refactor: Last batch of multi-goal proofs
This commit is contained in:
jstoobysmith
parent
b9479c904d
commit
c0499483a8
43 changed files with 910 additions and 955 deletions
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@ -77,9 +77,8 @@ lemma accCube_explicit (n : ℕ) (S : (PureU1Charges n).Charges) :
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rw [accCubeTriLinSymm]
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simp only [PureU1Charges_numberCharges, TriLinearSymm.mk₃_toFun_apply_apply]
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apply Finset.sum_congr
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simp only
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ring_nf
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simp
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· rfl
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· exact fun x _ => Eq.symm (pow_three' (S x))
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end PureU1
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@ -130,42 +129,40 @@ lemma pureU1_anomalyFree_ext {n : ℕ} {S T : (PureU1 n.succ).LinSols}
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apply ACCSystemLinear.LinSols.ext
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funext i
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by_cases hi : i ≠ Fin.last n
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have hiCast : ∃ j : Fin n, j.castSucc = i := by
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exact Fin.exists_castSucc_eq.mpr hi
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obtain ⟨j, hj⟩ := hiCast
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rw [← hj]
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exact h j
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simp at hi
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rw [hi]
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rw [pureU1_last, pureU1_last]
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simp only [neg_inj]
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apply Finset.sum_congr
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simp only
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intro j _
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exact h j
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· have hiCast : ∃ j : Fin n, j.castSucc = i := by
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exact Fin.exists_castSucc_eq.mpr hi
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obtain ⟨j, hj⟩ := hiCast
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rw [← hj]
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exact h j
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· simp at hi
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rw [hi, pureU1_last, pureU1_last]
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simp only [neg_inj]
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apply Finset.sum_congr
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· simp only
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· exact fun j _ => h j
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namespace PureU1
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lemma sum_of_charges {n : ℕ} (f : Fin k → (PureU1 n).Charges) (j : Fin n) :
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(∑ i : Fin k, (f i)) j = ∑ i : Fin k, (f i) j := by
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induction k
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simp
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rfl
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rename_i k hl
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rw [Fin.sum_univ_castSucc, Fin.sum_univ_castSucc]
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have hlt := hl (f ∘ Fin.castSucc)
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erw [← hlt]
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simp
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· simp only [univ_eq_empty, sum_empty]
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rfl
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· rename_i k hl
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rw [Fin.sum_univ_castSucc, Fin.sum_univ_castSucc]
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have hlt := hl (f ∘ Fin.castSucc)
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erw [← hlt]
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simp
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lemma sum_of_anomaly_free_linear {n : ℕ} (f : Fin k → (PureU1 n).LinSols) (j : Fin n) :
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(∑ i : Fin k, (f i)).1 j = (∑ i : Fin k, (f i).1 j) := by
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induction k
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simp
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rfl
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rename_i k hl
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rw [Fin.sum_univ_castSucc, Fin.sum_univ_castSucc]
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have hlt := hl (f ∘ Fin.castSucc)
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erw [← hlt]
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simp
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· simp only [univ_eq_empty, sum_empty, ACCSystemLinear.linSolsAddCommMonoid_zero_val]
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rfl
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· rename_i k hl
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rw [Fin.sum_univ_castSucc, Fin.sum_univ_castSucc]
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have hlt := hl (f ∘ Fin.castSucc)
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erw [← hlt]
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simp
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end PureU1
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@ -38,16 +38,16 @@ lemma asCharges_ne_castSucc {k j : Fin n} (h : k ≠ j) :
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asCharges k (Fin.castSucc j) = 0 := by
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simp [asCharges]
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split
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rename_i h1
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exact False.elim (h (id (Eq.symm h1)))
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split
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rename_i h1 h2
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rw [Fin.ext_iff] at h1 h2
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simp at h1 h2
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have hj : j.val < n := by
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exact j.prop
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simp_all
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rfl
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· rename_i h1
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exact False.elim (h (id (Eq.symm h1)))
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· split
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· rename_i h1 h2
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rw [Fin.ext_iff] at h1 h2
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simp at h1 h2
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have hj : j.val < n := by
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exact j.prop
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simp_all
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· rfl
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/-- The basis elements as `LinSols`. -/
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@[simps!]
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@ -61,16 +61,16 @@ def asLinSols (j : Fin n) : (PureU1 n.succ).LinSols :=
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LinearMap.coe_mk, AddHom.coe_mk, Fin.coe_eq_castSucc]
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rw [Fin.sum_univ_castSucc]
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rw [Finset.sum_eq_single j]
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simp only [asCharges, PureU1_numberCharges, ↓reduceIte]
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have hn : ¬ (Fin.last n = Fin.castSucc j) := Fin.ne_of_gt j.prop
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split
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rename_i ht
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exact (hn ht).elim
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rfl
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intro k _ hkj
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exact asCharges_ne_castSucc hkj.symm
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intro hk
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simp at hk⟩
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· simp only [asCharges, PureU1_numberCharges, ↓reduceIte]
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have hn : ¬ (Fin.last n = Fin.castSucc j) := Fin.ne_of_gt j.prop
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split
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· rename_i ht
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exact (hn ht).elim
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· rfl
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· intro k _ hkj
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exact asCharges_ne_castSucc hkj.symm
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· intro hk
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simp at hk⟩
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lemma sum_of_vectors {n : ℕ} (f : Fin k → (PureU1 n).LinSols) (j : Fin n) :
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(∑ i : Fin k, (f i)).1 j = (∑ i : Fin k, (f i).1 j) :=
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@ -94,11 +94,11 @@ def coordinateMap : (PureU1 n.succ).LinSols ≃ₗ[ℚ] Fin n →₀ ℚ where
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asLinSols_val, Equiv.toFun_as_coe,
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Fin.coe_eq_castSucc, mul_ite, mul_one, mul_neg, mul_zero, Equiv.invFun_as_coe]
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rw [Finset.sum_eq_single j]
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simp only [asCharges, PureU1_numberCharges, ↓reduceIte, mul_one]
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intro k _ hkj
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rw [asCharges_ne_castSucc hkj]
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exact Rat.mul_zero (S.val k.castSucc)
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simp
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· simp only [asCharges, PureU1_numberCharges, ↓reduceIte, mul_one]
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· intro k _ hkj
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rw [asCharges_ne_castSucc hkj]
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exact Rat.mul_zero (S.val k.castSucc)
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· simp
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right_inv f := by
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simp only [PureU1_numberCharges, Equiv.invFun_as_coe]
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ext
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@ -109,11 +109,11 @@ def coordinateMap : (PureU1 n.succ).LinSols ≃ₗ[ℚ] Fin n →₀ ℚ where
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asLinSols_val, Equiv.toFun_as_coe,
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Fin.coe_eq_castSucc, mul_ite, mul_one, mul_neg, mul_zero, Equiv.invFun_as_coe]
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rw [Finset.sum_eq_single j]
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simp only [asCharges, PureU1_numberCharges, ↓reduceIte, mul_one]
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intro k _ hkj
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rw [asCharges_ne_castSucc hkj]
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exact Rat.mul_zero (f k)
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simp
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· simp only [asCharges, PureU1_numberCharges, ↓reduceIte, mul_one]
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· intro k _ hkj
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rw [asCharges_ne_castSucc hkj]
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exact Rat.mul_zero (f k)
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· simp
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/-- The basis of `LinSols`. -/
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noncomputable
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@ -32,13 +32,11 @@ lemma constAbs_perm (S : (PureU1 n).Charges) (M :(FamilyPermutations n).group) :
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ConstAbs ((FamilyPermutations n).rep M S) ↔ ConstAbs S := by
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simp only [ConstAbs, PureU1_numberCharges, FamilyPermutations, PermGroup, permCharges,
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MonoidHom.coe_mk, OneHom.coe_mk, chargeMap_apply]
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apply Iff.intro
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intro h i j
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refine Iff.intro (fun h i j => ?_) (fun h i j => h (M.invFun i) (M.invFun j))
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have h2 := h (M.toFun i) (M.toFun j)
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simp at h2
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exact h2
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intro h i j
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exact h (M.invFun i) (M.invFun j)
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lemma constAbs_sort {S : (PureU1 n).Charges} (CA : ConstAbs S) : ConstAbs (sort S) := by
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rw [sort]
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@ -58,8 +56,8 @@ lemma lt_eq {k i : Fin n.succ} (hk : S k ≤ 0) (hik : i ≤ k) : S i = S k := b
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have ht := hS.1 i k
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rw [sq_eq_sq_iff_eq_or_eq_neg] at ht
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cases ht <;> rename_i h
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exact h
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linarith
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· exact h
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· linarith
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lemma val_le_zero {i : Fin n.succ} (hi : S i ≤ 0) : S i = S (0 : Fin n.succ) := by
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symm
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@ -71,8 +69,8 @@ lemma gt_eq {k i: Fin n.succ} (hk : 0 ≤ S k) (hik : k ≤ i) : S i = S k := by
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have ht := hS.1 i k
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rw [sq_eq_sq_iff_eq_or_eq_neg] at ht
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cases ht <;> rename_i h
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exact h
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linarith
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· exact h
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· linarith
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lemma zero_gt (h0 : 0 ≤ S (0 : Fin n.succ)) (i : Fin n.succ) : S (0 : Fin n.succ) = S i := by
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symm
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@ -83,9 +81,9 @@ lemma opposite_signs_eq_neg {i j : Fin n.succ} (hi : S i ≤ 0) (hj : 0 ≤ S j)
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have hSS := hS.1 i j
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rw [sq_eq_sq_iff_eq_or_eq_neg] at hSS
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cases' hSS with h h
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simp_all
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linarith
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exact h
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· simp_all
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linarith
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· exact h
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lemma is_zero (h0 : S (0 : Fin n.succ) = 0) : S = 0 := by
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funext i
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@ -115,10 +113,10 @@ lemma boundary_accGrav' (k : Fin n) : accGrav n.succ S =
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∑ i : Fin (k.succ.val + (n.succ - k.succ.val)), S (Fin.cast (boundary_split k) i) := by
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simp [accGrav]
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erw [Finset.sum_equiv (Fin.castOrderIso (boundary_split k)).toEquiv]
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intro i
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simp only [Fin.val_succ, mem_univ, RelIso.coe_fn_toEquiv]
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intro i
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simp
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· intro i
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simp only [Fin.val_succ, mem_univ, RelIso.coe_fn_toEquiv]
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· intro i
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simp
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lemma boundary_accGrav'' (k : Fin n) (hk : Boundary S k) :
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accGrav n.succ S = (2 * ↑↑k + 1 - ↑n) * S (0 : Fin n.succ) := by
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@ -146,19 +144,19 @@ lemma not_hasBoundary_zero_le (hnot : ¬ (HasBoundary S)) (h0 : S (0 : Fin n.suc
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intro ⟨i, hi⟩
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simp at hnot
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induction i
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rfl
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rename_i i hii
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have hnott := hnot ⟨i, by {simp at hi; linarith}⟩
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have hii := hii (by omega)
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erw [← hii] at hnott
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exact (val_le_zero hS (hnott h0)).symm
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· rfl
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· rename_i i hii
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have hnott := hnot ⟨i, by {simp at hi; linarith}⟩
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have hii := hii (by omega)
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erw [← hii] at hnott
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exact (val_le_zero hS (hnott h0)).symm
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lemma not_hasBoundry_zero (hnot : ¬ (HasBoundary S)) (i : Fin n.succ) :
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S (0 : Fin n.succ) = S i := by
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by_cases hi : S (0 : Fin n.succ) < 0
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exact not_hasBoundary_zero_le hS hnot hi i
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simp at hi
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exact zero_gt hS hi i
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· exact not_hasBoundary_zero_le hS hnot hi i
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· simp at hi
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exact zero_gt hS hi i
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lemma not_hasBoundary_grav (hnot : ¬ (HasBoundary S)) :
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accGrav n.succ S = n.succ * S (0 : Fin n.succ) := by
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@ -171,8 +169,8 @@ lemma AFL_hasBoundary (h : A.val (0 : Fin n.succ) ≠ 0) : HasBoundary A.val :=
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erw [pureU1_linear A] at h0
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simp at h0
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cases' h0
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linarith
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simp_all
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· linarith
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· simp_all
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lemma AFL_odd_noBoundary {A : (PureU1 (2 * n + 1)).LinSols} (h : ConstAbsSorted A.val)
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(hA : A.val (0 : Fin (2*n +1)) ≠ 0) : ¬ HasBoundary A.val := by
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@ -224,9 +222,9 @@ lemma AFL_even_below (A : (PureU1 (2 * n.succ)).LinSols) (h : ConstAbsSorted A.v
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A.val (Fin.cast (split_equal n.succ) (Fin.castAdd n.succ i))
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= A.val (0 : Fin (2*n.succ)) := by
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by_cases hA : A.val (0 : Fin (2*n.succ)) = 0
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rw [is_zero h hA]
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rfl
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exact AFL_even_below' h hA i
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· rw [is_zero h hA]
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rfl
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· exact AFL_even_below' h hA i
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lemma AFL_even_above' {A : (PureU1 (2 * n.succ)).LinSols} (h : ConstAbsSorted A.val)
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(hA : A.val (0 : Fin (2*n.succ)) ≠ 0) (i : Fin n.succ) :
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@ -245,9 +243,9 @@ lemma AFL_even_above (A : (PureU1 (2 * n.succ)).LinSols) (h : ConstAbsSorted A.v
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A.val (Fin.cast (split_equal n.succ) (Fin.natAdd n.succ i)) =
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- A.val (0 : Fin (2 * n.succ)) := by
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by_cases hA : A.val (0 : Fin (2 * n.succ)) = 0
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rw [is_zero h hA]
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rfl
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exact AFL_even_above' h hA i
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· rw [is_zero h hA]
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rfl
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· exact AFL_even_above' h hA i
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end charges
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@ -54,19 +54,19 @@ lemma ext_δ (S T : Fin (2 * n.succ) → ℚ) (h1 : ∀ i, S (δ₁ i) = T (δ
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(h2 : ∀ i, S (δ₂ i) = T (δ₂ i)) : S = T := by
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funext i
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by_cases hi : i.val < n.succ
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let j : Fin n.succ := ⟨i, hi⟩
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have h2 := h1 j
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have h3 : δ₁ j = i := by
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simp [δ₁, Fin.ext_iff]
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rw [h3] at h2
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exact h2
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let j : Fin n.succ := ⟨i - n.succ, by omega⟩
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have h2 := h2 j
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have h3 : δ₂ j = i := by
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simp [δ₂, Fin.ext_iff]
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omega
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rw [h3] at h2
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exact h2
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· let j : Fin n.succ := ⟨i, hi⟩
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have h2 := h1 j
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have h3 : δ₁ j = i := by
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simp [δ₁, Fin.ext_iff]
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rw [h3] at h2
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exact h2
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· let j : Fin n.succ := ⟨i - n.succ, by omega⟩
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have h2 := h2 j
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have h3 : δ₂ j = i := by
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simp [δ₂, Fin.ext_iff]
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omega
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rw [h3] at h2
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exact h2
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lemma sum_δ₁_δ₂ (S : Fin (2 * n.succ) → ℚ) :
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∑ i, S i = ∑ i : Fin n.succ, ((S ∘ δ₁) i + (S ∘ δ₂) i) := by
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@ -167,18 +167,18 @@ lemma basis_on_δ₁_other {k j : Fin n.succ} (h : k ≠ j) :
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simp [basisAsCharges]
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simp [δ₁, δ₂]
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split
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rename_i h1
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rw [Fin.ext_iff] at h1
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simp_all
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rw [Fin.ext_iff] at h
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simp_all
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split
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rename_i h1 h2
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simp_all
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rw [Fin.ext_iff] at h2
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simp at h2
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omega
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rfl
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· rename_i h1
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rw [Fin.ext_iff] at h1
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simp_all
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rw [Fin.ext_iff] at h
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simp_all
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· split
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· rename_i h1 h2
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simp_all
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rw [Fin.ext_iff] at h2
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simp at h2
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omega
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· rfl
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lemma basis_on_other {k : Fin n.succ} {j : Fin (2 * n.succ)} (h1 : j ≠ δ₁ k) (h2 : j ≠ δ₂ k) :
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basisAsCharges k j = 0 := by
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@ -195,18 +195,18 @@ lemma basis!_on_δ!₁_other {k j : Fin n} (h : k ≠ j) :
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simp [basis!AsCharges]
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simp [δ!₁, δ!₂]
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split
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rename_i h1
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rw [Fin.ext_iff] at h1
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simp_all
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rw [Fin.ext_iff] at h
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simp_all
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split
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rename_i h1 h2
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simp_all
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rw [Fin.ext_iff] at h2
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simp at h2
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omega
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rfl
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· rename_i h1
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rw [Fin.ext_iff] at h1
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simp_all
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rw [Fin.ext_iff] at h
|
||||
simp_all
|
||||
· split
|
||||
· rename_i h1 h2
|
||||
simp_all
|
||||
rw [Fin.ext_iff] at h2
|
||||
simp at h2
|
||||
omega
|
||||
· rfl
|
||||
|
||||
lemma basis_δ₂_eq_minus_δ₁ (j i : Fin n.succ) :
|
||||
basisAsCharges j (δ₂ i) = - basisAsCharges j (δ₁ i) := by
|
||||
|
|
@ -233,8 +233,8 @@ lemma basis!_δ!₂_eq_minus_δ!₁ (j i : Fin n) :
|
|||
all_goals rename_i h1 h2
|
||||
all_goals rw [Fin.ext_iff] at h1 h2
|
||||
all_goals simp_all
|
||||
subst h1
|
||||
exact Fin.elim0 i
|
||||
· subst h1
|
||||
exact Fin.elim0 i
|
||||
all_goals rename_i h3
|
||||
all_goals rw [Fin.ext_iff] at h3
|
||||
all_goals simp_all
|
||||
|
|
@ -256,27 +256,27 @@ lemma basis!_on_δ!₂_other {k j : Fin n} (h : k ≠ j) : basis!AsCharges k (δ
|
|||
|
||||
lemma basis!_on_δ!₃ (j : Fin n) : basis!AsCharges j δ!₃ = 0 := by
|
||||
simp [basis!AsCharges]
|
||||
split <;> rename_i h
|
||||
rw [Fin.ext_iff] at h
|
||||
simp [δ!₃, δ!₁] at h
|
||||
omega
|
||||
split <;> rename_i h2
|
||||
rw [Fin.ext_iff] at h2
|
||||
simp [δ!₃, δ!₂] at h2
|
||||
omega
|
||||
rfl
|
||||
split<;> rename_i h
|
||||
· simp only [δ!₃, succ_eq_add_one, Fin.isValue, δ!₁, Fin.ext_iff, Fin.coe_cast, Fin.coe_castAdd,
|
||||
Fin.coe_fin_one, Fin.coe_natAdd] at h
|
||||
omega
|
||||
· split <;> rename_i h2
|
||||
· simp only [δ!₃, succ_eq_add_one, Fin.isValue, δ!₂, Fin.ext_iff, Fin.coe_cast,
|
||||
Fin.coe_castAdd, Fin.coe_fin_one, Fin.coe_natAdd] at h2
|
||||
omega
|
||||
· rfl
|
||||
|
||||
lemma basis!_on_δ!₄ (j : Fin n) : basis!AsCharges j δ!₄ = 0 := by
|
||||
simp [basis!AsCharges]
|
||||
split <;> rename_i h
|
||||
rw [Fin.ext_iff] at h
|
||||
simp [δ!₄, δ!₁] at h
|
||||
omega
|
||||
split <;> rename_i h2
|
||||
rw [Fin.ext_iff] at h2
|
||||
simp [δ!₄, δ!₂] at h2
|
||||
omega
|
||||
rfl
|
||||
· rw [Fin.ext_iff] at h
|
||||
simp [δ!₄, δ!₁] at h
|
||||
omega
|
||||
· split <;> rename_i h2
|
||||
· rw [Fin.ext_iff] at h2
|
||||
simp [δ!₄, δ!₂] at h2
|
||||
omega
|
||||
· rfl
|
||||
|
||||
lemma basis_linearACC (j : Fin n.succ) : (accGrav (2 * n.succ)) (basisAsCharges j) = 0 := by
|
||||
rw [accGrav]
|
||||
|
|
@ -343,16 +343,15 @@ lemma swap!_as_add {S S' : (PureU1 (2 * n.succ)).LinSols} (j : Fin n)
|
|||
funext i
|
||||
rw [← hS, FamilyPermutations_anomalyFreeLinear_apply]
|
||||
by_cases hi : i = δ!₁ j
|
||||
subst hi
|
||||
simp [HSMul.hSMul, basis!_on_δ!₁_self, pairSwap_inv_fst]
|
||||
by_cases hi2 : i = δ!₂ j
|
||||
subst hi2
|
||||
simp [HSMul.hSMul, basis!_on_δ!₂_self, pairSwap_inv_snd]
|
||||
simp [HSMul.hSMul]
|
||||
rw [basis!_on_other hi hi2]
|
||||
change S.val ((pairSwap (δ!₁ j) (δ!₂ j)).invFun i) =_
|
||||
erw [pairSwap_inv_other (Ne.symm hi) (Ne.symm hi2)]
|
||||
simp
|
||||
· subst hi
|
||||
simp [HSMul.hSMul, basis!_on_δ!₁_self, pairSwap_inv_fst]
|
||||
· by_cases hi2 : i = δ!₂ j
|
||||
· simp [HSMul.hSMul, hi2, basis!_on_δ!₂_self, pairSwap_inv_snd]
|
||||
· simp [HSMul.hSMul]
|
||||
rw [basis!_on_other hi hi2]
|
||||
change S.val ((pairSwap (δ!₁ j) (δ!₂ j)).invFun i) =_
|
||||
erw [pairSwap_inv_other (Ne.symm hi) (Ne.symm hi2)]
|
||||
simp
|
||||
|
||||
/-- A point in the span of the first part of the basis as a charge. -/
|
||||
def P (f : Fin n.succ → ℚ) : (PureU1 (2 * n.succ)).Charges := ∑ i, f i • basisAsCharges i
|
||||
|
|
@ -367,23 +366,23 @@ lemma P_δ₁ (f : Fin n.succ → ℚ) (j : Fin n.succ) : P f (δ₁ j) = f j :=
|
|||
rw [P, sum_of_charges]
|
||||
simp [HSMul.hSMul, SMul.smul]
|
||||
rw [Finset.sum_eq_single j]
|
||||
rw [basis_on_δ₁_self]
|
||||
simp only [mul_one]
|
||||
intro k _ hkj
|
||||
rw [basis_on_δ₁_other hkj]
|
||||
simp only [mul_zero]
|
||||
simp only [mem_univ, not_true_eq_false, _root_.mul_eq_zero, IsEmpty.forall_iff]
|
||||
· rw [basis_on_δ₁_self]
|
||||
simp only [mul_one]
|
||||
· intro k _ hkj
|
||||
rw [basis_on_δ₁_other hkj]
|
||||
simp only [mul_zero]
|
||||
· simp only [mem_univ, not_true_eq_false, _root_.mul_eq_zero, IsEmpty.forall_iff]
|
||||
|
||||
lemma P!_δ!₁ (f : Fin n → ℚ) (j : Fin n) : P! f (δ!₁ j) = f j := by
|
||||
rw [P!, sum_of_charges]
|
||||
simp [HSMul.hSMul, SMul.smul]
|
||||
rw [Finset.sum_eq_single j]
|
||||
rw [basis!_on_δ!₁_self]
|
||||
simp only [mul_one]
|
||||
intro k _ hkj
|
||||
rw [basis!_on_δ!₁_other hkj]
|
||||
simp only [mul_zero]
|
||||
simp only [mem_univ, not_true_eq_false, _root_.mul_eq_zero, IsEmpty.forall_iff]
|
||||
· rw [basis!_on_δ!₁_self]
|
||||
simp only [mul_one]
|
||||
· intro k _ hkj
|
||||
rw [basis!_on_δ!₁_other hkj]
|
||||
simp only [mul_zero]
|
||||
· simp only [mem_univ, not_true_eq_false, _root_.mul_eq_zero, IsEmpty.forall_iff]
|
||||
|
||||
lemma Pa_δ!₁ (f : Fin n.succ → ℚ) (g : Fin n → ℚ) (j : Fin n) :
|
||||
Pa f g (δ!₁ j) = f j.succ + g j := by
|
||||
|
|
@ -395,23 +394,21 @@ lemma P_δ₂ (f : Fin n.succ → ℚ) (j : Fin n.succ) : P f (δ₂ j) = - f j
|
|||
rw [P, sum_of_charges]
|
||||
simp [HSMul.hSMul, SMul.smul]
|
||||
rw [Finset.sum_eq_single j]
|
||||
rw [basis_on_δ₂_self]
|
||||
simp only [mul_neg, mul_one]
|
||||
intro k _ hkj
|
||||
rw [basis_on_δ₂_other hkj]
|
||||
simp only [mul_zero]
|
||||
simp
|
||||
· simp only [basis_on_δ₂_self, mul_neg, mul_one]
|
||||
· intro k _ hkj
|
||||
simp only [basis_on_δ₂_other hkj, mul_zero]
|
||||
· simp
|
||||
|
||||
lemma P!_δ!₂ (f : Fin n → ℚ) (j : Fin n) : P! f (δ!₂ j) = - f j := by
|
||||
rw [P!, sum_of_charges]
|
||||
simp [HSMul.hSMul, SMul.smul]
|
||||
rw [Finset.sum_eq_single j]
|
||||
rw [basis!_on_δ!₂_self]
|
||||
simp only [mul_neg, mul_one]
|
||||
intro k _ hkj
|
||||
rw [basis!_on_δ!₂_other hkj]
|
||||
simp only [mul_zero]
|
||||
simp
|
||||
· rw [basis!_on_δ!₂_self]
|
||||
simp only [mul_neg, mul_one]
|
||||
· intro k _ hkj
|
||||
rw [basis!_on_δ!₂_other hkj]
|
||||
simp only [mul_zero]
|
||||
· simp
|
||||
|
||||
lemma Pa_δ!₂ (f : Fin n.succ → ℚ) (g : Fin n → ℚ) (j : Fin n) :
|
||||
Pa f g (δ!₂ j) = - f j.castSucc - g j := by
|
||||
|
|
@ -474,13 +471,13 @@ lemma P_P_P!_accCube (g : Fin n.succ → ℚ) (j : Fin n) :
|
|||
rw [sum_δ!₁_δ!₂, basis!_on_δ!₃, basis!_on_δ!₄]
|
||||
simp only [mul_zero, add_zero, Function.comp_apply, zero_add]
|
||||
rw [Finset.sum_eq_single j, basis!_on_δ!₁_self, basis!_on_δ!₂_self]
|
||||
simp [δ!₁_δ₁, δ!₂_δ₂]
|
||||
rw [P_δ₁, P_δ₂]
|
||||
ring
|
||||
intro k _ hkj
|
||||
erw [basis!_on_δ!₁_other hkj.symm, basis!_on_δ!₂_other hkj.symm]
|
||||
simp only [mul_zero, add_zero]
|
||||
simp
|
||||
· simp [δ!₁_δ₁, δ!₂_δ₂]
|
||||
rw [P_δ₁, P_δ₂]
|
||||
ring
|
||||
· intro k _ hkj
|
||||
erw [basis!_on_δ!₁_other hkj.symm, basis!_on_δ!₂_other hkj.symm]
|
||||
simp only [mul_zero, add_zero]
|
||||
· simp
|
||||
|
||||
lemma P_P!_P!_accCube (g : Fin n → ℚ) (j : Fin n.succ) :
|
||||
accCubeTriLinSymm (P! g) (P! g) (basisAsCharges j)
|
||||
|
|
@ -489,12 +486,12 @@ lemma P_P!_P!_accCube (g : Fin n → ℚ) (j : Fin n.succ) :
|
|||
rw [sum_δ₁_δ₂]
|
||||
simp only [Function.comp_apply]
|
||||
rw [Finset.sum_eq_single j, basis_on_δ₁_self, basis_on_δ₂_self]
|
||||
simp [δ!₁_δ₁, δ!₂_δ₂]
|
||||
ring
|
||||
intro k _ hkj
|
||||
erw [basis_on_δ₁_other hkj.symm, basis_on_δ₂_other hkj.symm]
|
||||
simp only [mul_zero, add_zero]
|
||||
simp
|
||||
· simp [δ!₁_δ₁, δ!₂_δ₂]
|
||||
ring
|
||||
· intro k _ hkj
|
||||
erw [basis_on_δ₁_other hkj.symm, basis_on_δ₂_other hkj.symm]
|
||||
simp only [mul_zero, add_zero]
|
||||
· simp
|
||||
|
||||
lemma P_zero (f : Fin n.succ → ℚ) (h : P f = 0) : ∀ i, f i = 0 := by
|
||||
intro i
|
||||
|
|
@ -601,27 +598,24 @@ theorem basisa_linear_independent : LinearIndependent ℚ (@basisa n) := by
|
|||
intro i
|
||||
simp_all
|
||||
cases i
|
||||
simp_all
|
||||
simp_all
|
||||
· simp_all
|
||||
· simp_all
|
||||
|
||||
lemma Pa'_eq (f f' : (Fin n.succ) ⊕ (Fin n) → ℚ) : Pa' f = Pa' f' ↔ f = f' := by
|
||||
apply Iff.intro
|
||||
intro h
|
||||
funext i
|
||||
rw [Pa', Pa'] at h
|
||||
have h1 : ∑ i : Fin (succ n) ⊕ Fin n, (f i + (- f' i)) • basisa i = 0 := by
|
||||
simp only [add_smul, neg_smul]
|
||||
rw [Finset.sum_add_distrib]
|
||||
rw [h]
|
||||
rw [← Finset.sum_add_distrib]
|
||||
simp
|
||||
have h2 : ∀ i, (f i + (- f' i)) = 0 := by
|
||||
exact Fintype.linearIndependent_iff.mp (@basisa_linear_independent n)
|
||||
(fun i => f i + -f' i) h1
|
||||
have h2i := h2 i
|
||||
linarith
|
||||
intro h
|
||||
rw [h]
|
||||
refine Iff.intro (fun h => (funext (fun i => ?_))) (fun h => ?_)
|
||||
· rw [Pa', Pa'] at h
|
||||
have h1 : ∑ i : Fin (succ n) ⊕ Fin n, (f i + (- f' i)) • basisa i = 0 := by
|
||||
simp only [add_smul, neg_smul]
|
||||
rw [Finset.sum_add_distrib]
|
||||
rw [h]
|
||||
rw [← Finset.sum_add_distrib]
|
||||
simp
|
||||
have h2 : ∀ i, (f i + (- f' i)) = 0 := by
|
||||
exact Fintype.linearIndependent_iff.mp (@basisa_linear_independent n)
|
||||
(fun i => f i + -f' i) h1
|
||||
have h2i := h2 i
|
||||
linarith
|
||||
· rw [h]
|
||||
|
||||
/-! TODO: Replace the definition of `join` with a Mathlib definition, most likely `Sum.elim`. -/
|
||||
/-- A helper function for what follows. -/
|
||||
|
|
@ -632,33 +626,24 @@ def join (g : Fin n.succ → ℚ) (f : Fin n → ℚ) : (Fin n.succ) ⊕ (Fin n)
|
|||
|
||||
lemma join_ext (g g' : Fin n.succ → ℚ) (f f' : Fin n → ℚ) :
|
||||
join g f = join g' f' ↔ g = g' ∧ f = f' := by
|
||||
apply Iff.intro
|
||||
intro h
|
||||
apply And.intro
|
||||
funext i
|
||||
exact congr_fun h (Sum.inl i)
|
||||
funext i
|
||||
exact congr_fun h (Sum.inr i)
|
||||
intro h
|
||||
rw [h.left, h.right]
|
||||
refine Iff.intro (fun h => ?_) (fun h => ?_)
|
||||
· apply And.intro (funext (fun i => congr_fun h (Sum.inl i)))
|
||||
(funext (fun i => congr_fun h (Sum.inr i)))
|
||||
· rw [h.left, h.right]
|
||||
|
||||
lemma join_Pa (g g' : Fin n.succ → ℚ) (f f' : Fin n → ℚ) :
|
||||
Pa' (join g f) = Pa' (join g' f') ↔ Pa g f = Pa g' f' := by
|
||||
apply Iff.intro
|
||||
intro h
|
||||
rw [Pa'_eq] at h
|
||||
rw [join_ext] at h
|
||||
rw [h.left, h.right]
|
||||
intro h
|
||||
apply ACCSystemLinear.LinSols.ext
|
||||
rw [Pa'_P'_P!', Pa'_P'_P!']
|
||||
simp [P'_val, P!'_val]
|
||||
exact h
|
||||
refine Iff.intro (fun h => ?_) (fun h => ?_)
|
||||
· rw [Pa'_eq, join_ext] at h
|
||||
rw [h.left, h.right]
|
||||
· apply ACCSystemLinear.LinSols.ext
|
||||
rw [Pa'_P'_P!', Pa'_P'_P!']
|
||||
simp only [succ_eq_add_one, ACCSystemLinear.linSolsAddCommMonoid_add_val, P'_val, P!'_val]
|
||||
exact h
|
||||
|
||||
lemma Pa_eq (g g' : Fin n.succ → ℚ) (f f' : Fin n → ℚ) :
|
||||
Pa g f = Pa g' f' ↔ g = g' ∧ f = f' := by
|
||||
rw [← join_Pa]
|
||||
rw [← join_ext]
|
||||
rw [← join_Pa, ← join_ext]
|
||||
exact Pa'_eq _ _
|
||||
|
||||
lemma basisa_card : Fintype.card ((Fin n.succ) ⊕ (Fin n)) =
|
||||
|
|
@ -694,8 +679,8 @@ lemma smul_basis!AsCharges_in_span (S : (PureU1 (2 * n.succ)).LinSols) (j : Fin
|
|||
Submodule.span ℚ (Set.range basis!AsCharges) := by
|
||||
apply Submodule.smul_mem
|
||||
apply SetLike.mem_of_subset
|
||||
apply Submodule.subset_span
|
||||
simp_all only [Set.mem_range, exists_apply_eq_apply]
|
||||
· exact Submodule.subset_span
|
||||
· simp_all only [Set.mem_range, exists_apply_eq_apply]
|
||||
|
||||
lemma span_basis_swap! {S : (PureU1 (2 * n.succ)).LinSols} (j : Fin n)
|
||||
(hS : ((FamilyPermutations (2 * n.succ)).linSolRep
|
||||
|
|
@ -732,19 +717,19 @@ lemma vectorLikeEven_in_span (S : (PureU1 (2 * n.succ)).LinSols)
|
|||
rw [sortAFL_val]
|
||||
erw [P'_val]
|
||||
apply ext_δ
|
||||
intro i
|
||||
rw [P_δ₁]
|
||||
rfl
|
||||
intro i
|
||||
rw [P_δ₂]
|
||||
have ht := hS i
|
||||
change sort S.val (δ₁ i) = - sort S.val (δ₂ i) at ht
|
||||
have h : sort S.val (δ₂ i) = - sort S.val (δ₁ i) := by
|
||||
rw [ht]
|
||||
ring
|
||||
rw [h]
|
||||
simp
|
||||
rfl
|
||||
· intro i
|
||||
rw [P_δ₁]
|
||||
rfl
|
||||
· intro i
|
||||
rw [P_δ₂]
|
||||
have ht := hS i
|
||||
change sort S.val (δ₁ i) = - sort S.val (δ₂ i) at ht
|
||||
have h : sort S.val (δ₂ i) = - sort S.val (δ₁ i) := by
|
||||
rw [ht]
|
||||
ring
|
||||
rw [h]
|
||||
simp
|
||||
rfl
|
||||
|
||||
end VectorLikeEvenPlane
|
||||
|
||||
|
|
|
|||
|
|
@ -135,26 +135,22 @@ lemma lineInCubicPerm_last_cond {S : (PureU1 (2 * n.succ.succ)).LinSols}
|
|||
rw [P_P_P!_accCube' g f hfg] at h1
|
||||
simp at h1
|
||||
cases h1 <;> rename_i h1
|
||||
apply Or.inl
|
||||
linear_combination h1
|
||||
cases h1 <;> rename_i h1
|
||||
apply Or.inr
|
||||
apply Or.inl
|
||||
linear_combination -(1 * h1)
|
||||
apply Or.inr
|
||||
apply Or.inr
|
||||
exact h1
|
||||
· apply Or.inl
|
||||
linear_combination h1
|
||||
· cases h1 <;> rename_i h1
|
||||
· refine Or.inr (Or.inl ?_)
|
||||
linear_combination -(1 * h1)
|
||||
· exact Or.inr (Or.inr h1)
|
||||
|
||||
lemma lineInCubicPerm_last_perm {S : (PureU1 (2 * n.succ.succ)).LinSols}
|
||||
(LIC : LineInCubicPerm S) : LineInPlaneCond S := by
|
||||
refine @Prop_three (2 * n.succ.succ) LineInPlaneProp S (δ!₂ (Fin.last n)) (δ!₁ (Fin.last n))
|
||||
δ!₄ ?_ ?_ ?_ ?_
|
||||
simp [Fin.ext_iff, δ!₂, δ!₁]
|
||||
simp [Fin.ext_iff, δ!₂, δ!₄]
|
||||
simp [Fin.ext_iff, δ!₁, δ!₄]
|
||||
omega
|
||||
intro M
|
||||
exact lineInCubicPerm_last_cond (lineInCubicPerm_permute LIC M)
|
||||
· simp [Fin.ext_iff, δ!₂, δ!₁]
|
||||
· simp [Fin.ext_iff, δ!₂, δ!₄]
|
||||
· simp [Fin.ext_iff, δ!₁, δ!₄]
|
||||
omega
|
||||
· exact fun M => lineInCubicPerm_last_cond (lineInCubicPerm_permute LIC M)
|
||||
|
||||
lemma lineInCubicPerm_constAbs {S : (PureU1 (2 * n.succ.succ)).Sols}
|
||||
(LIC : LineInCubicPerm S.1.1) : ConstAbs S.val :=
|
||||
|
|
|
|||
|
|
@ -72,9 +72,7 @@ lemma anomalyFree_param {S : (PureU1 (2 * n.succ)).Sols}
|
|||
have hC := S.cubicSol
|
||||
rw [hS] at hC
|
||||
change (accCube (2 * n.succ)) (P g + P! f) = 0 at hC
|
||||
erw [TriLinearSymm.toCubic_add] at hC
|
||||
erw [P_accCube] at hC
|
||||
erw [P!_accCube] at hC
|
||||
erw [TriLinearSymm.toCubic_add, P_accCube, P!_accCube] at hC
|
||||
linear_combination hC / 3
|
||||
|
||||
/-- A proposition on a solution which is true if `accCubeTriLinSymm (P g, P g, P! f) ≠ 0`.
|
||||
|
|
@ -114,9 +112,9 @@ lemma generic_or_special (S : (PureU1 (2 * n.succ)).Sols) :
|
|||
have h1 : accCubeTriLinSymm (P g) (P g) (P! f) ≠ 0 ∨
|
||||
accCubeTriLinSymm (P g) (P g) (P! f) = 0 := by
|
||||
exact ne_or_eq _ _
|
||||
cases h1 <;> rename_i h1
|
||||
exact Or.inl (genericCase_exists S ⟨g, f, h, h1⟩)
|
||||
exact Or.inr (specialCase_exists S ⟨g, f, h, h1⟩)
|
||||
rcases h1 with h1 | h1
|
||||
· exact Or.inl (genericCase_exists S ⟨g, f, h, h1⟩)
|
||||
· exact Or.inr (specialCase_exists S ⟨g, f, h, h1⟩)
|
||||
|
||||
theorem generic_case {S : (PureU1 (2 * n.succ)).Sols} (h : GenericCase S) :
|
||||
∃ g f a, S = parameterization g f a := by
|
||||
|
|
@ -128,11 +126,11 @@ theorem generic_case {S : (PureU1 (2 * n.succ)).Sols} (h : GenericCase S) :
|
|||
change S.val = _ • (_ • P g + _• P! f)
|
||||
rw [anomalyFree_param _ _ hS]
|
||||
rw [neg_neg, ← smul_add, smul_smul, inv_mul_cancel, one_smul]
|
||||
exact hS
|
||||
have h := h g f hS
|
||||
rw [anomalyFree_param _ _ hS] at h
|
||||
simp at h
|
||||
exact h
|
||||
· exact hS
|
||||
· have h := h g f hS
|
||||
rw [anomalyFree_param _ _ hS] at h
|
||||
simp at h
|
||||
exact h
|
||||
|
||||
lemma special_case_lineInCubic {S : (PureU1 (2 * n.succ)).Sols}
|
||||
(h : SpecialCase S) : LineInCubic S.1.1 := by
|
||||
|
|
@ -154,9 +152,8 @@ lemma special_case_lineInCubic {S : (PureU1 (2 * n.succ)).Sols}
|
|||
lemma special_case_lineInCubic_perm {S : (PureU1 (2 * n.succ)).Sols}
|
||||
(h : ∀ (M : (FamilyPermutations (2 * n.succ)).group),
|
||||
SpecialCase ((FamilyPermutations (2 * n.succ)).solAction.toFun S M)) :
|
||||
LineInCubicPerm S.1.1 := by
|
||||
intro M
|
||||
exact special_case_lineInCubic (h M)
|
||||
LineInCubicPerm S.1.1 :=
|
||||
fun M => special_case_lineInCubic (h M)
|
||||
|
||||
theorem special_case {S : (PureU1 (2 * n.succ.succ)).Sols}
|
||||
(h : ∀ (M : (FamilyPermutations (2 * n.succ.succ)).group),
|
||||
|
|
|
|||
|
|
@ -110,9 +110,9 @@ theorem linesInPlane_constAbs {S : (PureU1 (n.succ.succ.succ.succ.succ)).LinSols
|
|||
(hS : LineInPlaneCond S) : ConstAbs S.val := by
|
||||
intro i j
|
||||
by_cases hij : i ≠ j
|
||||
exact linesInPlane_eq_sq hS i j hij
|
||||
simp at hij
|
||||
rw [hij]
|
||||
· exact linesInPlane_eq_sq hS i j hij
|
||||
· simp at hij
|
||||
rw [hij]
|
||||
|
||||
lemma linesInPlane_four (S : (PureU1 4).Sols) (hS : LineInPlaneCond S.1.1) :
|
||||
ConstAbsProp (S.val (0 : Fin 4), S.val (1 : Fin 4)) := by
|
||||
|
|
@ -131,29 +131,29 @@ lemma linesInPlane_four (S : (PureU1 4).Sols) (hS : LineInPlaneCond S.1.1) :
|
|||
have h1 : S.val (2 : Fin 4) = S.val (3 : Fin 4) := by
|
||||
linear_combination l012 / 2 + -1 * l013 / 2
|
||||
by_cases h2 : S.val (0 : Fin 4) = S.val (2 : Fin 4)
|
||||
simp_all
|
||||
have h3 : S.val (1 : Fin 4) = - 3 * S.val (2 : Fin 4) := by
|
||||
linear_combination l012 + 3 * h1
|
||||
rw [← h1, h3] at hcube
|
||||
have h4 : S.val (2 : Fin 4) ^ 3 = 0 := by
|
||||
linear_combination -1 * hcube / 24
|
||||
simp at h4
|
||||
simp_all
|
||||
by_cases h3 : S.val (0 : Fin 4) = - S.val (2 : Fin 4)
|
||||
simp_all
|
||||
have h4 : S.val (1 : Fin 4) = - S.val (2 : Fin 4) := by
|
||||
linear_combination l012 + h1
|
||||
simp_all
|
||||
simp_all
|
||||
have h4 : S.val (0 : Fin 4) = - 3 * S.val (3 : Fin 4) := by
|
||||
linear_combination l023
|
||||
have h5 : S.val (1 : Fin 4) = S.val (3 : Fin 4) := by
|
||||
linear_combination l013 - 1 * h4
|
||||
rw [h4, h5] at hcube
|
||||
have h6 : S.val (3 : Fin 4) ^ 3 = 0 := by
|
||||
linear_combination -1 * hcube / 24
|
||||
simp at h6
|
||||
simp_all
|
||||
· simp_all
|
||||
have h3 : S.val (1 : Fin 4) = - 3 * S.val (2 : Fin 4) := by
|
||||
linear_combination l012 + 3 * h1
|
||||
rw [← h1, h3] at hcube
|
||||
have h4 : S.val (2 : Fin 4) ^ 3 = 0 := by
|
||||
linear_combination -1 * hcube / 24
|
||||
simp at h4
|
||||
simp_all
|
||||
· by_cases h3 : S.val (0 : Fin 4) = - S.val (2 : Fin 4)
|
||||
· simp_all
|
||||
have h4 : S.val (1 : Fin 4) = - S.val (2 : Fin 4) := by
|
||||
linear_combination l012 + h1
|
||||
simp_all
|
||||
· simp_all
|
||||
have h4 : S.val (0 : Fin 4) = - 3 * S.val (3 : Fin 4) := by
|
||||
linear_combination l023
|
||||
have h5 : S.val (1 : Fin 4) = S.val (3 : Fin 4) := by
|
||||
linear_combination l013 - 1 * h4
|
||||
rw [h4, h5] at hcube
|
||||
have h6 : S.val (3 : Fin 4) ^ 3 = 0 := by
|
||||
linear_combination -1 * hcube / 24
|
||||
simp at h6
|
||||
simp_all
|
||||
|
||||
lemma linesInPlane_eq_sq_four {S : (PureU1 4).Sols}
|
||||
(hS : LineInPlaneCond S.1.1) : ∀ (i j : Fin 4) (_ : i ≠ j),
|
||||
|
|
@ -169,14 +169,14 @@ lemma linesInPlane_constAbs_four (S : (PureU1 4).Sols)
|
|||
(hS : LineInPlaneCond S.1.1) : ConstAbs S.val := by
|
||||
intro i j
|
||||
by_cases hij : i ≠ j
|
||||
exact linesInPlane_eq_sq_four hS i j hij
|
||||
simp at hij
|
||||
rw [hij]
|
||||
· exact linesInPlane_eq_sq_four hS i j hij
|
||||
· simp at hij
|
||||
rw [hij]
|
||||
|
||||
theorem linesInPlane_constAbs_AF (S : (PureU1 (n.succ.succ.succ.succ)).Sols)
|
||||
(hS : LineInPlaneCond S.1.1) : ConstAbs S.val := by
|
||||
induction n
|
||||
exact linesInPlane_constAbs_four S hS
|
||||
exact linesInPlane_constAbs hS
|
||||
· exact linesInPlane_constAbs_four S hS
|
||||
· exact linesInPlane_constAbs hS
|
||||
|
||||
end PureU1
|
||||
|
|
|
|||
|
|
@ -137,25 +137,22 @@ lemma lineInCubicPerm_last_cond {S : (PureU1 (2 * n.succ.succ+1)).LinSols}
|
|||
rw [P_P_P!_accCube' g f hfg] at h1
|
||||
simp at h1
|
||||
cases h1 <;> rename_i h1
|
||||
apply Or.inl
|
||||
linear_combination h1
|
||||
cases h1 <;> rename_i h1
|
||||
apply Or.inr
|
||||
apply Or.inl
|
||||
linear_combination h1
|
||||
apply Or.inr
|
||||
apply Or.inr
|
||||
linear_combination h1
|
||||
· apply Or.inl
|
||||
linear_combination h1
|
||||
· cases h1 <;> rename_i h1
|
||||
· refine Or.inr (Or.inl ?_)
|
||||
linear_combination h1
|
||||
· refine Or.inr (Or.inr ?_)
|
||||
linear_combination h1
|
||||
|
||||
lemma lineInCubicPerm_last_perm {S : (PureU1 (2 * n.succ.succ + 1)).LinSols}
|
||||
(LIC : LineInCubicPerm S) : LineInPlaneCond S := by
|
||||
refine @Prop_three (2 * n.succ.succ + 1) LineInPlaneProp S (δ!₂ 0) (δ!₁ 0)
|
||||
δ!₃ ?_ ?_ ?_ ?_
|
||||
simp [Fin.ext_iff, δ!₂, δ!₁]
|
||||
simp [Fin.ext_iff, δ!₂, δ!₃]
|
||||
simp [Fin.ext_iff, δ!₁, δ!₃]
|
||||
intro M
|
||||
exact lineInCubicPerm_last_cond (lineInCubicPerm_permute LIC M)
|
||||
· simp [Fin.ext_iff, δ!₂, δ!₁]
|
||||
· simp [Fin.ext_iff, δ!₂, δ!₃]
|
||||
· simp [Fin.ext_iff, δ!₁, δ!₃]
|
||||
· exact fun M => lineInCubicPerm_last_cond (lineInCubicPerm_permute LIC M)
|
||||
|
||||
lemma lineInCubicPerm_constAbs {S : (PureU1 (2 * n.succ.succ + 1)).LinSols}
|
||||
(LIC : LineInCubicPerm S) : ConstAbs S.val :=
|
||||
|
|
|
|||
|
|
@ -149,19 +149,23 @@ lemma pairSwap_other {n : ℕ} (i j k : Fin n) (hik : i ≠ k) (hjk : j ≠ k) :
|
|||
(pairSwap i j).toFun k = k := by
|
||||
simp [pairSwap]
|
||||
split
|
||||
simp_all
|
||||
split
|
||||
simp_all
|
||||
rfl
|
||||
· rename_i h
|
||||
exact False.elim (hik (id (Eq.symm h)))
|
||||
· split
|
||||
· rename_i h
|
||||
exact False.elim (hjk (id (Eq.symm h)))
|
||||
· rfl
|
||||
|
||||
lemma pairSwap_inv_other {n : ℕ} {i j k : Fin n} (hik : i ≠ k) (hjk : j ≠ k) :
|
||||
(pairSwap i j).invFun k = k := by
|
||||
simp [pairSwap]
|
||||
split
|
||||
simp_all
|
||||
split
|
||||
simp_all
|
||||
rfl
|
||||
· rename_i h
|
||||
exact False.elim (hik (id (Eq.symm h)))
|
||||
· split
|
||||
· rename_i h
|
||||
exact False.elim (hjk (id (Eq.symm h)))
|
||||
· rfl
|
||||
|
||||
/-- A permutation of fermions which takes one ordered subset into another. -/
|
||||
noncomputable def permOfInjection (f g : Fin m ↪ Fin n) : (FamilyPermutations n).group :=
|
||||
|
|
|
|||
Loading…
Add table
Add a link
Reference in a new issue