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reactor: Removal of double spaces

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jstoobysmith 2024-07-12 11:23:02 -04:00
parent ce92e1d649
commit 13f62a50eb
64 changed files with 550 additions and 546 deletions
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42
HepLean/AnomalyCancellation/PureU1/Even/BasisLinear.lean
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@ -16,7 +16,7 @@ that splits into two planes on which every point is a solution to the ACCs.
universe v u
open Nat
open Finset
open Finset
open BigOperators
namespace PureU1
@ -51,7 +51,7 @@ def δ!₃ : Fin (2 * n.succ) := (Fin.cast (n_cond₂ n) (Fin.castAdd ((n + n) +
def δ!₄ : Fin (2 * n.succ) := (Fin.cast (n_cond₂ n) (Fin.natAdd 1 (Fin.natAdd (n + n) 0)))
lemma ext_δ (S T : Fin (2 * n.succ) → ℚ) (h1 : ∀ i, S (δ₁ i) = T (δ₁ i))
(h2 : ∀ i, S (δ₂ i) = T (δ₂ i)) : S = T := by
(h2 : ∀ i, S (δ₂ i) = T (δ₂ i)) : S = T := by
funext i
by_cases hi : i.val < n.succ
let j : Fin n.succ := ⟨i, hi⟩
@ -68,7 +68,7 @@ lemma ext_δ (S T : Fin (2 * n.succ) → ℚ) (h1 : ∀ i, S (δ₁ i) = T (δ
rw [h3] at h2
exact h2
lemma sum_δ₁_δ₂ (S : Fin (2 * n.succ) → ℚ) :
lemma sum_δ₁_δ₂ (S : Fin (2 * n.succ) → ℚ) :
∑ i, S i = ∑ i : Fin n.succ, ((S ∘ δ₁) i + (S ∘ δ₂) i) := by
have h1 : ∑ i, S i = ∑ i : Fin (n.succ + n.succ), S (Fin.cast (split_equal n.succ) i) := by
rw [Finset.sum_equiv (Fin.castOrderIso (split_equal n.succ)).symm.toEquiv]
@ -80,7 +80,7 @@ lemma sum_δ₁_δ₂ (S : Fin (2 * n.succ) → ℚ) :
rw [Fin.sum_univ_add, Finset.sum_add_distrib]
rfl
lemma sum_δ₁_δ₂' (S : Fin (2 * n.succ) → ℚ) :
lemma sum_δ₁_δ₂' (S : Fin (2 * n.succ) → ℚ) :
∑ i, S i = ∑ i : Fin n.succ, ((S ∘ δ₁) i + (S ∘ δ₂) i) := by
have h1 : ∑ i, S i = ∑ i : Fin (n.succ + n.succ), S (Fin.cast (split_equal n.succ) i) := by
rw [Finset.sum_equiv (Fin.castOrderIso (split_equal n.succ)).symm.toEquiv]
@ -92,8 +92,8 @@ lemma sum_δ₁_δ₂' (S : Fin (2 * n.succ) → ℚ) :
rw [Fin.sum_univ_add, Finset.sum_add_distrib]
rfl
lemma sum_δ!₁_δ!₂ (S : Fin (2 * n.succ) → ℚ) :
∑ i, S i = S δ!₃ + S δ!₄ + ∑ i : Fin n, ((S ∘ δ!₁) i + (S ∘ δ!₂) i) := by
lemma sum_δ!₁_δ!₂ (S : Fin (2 * n.succ) → ℚ) :
∑ i, S i = S δ!₃ + S δ!₄ + ∑ i : Fin n, ((S ∘ δ!₁) i + (S ∘ δ!₂) i) := by
have h1 : ∑ i, S i = ∑ i : Fin (1 + ((n + n) + 1)), S (Fin.cast (n_cond₂ n) i) := by
rw [Finset.sum_equiv (Fin.castOrderIso (n_cond₂ n)).symm.toEquiv]
intro i
@ -180,12 +180,12 @@ lemma basis_on_δ₁_other {k j : Fin n.succ} (h : k ≠ j) :
omega
rfl
lemma basis_on_other {k : Fin n.succ} {j : Fin (2 * n.succ)} (h1 : j ≠ δ₁ k) (h2 : j ≠ δ₂ k) :
lemma basis_on_other {k : Fin n.succ} {j : Fin (2 * n.succ)} (h1 : j ≠ δ₁ k) (h2 : j ≠ δ₂ k) :
basisAsCharges k j = 0 := by
simp [basisAsCharges]
simp_all only [ne_eq, ↓reduceIte]
lemma basis!_on_other {k : Fin n} {j : Fin (2 * n.succ)} (h1 : j ≠ δ!₁ k) (h2 : j ≠ δ!₂ k) :
lemma basis!_on_other {k : Fin n} {j : Fin (2 * n.succ)} (h1 : j ≠ δ!₁ k) (h2 : j ≠ δ!₂ k) :
basis!AsCharges k j = 0 := by
simp [basis!AsCharges]
simp_all only [ne_eq, ↓reduceIte]
@ -338,11 +338,11 @@ def basisa : (Fin n.succ) ⊕ (Fin n) → (PureU1 (2 * n.succ)).LinSols := fun i
/-- Swapping the elements δ!₁ j and δ!₂ j is equivalent to adding a vector basis!AsCharges j. -/
lemma swap!_as_add {S S' : (PureU1 (2 * n.succ)).LinSols} (j : Fin n)
(hS : ((FamilyPermutations (2 * n.succ)).linSolRep
(pairSwap (δ!₁ j) (δ!₂ j))) S = S') :
(pairSwap (δ!₁ j) (δ!₂ j))) S = S') :
S'.val = S.val + (S.val (δ!₂ j) - S.val (δ!₁ j)) • basis!AsCharges j := by
funext i
rw [← hS, FamilyPermutations_anomalyFreeLinear_apply]
by_cases hi : i = δ!₁ j
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
@ -350,7 +350,7 @@ lemma swap!_as_add {S S' : (PureU1 (2 * n.succ)).LinSols} (j : Fin n)
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) =_
change S.val ((pairSwap (δ!₁ j) (δ!₂ j)).invFun i) =_
erw [pairSwap_inv_other (Ne.symm hi) (Ne.symm hi2)]
simp
@ -374,7 +374,7 @@ lemma P_δ₁ (f : Fin n.succ → ℚ) (j : Fin n.succ) : P f (δ₁ j) = f j :=
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
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]
@ -424,7 +424,7 @@ lemma P!_δ!₃ (f : Fin n → ℚ) : P! f (δ!₃) = 0 := by
rw [P!, sum_of_charges]
simp [HSMul.hSMul, SMul.smul, basis!_on_δ!₃]
lemma Pa_δ!₃ (f : Fin n.succ → ℚ) (g : Fin n → ℚ) : Pa f g (δ!₃) = f 0 := by
lemma Pa_δ!₃ (f : Fin n.succ → ℚ) (g : Fin n → ℚ) : Pa f g (δ!₃) = f 0 := by
rw [Pa]
simp only [ACCSystemCharges.chargesAddCommMonoid_add]
rw [P!_δ!₃, δ!₃_δ₁0, P_δ₁]
@ -434,13 +434,13 @@ lemma P!_δ!₄ (f : Fin n → ℚ) : P! f (δ!₄) = 0 := by
rw [P!, sum_of_charges]
simp [HSMul.hSMul, SMul.smul, basis!_on_δ!₄]
lemma Pa_δ!₄ (f : Fin n.succ → ℚ) (g : Fin n → ℚ) : Pa f g (δ!₄) = - f (Fin.last n) := by
lemma Pa_δ!₄ (f : Fin n.succ → ℚ) (g : Fin n → ℚ) : Pa f g (δ!₄) = - f (Fin.last n) := by
rw [Pa]
simp only [ACCSystemCharges.chargesAddCommMonoid_add]
rw [P!_δ!₄, δ!₄_δ₂Last, P_δ₂]
simp
lemma P_δ₁_δ₂ (f : Fin n.succ → ℚ) : P f ∘ δ₂ = - P f ∘ δ₁ := by
lemma P_δ₁_δ₂ (f : Fin n.succ → ℚ) : P f ∘ δ₂ = - P f ∘ δ₁ := by
funext j
simp only [PureU1_numberCharges, Function.comp_apply, Pi.neg_apply]
rw [P_δ₁, P_δ₂]
@ -484,7 +484,7 @@ lemma P_P_P!_accCube (g : Fin n.succ → ℚ) (j : Fin n) :
lemma P_P!_P!_accCube (g : Fin n → ℚ) (j : Fin n.succ) :
accCubeTriLinSymm (P! g) (P! g) (basisAsCharges j)
= (P! g (δ₁ j))^2 - (P! g (δ₂ j))^2 := by
= (P! g (δ₁ j))^2 - (P! g (δ₂ j))^2 := by
simp [accCubeTriLinSymm]
rw [sum_δ₁_δ₂]
simp only [Function.comp_apply]
@ -604,7 +604,7 @@ theorem basisa_linear_independent : LinearIndependent ℚ (@basisa n) := by
simp_all
simp_all
lemma Pa'_eq (f f' : (Fin n.succ) ⊕ (Fin n) → ℚ) : Pa' f = Pa' f' ↔ f = f' := by
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
@ -625,7 +625,7 @@ lemma Pa'_eq (f f' : (Fin n.succ) ⊕ (Fin n) → ℚ) : Pa' f = Pa' f' ↔ f =
/-! TODO: Replace the definition of `join` with a Mathlib definition, most likely `Sum.elim`. -/
/-- A helper function for what follows. -/
def join (g : Fin n.succ → ℚ) (f : Fin n → ℚ) : (Fin n.succ) ⊕ (Fin n) → ℚ := fun i =>
def join (g : Fin n.succ → ℚ) (f : Fin n → ℚ) : (Fin n.succ) ⊕ (Fin n) → ℚ := fun i =>
match i with
| .inl i => g i
| .inr i => f i
@ -661,7 +661,7 @@ lemma Pa_eq (g g' : Fin n.succ → ℚ) (f f' : Fin n → ℚ) :
rw [← join_ext]
exact Pa'_eq _ _
lemma basisa_card : Fintype.card ((Fin n.succ) ⊕ (Fin n)) =
lemma basisa_card : Fintype.card ((Fin n.succ) ⊕ (Fin n)) =
FiniteDimensional.finrank ℚ (PureU1 (2 * n.succ)).LinSols := by
erw [BasisLinear.finrank_AnomalyFreeLinear]
simp only [Fintype.card_sum, Fintype.card_fin, mul_eq]
@ -673,7 +673,7 @@ noncomputable def basisaAsBasis :
basisOfLinearIndependentOfCardEqFinrank (@basisa_linear_independent n) basisa_card
lemma span_basis (S : (PureU1 (2 * n.succ)).LinSols) :
∃ (g : Fin n.succ → ℚ) (f : Fin n → ℚ), S.val = P g + P! f := by
∃ (g : Fin n.succ → ℚ) (f : Fin n → ℚ), S.val = P g + P! f := by
have h := (mem_span_range_iff_exists_fun ℚ).mp (Basis.mem_span basisaAsBasis S)
obtain ⟨f, hf⟩ := h
simp [basisaAsBasis] at hf
@ -706,7 +706,7 @@ lemma span_basis_swap! {S : (PureU1 (2 * n.succ)).LinSols} (j : Fin n)
S'.val = P g' + P! f' ∧ P! f' = P! f +
(S.val (δ!₂ j) - S.val (δ!₁ j)) • basis!AsCharges j ∧ g' = g := by
let X := P! f + (S.val (δ!₂ j) - S.val (δ!₁ j)) • basis!AsCharges j
have hX : X ∈ Submodule.span ℚ (Set.range (basis!AsCharges)) := by
have hX : X ∈ Submodule.span ℚ (Set.range (basis!AsCharges)) := by
apply Submodule.add_mem
exact (P!_in_span f)
exact (smul_basis!AsCharges_in_span S j)