reactor: Removal of double spaces
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jstoobysmith
parent
ce92e1d649
commit
13f62a50eb
64 changed files with 550 additions and 546 deletions
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@ -16,7 +16,7 @@ that splits into two planes on which every point is a solution to the ACCs.
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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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open BigOperators
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namespace PureU1
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@ -111,7 +111,7 @@ lemma δ₂_δ!₂ (j : Fin n) : δ₂ j = δ!₂ j := by
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simp [δ₂, δ!₂]
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omega
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lemma sum_δ (S : Fin (2 * n + 1) → ℚ) :
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lemma sum_δ (S : Fin (2 * n + 1) → ℚ) :
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∑ i, S i = S δ₃ + ∑ i : Fin n, ((S ∘ δ₁) i + (S ∘ δ₂) i) := by
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have h1 : ∑ i, S i = ∑ i : Fin (n + 1 + n), S (Fin.cast (split_odd n) i) := by
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rw [Finset.sum_equiv (Fin.castOrderIso (split_odd n)).symm.toEquiv]
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@ -127,7 +127,7 @@ lemma sum_δ (S : Fin (2 * n + 1) → ℚ) :
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rw [Finset.sum_add_distrib]
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rfl
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lemma sum_δ! (S : Fin (2 * n + 1) → ℚ) :
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lemma sum_δ! (S : Fin (2 * n + 1) → ℚ) :
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∑ i, S i = S δ!₃ + ∑ i : Fin n, ((S ∘ δ!₁) i + (S ∘ δ!₂) i) := by
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have h1 : ∑ i, S i = ∑ i : Fin ((1+n)+n), S (Fin.cast (split_odd! n) i) := by
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rw [Finset.sum_equiv (Fin.castOrderIso (split_odd! n)).symm.toEquiv]
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@ -210,12 +210,12 @@ lemma basis!_on_δ!₁_other {k j : Fin n} (h : k ≠ j) :
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omega
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rfl
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lemma basis_on_other {k : Fin n} {j : Fin (2 * n + 1)} (h1 : j ≠ δ₁ k) (h2 : j ≠ δ₂ k) :
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lemma basis_on_other {k : Fin n} {j : Fin (2 * n + 1)} (h1 : j ≠ δ₁ k) (h2 : j ≠ δ₂ k) :
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basisAsCharges k j = 0 := by
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simp [basisAsCharges]
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simp_all only [ne_eq, ↓reduceIte]
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lemma basis!_on_other {k : Fin n} {j : Fin (2 * n + 1)} (h1 : j ≠ δ!₁ k) (h2 : j ≠ δ!₂ k) :
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lemma basis!_on_other {k : Fin n} {j : Fin (2 * n + 1)} (h1 : j ≠ δ!₁ k) (h2 : j ≠ δ!₂ k) :
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basis!AsCharges k j = 0 := by
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simp [basis!AsCharges]
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simp_all only [ne_eq, ↓reduceIte]
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@ -333,11 +333,11 @@ end theBasisVectors
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/-- Swapping the elements δ!₁ j and δ!₂ j is equivalent to adding a vector basis!AsCharges j. -/
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lemma swap!_as_add {S S' : (PureU1 (2 * n + 1)).LinSols} (j : Fin n)
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(hS : ((FamilyPermutations (2 * n + 1)).linSolRep
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(pairSwap (δ!₁ j) (δ!₂ j))) S = S') :
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(pairSwap (δ!₁ j) (δ!₂ j))) S = S') :
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S'.val = S.val + (S.val (δ!₂ j) - S.val (δ!₁ j)) • basis!AsCharges j := by
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funext i
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rw [← hS, FamilyPermutations_anomalyFreeLinear_apply]
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by_cases hi : i = δ!₁ j
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by_cases hi : i = δ!₁ j
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subst hi
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simp [HSMul.hSMul, basis!_on_δ!₁_self, pairSwap_inv_fst]
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by_cases hi2 : i = δ!₂ j
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@ -345,7 +345,7 @@ lemma swap!_as_add {S S' : (PureU1 (2 * n + 1)).LinSols} (j : Fin n)
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simp [HSMul.hSMul,basis!_on_δ!₂_self, pairSwap_inv_snd]
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simp [HSMul.hSMul]
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rw [basis!_on_other hi hi2]
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change S.val ((pairSwap (δ!₁ j) (δ!₂ j)).invFun i) =_
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change S.val ((pairSwap (δ!₁ j) (δ!₂ j)).invFun i) =_
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erw [pairSwap_inv_other (Ne.symm hi) (Ne.symm hi2)]
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simp
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@ -369,7 +369,7 @@ lemma P_δ₁ (f : Fin n → ℚ) (j : Fin n) : P f (δ₁ j) = f j := by
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simp only [mul_zero]
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simp only [mem_univ, not_true_eq_false, _root_.mul_eq_zero, IsEmpty.forall_iff]
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lemma P!_δ!₁ (f : Fin n → ℚ) (j : Fin n) : P! f (δ!₁ j) = f j := by
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lemma P!_δ!₁ (f : Fin n → ℚ) (j : Fin n) : P! f (δ!₁ j) = f j := by
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rw [P!, sum_of_charges]
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simp [HSMul.hSMul, SMul.smul]
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rw [Finset.sum_eq_single j]
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@ -418,14 +418,14 @@ lemma Pa_δa₁ (f g : Fin n.succ → ℚ) : Pa f g δa₁ = f 0 := by
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rw [P_δ₁, P!_δ!₃]
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simp
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lemma Pa_δa₂ (f g : Fin n.succ → ℚ) (j : Fin n) : Pa f g (δa₂ j) = f j.succ + g j.castSucc := by
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lemma Pa_δa₂ (f g : Fin n.succ → ℚ) (j : Fin n) : Pa f g (δa₂ j) = f j.succ + g j.castSucc := by
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rw [Pa]
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simp only [ACCSystemCharges.chargesAddCommMonoid_add]
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nth_rewrite 1 [δa₂_δ₁]
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rw [δa₂_δ!₁]
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rw [P_δ₁, P!_δ!₁]
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lemma Pa_δa₃ (f g : Fin n.succ → ℚ) : Pa f g (δa₃) = g (Fin.last n) := by
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lemma Pa_δa₃ (f g : Fin n.succ → ℚ) : Pa f g (δa₃) = g (Fin.last n) := by
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rw [Pa]
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simp only [ACCSystemCharges.chargesAddCommMonoid_add]
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nth_rewrite 1 [δa₃_δ₃]
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@ -495,7 +495,7 @@ lemma P!_zero (f : Fin n → ℚ) (h : P! f = 0) : ∀ i, f i = 0 := by
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rw [h]
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rfl
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lemma Pa_zero (f g : Fin n.succ → ℚ) (h : Pa f g = 0) :
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lemma Pa_zero (f g : Fin n.succ → ℚ) (h : Pa f g = 0) :
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∀ i, f i = 0 := by
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have h₃ := Pa_δa₁ f g
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rw [h] at h₃
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@ -589,7 +589,7 @@ theorem basisa_linear_independent : LinearIndependent ℚ (@basisa n.succ) := by
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simp_all
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simp_all
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lemma Pa'_eq (f f' : (Fin n.succ) ⊕ (Fin n.succ) → ℚ) : Pa' f = Pa' f' ↔ f = f' := by
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lemma Pa'_eq (f f' : (Fin n.succ) ⊕ (Fin n.succ) → ℚ) : Pa' f = Pa' f' ↔ f = f' := by
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apply Iff.intro
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intro h
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funext i
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@ -610,7 +610,7 @@ lemma Pa'_eq (f f' : (Fin n.succ) ⊕ (Fin n.succ) → ℚ) : Pa' f = Pa' f'
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/-! TODO: Replace the definition of `join` with a Mathlib definition, most likely `Sum.elim`. -/
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/-- A helper function for what follows. -/
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def join (g f : Fin n → ℚ) : Fin n ⊕ Fin n → ℚ := fun i =>
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def join (g f : Fin n → ℚ) : Fin n ⊕ Fin n → ℚ := fun i =>
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match i with
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| .inl i => g i
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| .inr i => f i
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@ -646,7 +646,7 @@ lemma Pa_eq (g g' : Fin n.succ → ℚ) (f f' : Fin n.succ → ℚ) :
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rw [← join_ext]
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exact Pa'_eq _ _
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lemma basisa_card : Fintype.card ((Fin n.succ) ⊕ (Fin n.succ)) =
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lemma basisa_card : Fintype.card ((Fin n.succ) ⊕ (Fin n.succ)) =
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FiniteDimensional.finrank ℚ (PureU1 (2 * n.succ + 1)).LinSols := by
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erw [BasisLinear.finrank_AnomalyFreeLinear]
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simp only [Fintype.card_sum, Fintype.card_fin]
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@ -658,7 +658,7 @@ noncomputable def basisaAsBasis :
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basisOfLinearIndependentOfCardEqFinrank (@basisa_linear_independent n) basisa_card
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lemma span_basis (S : (PureU1 (2 * n.succ + 1)).LinSols) :
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∃ (g f : Fin n.succ → ℚ) , S.val = P g + P! f := by
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∃ (g f : Fin n.succ → ℚ) , S.val = P g + P! f := by
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have h := (mem_span_range_iff_exists_fun ℚ).mp (Basis.mem_span basisaAsBasis S)
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obtain ⟨f, hf⟩ := h
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simp [basisaAsBasis] at hf
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@ -686,7 +686,7 @@ lemma span_basis_swap! {S : (PureU1 (2 * n.succ + 1)).LinSols} (j : Fin n.succ)
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apply SetLike.mem_of_subset
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apply Submodule.subset_span
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simp_all only [Set.mem_range, exists_apply_eq_apply]
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have hX : X ∈ Submodule.span ℚ (Set.range (basis!AsCharges)) := by
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have hX : X ∈ Submodule.span ℚ (Set.range (basis!AsCharges)) := by
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apply Submodule.add_mem
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exact hf
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exact hP
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