ((r0 :: r).eraseIdx ↑n) ((listMEraseEquiv q n).symm a))) x := by
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simp only [optionEraseZ, List.get_eq_getElem, List.length_cons, Fin.coe_cast]
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simp [liftMCongrEquiv]
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congr
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sorry
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conv_lhs =>
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rhs
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intro a0
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rhs
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rhs
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enter [2, 2]
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intro a
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erw [hl a0 a]
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erw [hl n a0 a]
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rw [← Finset.sum_mul]
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conv_lhs =>
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lhs
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