reactor: Removal of double spaces
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jstoobysmith
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ce92e1d649
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13f62a50eb
64 changed files with 550 additions and 546 deletions
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@ -18,13 +18,13 @@ We further define the action on the ACC System.
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universe v u
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open Nat
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open Finset
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open Finset
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namespace PureU1
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/-- The permutation group of the n-fermions. -/
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@[simp]
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def PermGroup (n : ℕ) := Equiv.Perm (Fin n)
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def PermGroup (n : ℕ) := Equiv.Perm (Fin n)
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instance {n : ℕ} : Group (PermGroup n) := by
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simp [PermGroup]
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@ -35,7 +35,7 @@ section Charges
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/-- The image of an element of `permGroup` under the representation on charges. -/
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@[simps!]
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def chargeMap {n : ℕ} (f : PermGroup n) :
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(PureU1 n).Charges →ₗ[ℚ] (PureU1 n).Charges where
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(PureU1 n).Charges →ₗ[ℚ] (PureU1 n).Charges where
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toFun S := S ∘ f.toFun
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map_add' S T := by
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funext i
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@ -73,7 +73,7 @@ open BigOperators
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lemma accCube_invariant {n : ℕ} (f : (PermGroup n)) (S : (PureU1 n).Charges) :
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accCube n (permCharges f S) = accCube n S := by
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rw [accCube_explicit, accCube_explicit]
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change ∑ i : Fin n, ((((fun a => a^3) ∘ S) (f.symm i))) = _
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change ∑ i : Fin n, ((((fun a => a^3) ∘ S) (f.symm i))) = _
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apply (Equiv.Perm.sum_comp _ _ _ ?_)
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simp
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@ -130,22 +130,22 @@ def pairSwap {n : ℕ} (i j : Fin n) : (FamilyPermutations n).group where
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right_inv s := by
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aesop
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lemma pairSwap_self_inv {n : ℕ} (i j : Fin n) : (pairSwap i j)⁻¹ = (pairSwap i j) := by
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lemma pairSwap_self_inv {n : ℕ} (i j : Fin n) : (pairSwap i j)⁻¹ = (pairSwap i j) := by
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rfl
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lemma pairSwap_fst {n : ℕ} (i j : Fin n) : (pairSwap i j).toFun i = j := by
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lemma pairSwap_fst {n : ℕ} (i j : Fin n) : (pairSwap i j).toFun i = j := by
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simp [pairSwap]
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lemma pairSwap_snd {n : ℕ} (i j : Fin n) : (pairSwap i j).toFun j = i := by
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lemma pairSwap_snd {n : ℕ} (i j : Fin n) : (pairSwap i j).toFun j = i := by
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simp [pairSwap]
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lemma pairSwap_inv_fst {n : ℕ} (i j : Fin n) : (pairSwap i j).invFun i = j := by
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lemma pairSwap_inv_fst {n : ℕ} (i j : Fin n) : (pairSwap i j).invFun i = j := by
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simp [pairSwap]
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lemma pairSwap_inv_snd {n : ℕ} (i j : Fin n) : (pairSwap i j).invFun j = i := by
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lemma pairSwap_inv_snd {n : ℕ} (i j : Fin n) : (pairSwap i j).invFun j = i := by
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simp [pairSwap]
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lemma pairSwap_other {n : ℕ} (i j k : Fin n) (hik : i ≠ k) (hjk : j ≠ k) :
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lemma pairSwap_other {n : ℕ} (i j k : Fin n) (hik : i ≠ k) (hjk : j ≠ k) :
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(pairSwap i j).toFun k = k := by
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simp [pairSwap]
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split
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@ -154,7 +154,7 @@ lemma pairSwap_other {n : ℕ} (i j k : Fin n) (hik : i ≠ k) (hjk : j ≠ k)
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simp_all
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rfl
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lemma pairSwap_inv_other {n : ℕ} {i j k : Fin n} (hik : i ≠ k) (hjk : j ≠ k) :
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lemma pairSwap_inv_other {n : ℕ} {i j k : Fin n} (hik : i ≠ k) (hjk : j ≠ k) :
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(pairSwap i j).invFun k = k := by
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simp [pairSwap]
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split
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@ -164,12 +164,12 @@ lemma pairSwap_inv_other {n : ℕ} {i j k : Fin n} (hik : i ≠ k) (hjk : j
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rfl
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/-- A permutation of fermions which takes one ordered subset into another. -/
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noncomputable def permOfInjection (f g : Fin m ↪ Fin n) : (FamilyPermutations n).group :=
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noncomputable def permOfInjection (f g : Fin m ↪ Fin n) : (FamilyPermutations n).group :=
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Equiv.extendSubtype (g.toEquivRange.symm.trans f.toEquivRange)
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section permTwo
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variable {i j i' j' : Fin n} (hij : i ≠ j) (hij' : i' ≠ j')
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variable {i j i' j' : Fin n} (hij : i ≠ j) (hij' : i' ≠ j')
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/-- Given two distinct elements, an embedding of `Fin 2` into `Fin n`. -/
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def permTwoInj : Fin 2 ↪ Fin n where
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@ -185,7 +185,7 @@ lemma permTwoInj_fst : i ∈ Set.range ⇑(permTwoInj hij) := by
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rfl
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lemma permTwoInj_fst_apply :
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(Function.Embedding.toEquivRange (permTwoInj hij)).symm ⟨i, permTwoInj_fst hij⟩ = 0 := by
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(Function.Embedding.toEquivRange (permTwoInj hij)).symm ⟨i, permTwoInj_fst hij⟩ = 0 := by
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exact (Equiv.symm_apply_eq (Function.Embedding.toEquivRange (permTwoInj hij))).mpr rfl
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lemma permTwoInj_snd : j ∈ Set.range ⇑(permTwoInj hij) := by
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@ -195,12 +195,12 @@ lemma permTwoInj_snd : j ∈ Set.range ⇑(permTwoInj hij) := by
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lemma permTwoInj_snd_apply :
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(Function.Embedding.toEquivRange (permTwoInj hij)).symm
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⟨j, permTwoInj_snd hij⟩ = 1 := by
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⟨j, permTwoInj_snd hij⟩ = 1 := by
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exact (Equiv.symm_apply_eq (Function.Embedding.toEquivRange (permTwoInj hij))).mpr rfl
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/-- A permutation which swaps `i` with `i'` and `j` with `j'`. -/
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noncomputable def permTwo : (FamilyPermutations n).group :=
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permOfInjection (permTwoInj hij) (permTwoInj hij')
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permOfInjection (permTwoInj hij) (permTwoInj hij')
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lemma permTwo_fst : (permTwo hij hij').toFun i' = i := by
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rw [permTwo, permOfInjection]
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@ -243,7 +243,7 @@ lemma permThreeInj_fst : i ∈ Set.range ⇑(permThreeInj hij hjk hik) := by
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lemma permThreeInj_fst_apply :
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(Function.Embedding.toEquivRange (permThreeInj hij hjk hik)).symm
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⟨i, permThreeInj_fst hij hjk hik⟩ = 0 := by
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⟨i, permThreeInj_fst hij hjk hik⟩ = 0 := by
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exact (Equiv.symm_apply_eq (Function.Embedding.toEquivRange (permThreeInj hij hjk hik))).mpr rfl
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lemma permThreeInj_snd : j ∈ Set.range ⇑(permThreeInj hij hjk hik) := by
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@ -253,7 +253,7 @@ lemma permThreeInj_snd : j ∈ Set.range ⇑(permThreeInj hij hjk hik) := by
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lemma permThreeInj_snd_apply :
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(Function.Embedding.toEquivRange (permThreeInj hij hjk hik)).symm
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⟨j, permThreeInj_snd hij hjk hik⟩ = 1 := by
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⟨j, permThreeInj_snd hij hjk hik⟩ = 1 := by
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exact (Equiv.symm_apply_eq (Function.Embedding.toEquivRange (permThreeInj hij hjk hik))).mpr rfl
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lemma permThreeInj_thd : k ∈ Set.range ⇑(permThreeInj hij hjk hik) := by
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@ -268,7 +268,7 @@ lemma permThreeInj_thd_apply :
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/-- A permutation which swaps three distinct elements with another three. -/
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noncomputable def permThree : (FamilyPermutations n).group :=
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permOfInjection (permThreeInj hij hjk hik) (permThreeInj hij' hjk' hik')
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permOfInjection (permThreeInj hij hjk hik) (permThreeInj hij' hjk' hik')
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lemma permThree_fst : (permThree hij hjk hik hij' hjk' hik').toFun i' = i := by
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rw [permThree, permOfInjection]
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@ -310,7 +310,7 @@ lemma Prop_two (P : ℚ × ℚ → Prop) {S : (PureU1 n).LinSols}
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have h1 := h (permTwo hij hab ).symm
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rw [FamilyPermutations_anomalyFreeLinear_apply, FamilyPermutations_anomalyFreeLinear_apply] at h1
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simp at h1
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change P
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change P
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(S.val ((permTwo hij hab ).toFun a),
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S.val ((permTwo hij hab ).toFun b)) at h1
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erw [permTwo_fst,permTwo_snd] at h1
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@ -322,14 +322,14 @@ lemma Prop_three (P : ℚ × ℚ × ℚ → Prop) {S : (PureU1 n).LinSols}
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P ((((FamilyPermutations n).linSolRep f S).val a),(
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(((FamilyPermutations n).linSolRep f S).val b),
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(((FamilyPermutations n).linSolRep f S).val c)
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))) : ∀ (i j k : Fin n) (_ : i ≠ j) (_ : j ≠ k) (_ : i ≠ k),
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))) : ∀ (i j k : Fin n) (_ : i ≠ j) (_ : j ≠ k) (_ : i ≠ k),
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P (S.val i, (S.val j, S.val k)) := by
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intro i j k hij hjk hik
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have h1 := h (permThree hij hjk hik hab hbc hac).symm
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rw [FamilyPermutations_anomalyFreeLinear_apply, FamilyPermutations_anomalyFreeLinear_apply,
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FamilyPermutations_anomalyFreeLinear_apply] at h1
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simp at h1
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change P
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change P
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(S.val ((permThree hij hjk hik hab hbc hac).toFun a),
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S.val ((permThree hij hjk hik hab hbc hac).toFun b),
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S.val ((permThree hij hjk hik hab hbc hac).toFun c)) at h1
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