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refactor: Rename deltas in ACC conditions

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jstoobysmith 2024-11-28 12:50:34 +00:00
parent d26f010cb2
commit 6f83f5a623
8 changed files with 367 additions and 355 deletions
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HepLean/AnomalyCancellation/PureU1/Odd/BasisLinear.lean
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@ -25,22 +25,22 @@ variable {n : ℕ}
namespace VectorLikeOddPlane
lemma split_odd! (n : ℕ) : (1 + n) + n = 2 * n +1 := by
lemma odd_shift_eq (n : ℕ) : (1 + n) + n = 2 * n +1 := by
omega
lemma split_adda (n : ℕ) : ((1+n)+1) + n.succ = 2 * n.succ + 1 := by
lemma odd_shift_shift_eq (n : ℕ) : ((1+n)+1) + n.succ = 2 * n.succ + 1 := by
omega
section theDeltas
/-- The inclusion of `Fin n` into `Fin ((n + 1) + n)` via the first `n`.
This is then casted to `Fin (2 * n + 1)`. -/
def oddFstEmbed (j : Fin n) : Fin (2 * n + 1) :=
def oddFst (j : Fin n) : Fin (2 * n + 1) :=
Fin.cast (split_odd n) (Fin.castAdd n (Fin.castAdd 1 j))
/-- The inclusion of `Fin n` into `Fin ((n + 1) + n)` via the second `n`.
This is then casted to `Fin (2 * n + 1)`. -/
def oddSndEmbed (j : Fin n) : Fin (2 * n + 1) :=
def oddSnd (j : Fin n) : Fin (2 * n + 1) :=
Fin.cast (split_odd n) (Fin.natAdd (n+1) j)
/-- The element representing `1` in `Fin ((n + 1) + n)`.
@ -50,76 +50,80 @@ def oddMid : Fin (2 * n + 1) :=
/-- The inclusion of `Fin n` into `Fin (1 + n + n)` via the first `n`.
This is then casted to `Fin (2 * n + 1)`. -/
def oddShiftFstEmbed (j : Fin n) : Fin (2 * n + 1) :=
Fin.cast (split_odd! n) (Fin.castAdd n (Fin.natAdd 1 j))
def oddShiftFst (j : Fin n) : Fin (2 * n + 1) :=
Fin.cast (odd_shift_eq n) (Fin.castAdd n (Fin.natAdd 1 j))
/-- The inclusion of `Fin n` into `Fin (1 + n + n)` via the second `n`.
This is then casted to `Fin (2 * n + 1)`. -/
def oddShiftSndEmbed (j : Fin n) : Fin (2 * n + 1) :=
Fin.cast (split_odd! n) (Fin.natAdd (1 + n) j)
def oddShiftSnd (j : Fin n) : Fin (2 * n + 1) :=
Fin.cast (odd_shift_eq n) (Fin.natAdd (1 + n) j)
/-- The element representing the `1` in `Fin (1 + n + n)`.
This is then casted to `Fin (2 * n + 1)`. -/
def oddShiftZero : Fin (2 * n + 1) :=
Fin.cast (split_odd! n) (Fin.castAdd n (Fin.castAdd n 1))
Fin.cast (odd_shift_eq n) (Fin.castAdd n (Fin.castAdd n 1))
/-- The element representing the first `1` in `Fin (1 + n + 1 + n.succ)` casted
to `Fin (2 * n.succ + 1)`. -/
def δa₁ : Fin (2 * n.succ + 1) :=
Fin.cast (split_adda n) (Fin.castAdd n.succ (Fin.castAdd 1 (Fin.castAdd n 1)))
def oddShiftShiftZero : Fin (2 * n.succ + 1) :=
Fin.cast (odd_shift_shift_eq n) (Fin.castAdd n.succ (Fin.castAdd 1 (Fin.castAdd n 1)))
/-- The inclusion of `Fin n` into `Fin (1 + n + 1 + n.succ)` via the first `n` and casted
to `Fin (2 * n.succ + 1)`. -/
def δa₂ (j : Fin n) : Fin (2 * n.succ + 1) :=
Fin.cast (split_adda n) (Fin.castAdd n.succ (Fin.castAdd 1 (Fin.natAdd 1 j)))
def oddShiftShiftFst (j : Fin n) : Fin (2 * n.succ + 1) :=
Fin.cast (odd_shift_shift_eq n) (Fin.castAdd n.succ (Fin.castAdd 1 (Fin.natAdd 1 j)))
/-- The element representing the second `1` in `Fin (1 + n + 1 + n.succ)` casted
to `2 * n.succ + 1`. -/
def δa₃ : Fin (2 * n.succ + 1) :=
Fin.cast (split_adda n) (Fin.castAdd n.succ (Fin.natAdd (1+n) 1))
def oddShiftShiftMid : Fin (2 * n.succ + 1) :=
Fin.cast (odd_shift_shift_eq n) (Fin.castAdd n.succ (Fin.natAdd (1+n) 1))
/-- The inclusion of `Fin n.succ` into `Fin (1 + n + 1 + n.succ)` via the `n.succ` and casted
to `Fin (2 * n.succ + 1)`. -/
def δa₄ (j : Fin n.succ) : Fin (2 * n.succ + 1) :=
Fin.cast (split_adda n) (Fin.natAdd ((1+n)+1) j)
def oddShiftShiftSnd (j : Fin n.succ) : Fin (2 * n.succ + 1) :=
Fin.cast (odd_shift_shift_eq n) (Fin.natAdd ((1+n)+1) j)
lemma δa₁_δ₁ : @δa₁ n = oddFstEmbed 0 := Fin.rev_inj.mp rfl
lemma oddShiftShiftZero_eq_oddFst_zero : @oddShiftShiftZero n = oddFst 0 :=
Fin.rev_inj.mp rfl
lemma δa₁_δ!₃ : @δa₁ n = oddShiftZero := rfl
lemma oddShiftShiftZero_eq_oddShiftZero : @oddShiftShiftZero n = oddShiftZero := rfl
lemma δa₂_δ₁ (j : Fin n) : δa₂ j = oddFstEmbed j.succ := by
lemma oddShiftShiftFst_eq_oddFst_succ (j : Fin n) :
oddShiftShiftFst j = oddFst j.succ := by
rw [Fin.ext_iff]
simp only [succ_eq_add_one, δa₂, Fin.coe_cast, Fin.coe_castAdd, Fin.coe_natAdd, oddFstEmbed, Fin.val_succ]
simp only [succ_eq_add_one, oddShiftShiftFst, Fin.coe_cast, Fin.coe_castAdd, Fin.coe_natAdd,
oddFst, Fin.val_succ]
exact Nat.add_comm 1 ↑j
lemma δa₂_δ!₁ (j : Fin n) : δa₂ j = oddShiftFstEmbed j.castSucc := by
lemma oddShiftShiftFst_eq_oddShiftFst_castSucc (j : Fin n) :
oddShiftShiftFst j = oddShiftFst j.castSucc := by
rfl
lemma δa₃_δ₃ : @δa₃ n = oddMid := by
lemma oddShiftShiftMid_eq_oddMid : @oddShiftShiftMid n = oddMid := by
rw [Fin.ext_iff]
simp only [succ_eq_add_one, δa₃, Fin.isValue, Fin.coe_cast, Fin.coe_castAdd, Fin.coe_natAdd,
Fin.val_eq_zero, add_zero, oddMid]
simp only [succ_eq_add_one, oddShiftShiftMid, Fin.isValue, Fin.coe_cast, Fin.coe_castAdd,
Fin.coe_natAdd, Fin.val_eq_zero, add_zero, oddMid]
exact Nat.add_comm 1 n
lemma δa₃_δ!₁ : δa₃ = oddShiftFstEmbed (Fin.last n) := by
lemma oddShiftShiftMid_eq_oddShiftFst_last : oddShiftShiftMid = oddShiftFst (Fin.last n) := by
rfl
lemma δa₄_δ₂ (j : Fin n.succ) : δa₄ j = oddSndEmbed j := by
lemma oddShiftShiftSnd_eq_oddSnd (j : Fin n.succ) : oddShiftShiftSnd j = oddSnd j := by
rw [Fin.ext_iff]
simp only [succ_eq_add_one, δa₄, Fin.coe_cast, Fin.coe_natAdd, oddSndEmbed, add_left_inj]
simp only [succ_eq_add_one, oddShiftShiftSnd, Fin.coe_cast, Fin.coe_natAdd, oddSnd, add_left_inj]
exact Nat.add_comm 1 n
lemma δa₄_δ!₂ (j : Fin n.succ) : δa₄ j = oddShiftSndEmbed j := by
lemma oddShiftShiftSnd_eq_oddShiftSnd (j : Fin n.succ) : oddShiftShiftSnd j = oddShiftSnd j := by
rw [Fin.ext_iff]
rfl
lemma δ₂_δ!₂ (j : Fin n) : oddSndEmbed j = oddShiftSndEmbed j := by
lemma oddSnd_eq_oddShiftSnd (j : Fin n) : oddSnd j = oddShiftSnd j := by
rw [Fin.ext_iff]
simp only [oddSndEmbed, Fin.coe_cast, Fin.coe_natAdd, oddShiftSndEmbed, add_left_inj]
simp only [oddSnd, Fin.coe_cast, Fin.coe_natAdd, oddShiftSnd, add_left_inj]
exact Nat.add_comm n 1
lemma sum_δ (S : Fin (2 * n + 1) → ℚ) :
∑ i, S i = S oddMid + ∑ i : Fin n, ((S ∘ oddFstEmbed) i + (S ∘ oddSndEmbed) i) := by
lemma sum_odd (S : Fin (2 * n + 1) → ℚ) :
∑ i, S i = S oddMid + ∑ i : Fin n, ((S ∘ oddFst) i + (S ∘ oddSnd) i) := by
have h1 : ∑ i, S i = ∑ i : Fin (n + 1 + n), S (Fin.cast (split_odd n) i) := by
rw [Finset.sum_equiv (Fin.castOrderIso (split_odd n)).symm.toEquiv]
· intro i
@ -133,10 +137,10 @@ lemma sum_δ (S : Fin (2 * n + 1) → ℚ) :
rw [Finset.sum_add_distrib]
rfl
lemma sum_δ! (S : Fin (2 * n + 1) → ℚ) :
∑ i, S i = S oddShiftZero + ∑ i : Fin n, ((S ∘ oddShiftFstEmbed) i + (S ∘ oddShiftSndEmbed) i) := by
have h1 : ∑ i, S i = ∑ i : Fin ((1+n)+n), S (Fin.cast (split_odd! n) i) := by
rw [Finset.sum_equiv (Fin.castOrderIso (split_odd! n)).symm.toEquiv]
lemma sum_oddShift (S : Fin (2 * n + 1) → ℚ) :
∑ i, S i = S oddShiftZero + ∑ i : Fin n, ((S ∘ oddShiftFst) i + (S ∘ oddShiftSnd) i) := by
have h1 : ∑ i, S i = ∑ i : Fin ((1+n)+n), S (Fin.cast (odd_shift_eq n) i) := by
rw [Finset.sum_equiv (Fin.castOrderIso (odd_shift_eq n)).symm.toEquiv]
· intro i
simp only [mem_univ, Fin.castOrderIso, RelIso.coe_fn_toEquiv]
· exact fun _ _ => rfl
@ -152,10 +156,10 @@ section theBasisVectors
/-- The first part of the basis as charge assignments. -/
def basisAsCharges (j : Fin n) : (PureU1 (2 * n + 1)).Charges :=
fun i =>
if i = oddFstEmbed j then
if i = oddFst j then
1
else
if i = oddSndEmbed j then
if i = oddSnd j then
- 1
else
0
@ -163,24 +167,24 @@ def basisAsCharges (j : Fin n) : (PureU1 (2 * n + 1)).Charges :=
/-- The second part of the basis as charge assignments. -/
def basis!AsCharges (j : Fin n) : (PureU1 (2 * n + 1)).Charges :=
fun i =>
if i = oddShiftFstEmbed j then
if i = oddShiftFst j then
1
else
if i = oddShiftSndEmbed j then
if i = oddShiftSnd j then
- 1
else
0
lemma basis_on_δ₁_self (j : Fin n) : basisAsCharges j (oddFstEmbed j) = 1 := by
lemma basis_on_oddFst_self (j : Fin n) : basisAsCharges j (oddFst j) = 1 := by
simp [basisAsCharges]
lemma basis!_on_δ!₁_self (j : Fin n) : basis!AsCharges j (oddShiftFstEmbed j) = 1 := by
lemma basis!_on_oddShiftFst_self (j : Fin n) : basis!AsCharges j (oddShiftFst j) = 1 := by
simp [basis!AsCharges]
lemma basis_on_δ₁_other {k j : Fin n} (h : k ≠ j) :
basisAsCharges k (oddFstEmbed j) = 0 := by
lemma basis_on_oddFst_other {k j : Fin n} (h : k ≠ j) :
basisAsCharges k (oddFst j) = 0 := by
simp only [basisAsCharges, PureU1_numberCharges]
simp only [oddFstEmbed, oddSndEmbed]
simp only [oddFst, oddSnd]
split
· rename_i h1
rw [Fin.ext_iff] at h1
@ -195,10 +199,10 @@ lemma basis_on_δ₁_other {k j : Fin n} (h : k ≠ j) :
omega
· rfl
lemma basis!_on_δ!₁_other {k j : Fin n} (h : k ≠ j) :
basis!AsCharges k (oddShiftFstEmbed j) = 0 := by
lemma basis!_on_oddShiftFst_other {k j : Fin n} (h : k ≠ j) :
basis!AsCharges k (oddShiftFst j) = 0 := by
simp only [basis!AsCharges, PureU1_numberCharges]
simp only [oddShiftFstEmbed, oddShiftSndEmbed]
simp only [oddShiftFst, oddShiftSnd]
split
· rename_i h1
rw [Fin.ext_iff] at h1
@ -213,19 +217,20 @@ lemma basis!_on_δ!₁_other {k j : Fin n} (h : k ≠ j) :
omega
rfl
lemma basis_on_other {k : Fin n} {j : Fin (2 * n + 1)} (h1 : j ≠ oddFstEmbed k) (h2 : j ≠ oddSndEmbed k) :
lemma basis_on_other {k : Fin n} {j : Fin (2 * n + 1)} (h1 : j ≠ oddFst k) (h2 : j ≠ oddSnd k) :
basisAsCharges k j = 0 := by
simp only [basisAsCharges, PureU1_numberCharges]
simp_all only [ne_eq, ↓reduceIte]
lemma basis!_on_other {k : Fin n} {j : Fin (2 * n + 1)} (h1 : j ≠ oddShiftFstEmbed k) (h2 : j ≠ oddShiftSndEmbed k) :
lemma basis!_on_other {k : Fin n} {j : Fin (2 * n + 1)}
(h1 : j ≠ oddShiftFst k) (h2 : j ≠ oddShiftSnd k) :
basis!AsCharges k j = 0 := by
simp only [basis!AsCharges, PureU1_numberCharges]
simp_all only [ne_eq, ↓reduceIte]
lemma basis_δ₂_eq_minus_δ₁ (j i : Fin n) :
basisAsCharges j (oddSndEmbed i) = - basisAsCharges j (oddFstEmbed i) := by
simp only [basisAsCharges, PureU1_numberCharges, oddSndEmbed, oddFstEmbed]
lemma basis_oddSnd_eq_minus_oddFst (j i : Fin n) :
basisAsCharges j (oddSnd i) = - basisAsCharges j (oddFst i) := by
simp only [basisAsCharges, PureU1_numberCharges, oddSnd, oddFst]
split <;> split
any_goals split
any_goals split
@ -238,9 +243,9 @@ lemma basis_δ₂_eq_minus_δ₁ (j i : Fin n) :
all_goals simp_all
all_goals omega
lemma basis!_δ!₂_eq_minus_δ!₁ (j i : Fin n) :
basis!AsCharges j (oddShiftSndEmbed i) = - basis!AsCharges j (oddShiftFstEmbed i) := by
simp only [basis!AsCharges, PureU1_numberCharges, oddShiftSndEmbed, oddShiftFstEmbed]
lemma basis!_oddShiftSnd_eq_minus_oddShiftFst (j i : Fin n) :
basis!AsCharges j (oddShiftSnd i) = - basis!AsCharges j (oddShiftFst i) := by
simp only [basis!AsCharges, PureU1_numberCharges, oddShiftSnd, oddShiftFst]
split <;> split
any_goals split
any_goals split
@ -255,59 +260,60 @@ lemma basis!_δ!₂_eq_minus_δ!₁ (j i : Fin n) :
all_goals simp_all
all_goals omega
lemma basis_on_δ₂_self (j : Fin n) : basisAsCharges j (oddSndEmbed j) = - 1 := by
rw [basis_δ₂_eq_minus_δ₁, basis_on_δ₁_self]
lemma basis_on_oddSnd_self (j : Fin n) : basisAsCharges j (oddSnd j) = - 1 := by
rw [basis_oddSnd_eq_minus_oddFst, basis_on_oddFst_self]
lemma basis!_on_δ!₂_self (j : Fin n) : basis!AsCharges j (oddShiftSndEmbed j) = - 1 := by
rw [basis!_δ!₂_eq_minus_δ!₁, basis!_on_δ!₁_self]
lemma basis!_on_oddShiftSnd_self (j : Fin n) : basis!AsCharges j (oddShiftSnd j) = - 1 := by
rw [basis!_oddShiftSnd_eq_minus_oddShiftFst, basis!_on_oddShiftFst_self]
lemma basis_on_δ₂_other {k j : Fin n} (h : k ≠ j) : basisAsCharges k (oddSndEmbed j) = 0 := by
rw [basis_δ₂_eq_minus_δ₁, basis_on_δ₁_other h]
lemma basis_on_oddSnd_other {k j : Fin n} (h : k ≠ j) : basisAsCharges k (oddSnd j) = 0 := by
rw [basis_oddSnd_eq_minus_oddFst, basis_on_oddFst_other h]
rfl
lemma basis!_on_δ!₂_other {k j : Fin n} (h : k ≠ j) : basis!AsCharges k (oddShiftSndEmbed j) = 0 := by
rw [basis!_δ!₂_eq_minus_δ!₁, basis!_on_δ!₁_other h]
lemma basis!_on_oddShiftSnd_other {k j : Fin n} (h : k ≠ j) :
basis!AsCharges k (oddShiftSnd j) = 0 := by
rw [basis!_oddShiftSnd_eq_minus_oddShiftFst, basis!_on_oddShiftFst_other h]
rfl
lemma basis_on_δ₃ (j : Fin n) : basisAsCharges j oddMid = 0 := by
lemma basis_on_oddMid (j : Fin n) : basisAsCharges j oddMid = 0 := by
simp only [basisAsCharges, PureU1_numberCharges]
split <;> rename_i h
· rw [Fin.ext_iff] at h
simp only [oddMid, Fin.isValue, Fin.coe_cast, Fin.coe_castAdd, Fin.coe_natAdd, Fin.val_eq_zero,
add_zero, oddFstEmbed] at h
add_zero, oddFst] at h
omega
· split <;> rename_i h2
· rw [Fin.ext_iff] at h2
simp only [oddMid, Fin.isValue, Fin.coe_cast, Fin.coe_castAdd, Fin.coe_natAdd, Fin.val_eq_zero,
add_zero, oddSndEmbed] at h2
simp only [oddMid, Fin.isValue, Fin.coe_cast, Fin.coe_castAdd, Fin.coe_natAdd,
Fin.val_eq_zero, add_zero, oddSnd] at h2
omega
· rfl
lemma basis!_on_δ!₃ (j : Fin n) : basis!AsCharges j oddShiftZero = 0 := by
lemma basis!_on_oddShiftZero (j : Fin n) : basis!AsCharges j oddShiftZero = 0 := by
simp only [basis!AsCharges, PureU1_numberCharges]
split <;> rename_i h
· rw [Fin.ext_iff] at h
simp only [oddShiftZero, Fin.isValue, Fin.coe_cast, Fin.coe_castAdd, Fin.val_eq_zero, oddShiftFstEmbed,
Fin.coe_natAdd] at h
simp only [oddShiftZero, Fin.isValue, Fin.coe_cast, Fin.coe_castAdd, Fin.val_eq_zero,
oddShiftFst, Fin.coe_natAdd] at h
omega
· split <;> rename_i h2
· rw [Fin.ext_iff] at h2
simp only [oddShiftZero, Fin.isValue, Fin.coe_cast, Fin.coe_castAdd, Fin.val_eq_zero, oddShiftSndEmbed,
Fin.coe_natAdd] at h2
simp only [oddShiftZero, Fin.isValue, Fin.coe_cast, Fin.coe_castAdd, Fin.val_eq_zero,
oddShiftSnd, Fin.coe_natAdd] at h2
omega
· rfl
lemma basis_linearACC (j : Fin n) : (accGrav (2 * n + 1)) (basisAsCharges j) = 0 := by
rw [accGrav]
simp only [LinearMap.coe_mk, AddHom.coe_mk]
erw [sum_δ]
simp [basis_δ₂_eq_minus_δ₁, basis_on_δ₃]
erw [sum_odd]
simp [basis_oddSnd_eq_minus_oddFst, basis_on_oddMid]
lemma basis!_linearACC (j : Fin n) : (accGrav (2 * n + 1)) (basis!AsCharges j) = 0 := by
rw [accGrav]
simp only [LinearMap.coe_mk, AddHom.coe_mk]
rw [sum_δ!, basis!_on_δ!₃]
simp [basis!_δ!₂_eq_minus_δ!₁]
rw [sum_oddShift, basis!_on_oddShiftZero]
simp [basis!_oddShiftSnd_eq_minus_oddShiftFst]
/-- The first part of the basis as `LinSols`. -/
@[simps!]
@ -337,23 +343,24 @@ def basisa : Fin n ⊕ Fin n → (PureU1 (2 * n + 1)).LinSols := fun i =>
end theBasisVectors
/-- Swapping the elements δ!₁ j and δ!₂ j is equivalent to adding a vector basis!AsCharges j. -/
/-- Swapping the elements oddShiftFst j and oddShiftSnd j is equivalent to adding a vector
basis!AsCharges j. -/
lemma swap!_as_add {S S' : (PureU1 (2 * n + 1)).LinSols} (j : Fin n)
(hS : ((FamilyPermutations (2 * n + 1)).linSolRep
(pairSwap (oddShiftFstEmbed j) (oddShiftSndEmbed j))) S = S') :
S'.val = S.val + (S.val (oddShiftSndEmbed j) - S.val (oddShiftFstEmbed j)) • basis!AsCharges j := by
(pairSwap (oddShiftFst j) (oddShiftSnd j))) S = S') :
S'.val = S.val + (S.val (oddShiftSnd j) - S.val (oddShiftFst j)) • basis!AsCharges j := by
funext i
rw [← hS, FamilyPermutations_anomalyFreeLinear_apply]
by_cases hi : i = oddShiftFstEmbed j
by_cases hi : i = oddShiftFst j
· subst hi
simp [HSMul.hSMul, basis!_on_δ!₁_self, pairSwap_inv_fst]
· by_cases hi2 : i = oddShiftSndEmbed j
simp [HSMul.hSMul, basis!_on_oddShiftFst_self, pairSwap_inv_fst]
· by_cases hi2 : i = oddShiftSnd j
· subst hi2
simp [HSMul.hSMul,basis!_on_δ!₂_self, pairSwap_inv_snd]
simp [HSMul.hSMul,basis!_on_oddShiftSnd_self, pairSwap_inv_snd]
· simp only [Equiv.invFun_as_coe, HSMul.hSMul, ACCSystemCharges.chargesAddCommMonoid_add,
ACCSystemCharges.chargesModule_smul]
rw [basis!_on_other hi hi2]
change S.val ((pairSwap (oddShiftFstEmbed j) (oddShiftSndEmbed j)).invFun i) =_
change S.val ((pairSwap (oddShiftFst j) (oddShiftSnd j)).invFun i) =_
erw [pairSwap_inv_other (Ne.symm hi) (Ne.symm hi2)]
simp
@ -366,146 +373,148 @@ def P! (f : Fin n → ℚ) : (PureU1 (2 * n + 1)).Charges := ∑ i, f i • basi
/-- A point in the span of the basis as a charge. -/
def Pa (f : Fin n → ℚ) (g : Fin n → ℚ) : (PureU1 (2 * n + 1)).Charges := P f + P! g
lemma P_δ₁ (f : Fin n → ℚ) (j : Fin n) : P f (oddFstEmbed j) = f j := by
lemma P_oddFst (f : Fin n → ℚ) (j : Fin n) : P f (oddFst j) = f j := by
rw [P, sum_of_charges]
simp only [HSMul.hSMul, SMul.smul]
rw [Finset.sum_eq_single j]
· rw [basis_on_δ₁_self]
· rw [basis_on_oddFst_self]
exact Rat.mul_one (f j)
· intro k _ hkj
rw [basis_on_δ₁_other hkj]
rw [basis_on_oddFst_other hkj]
exact Rat.mul_zero (f k)
· 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 (oddShiftFstEmbed j) = f j := by
lemma P!_oddShiftFst (f : Fin n → ℚ) (j : Fin n) : P! f (oddShiftFst j) = f j := by
rw [P!, sum_of_charges]
simp only [HSMul.hSMul, SMul.smul]
rw [Finset.sum_eq_single j]
· rw [basis!_on_δ!₁_self]
· rw [basis!_on_oddShiftFst_self]
exact Rat.mul_one (f j)
· intro k _ hkj
rw [basis!_on_δ!₁_other hkj]
rw [basis!_on_oddShiftFst_other hkj]
exact Rat.mul_zero (f k)
· simp only [mem_univ, not_true_eq_false, _root_.mul_eq_zero, IsEmpty.forall_iff]
lemma P_δ₂ (f : Fin n → ℚ) (j : Fin n) : P f (oddSndEmbed j) = - f j := by
lemma P_oddSnd (f : Fin n → ℚ) (j : Fin n) : P f (oddSnd j) = - f j := by
rw [P, sum_of_charges]
simp only [HSMul.hSMul, SMul.smul]
rw [Finset.sum_eq_single j]
· rw [basis_on_δ₂_self]
· rw [basis_on_oddSnd_self]
exact mul_neg_one (f j)
· intro k _ hkj
rw [basis_on_δ₂_other hkj]
rw [basis_on_oddSnd_other hkj]
exact Rat.mul_zero (f k)
· simp
lemma P!_δ!₂ (f : Fin n → ℚ) (j : Fin n) : P! f (oddShiftSndEmbed j) = - f j := by
lemma P!_oddShiftSnd (f : Fin n → ℚ) (j : Fin n) : P! f (oddShiftSnd j) = - f j := by
rw [P!, sum_of_charges]
simp only [HSMul.hSMul, SMul.smul]
rw [Finset.sum_eq_single j]
· rw [basis!_on_δ!₂_self]
· rw [basis!_on_oddShiftSnd_self]
exact mul_neg_one (f j)
· intro k _ hkj
rw [basis!_on_δ!₂_other hkj]
rw [basis!_on_oddShiftSnd_other hkj]
exact Rat.mul_zero (f k)
· simp
lemma P_δ₃ (f : Fin n → ℚ) : P f (oddMid) = 0 := by
lemma P_oddMid (f : Fin n → ℚ) : P f (oddMid) = 0 := by
rw [P, sum_of_charges]
simp [HSMul.hSMul, SMul.smul, basis_on_δ₃]
simp [HSMul.hSMul, SMul.smul, basis_on_oddMid]
lemma P!_δ!₃ (f : Fin n → ℚ) : P! f (oddShiftZero) = 0 := by
lemma P!_oddShiftZero (f : Fin n → ℚ) : P! f (oddShiftZero) = 0 := by
rw [P!, sum_of_charges]
simp [HSMul.hSMul, SMul.smul, basis!_on_δ!₃]
simp [HSMul.hSMul, SMul.smul, basis!_on_oddShiftZero]
lemma Pa_δa₁ (f g : Fin n.succ → ℚ) : Pa f g δa₁ = f 0 := by
lemma Pa_oddShiftShiftZero (f g : Fin n.succ → ℚ) : Pa f g oddShiftShiftZero = f 0 := by
rw [Pa]
simp only [ACCSystemCharges.chargesAddCommMonoid_add]
nth_rewrite 1 [δa₁_δ₁]
rw [δa₁_δ!₃]
rw [P_δ₁, P!_δ!₃]
nth_rewrite 1 [oddShiftShiftZero_eq_oddFst_zero]
rw [oddShiftShiftZero_eq_oddShiftZero]
rw [P_oddFst, P!_oddShiftZero]
exact Rat.add_zero (f 0)
lemma Pa_δa₂ (f g : Fin n.succ → ℚ) (j : Fin n) : Pa f g (δa₂ j) = f j.succ + g j.castSucc := by
lemma Pa_oddShiftShiftFst (f g : Fin n.succ → ℚ) (j : Fin n) :
Pa f g (oddShiftShiftFst j) = f j.succ + g j.castSucc := by
rw [Pa]
simp only [ACCSystemCharges.chargesAddCommMonoid_add]
nth_rewrite 1 [δa₂_δ₁]
rw [δa₂_δ!₁]
rw [P_δ₁, P!_δ!₁]
nth_rewrite 1 [oddShiftShiftFst_eq_oddFst_succ]
rw [oddShiftShiftFst_eq_oddShiftFst_castSucc]
rw [P_oddFst, P!_oddShiftFst]
lemma Pa_δa₃ (f g : Fin n.succ → ℚ) : Pa f g (δa₃) = g (Fin.last n) := by
lemma Pa_oddShiftShiftMid (f g : Fin n.succ → ℚ) : Pa f g (oddShiftShiftMid) = g (Fin.last n) := by
rw [Pa]
simp only [ACCSystemCharges.chargesAddCommMonoid_add]
nth_rewrite 1 [δa₃_δ₃]
rw [δa₃_δ!₁]
rw [P_δ₃, P!_δ!₁]
nth_rewrite 1 [oddShiftShiftMid_eq_oddMid]
rw [oddShiftShiftMid_eq_oddShiftFst_last]
rw [P_oddMid, P!_oddShiftFst]
exact Rat.zero_add (g (Fin.last n))
lemma Pa_δa₄ (f g : Fin n.succ → ℚ) (j : Fin n.succ) : Pa f g (δa₄ j) = - f j - g j := by
lemma Pa_oddShiftShiftSnd (f g : Fin n.succ → ℚ) (j : Fin n.succ) :
Pa f g (oddShiftShiftSnd j) = - f j - g j := by
rw [Pa]
simp only [ACCSystemCharges.chargesAddCommMonoid_add]
nth_rewrite 1 [δa₄_δ₂]
rw [δa₄_δ!₂]
rw [P_δ₂, P!_δ!₂]
nth_rewrite 1 [oddShiftShiftSnd_eq_oddSnd]
rw [oddShiftShiftSnd_eq_oddShiftSnd]
rw [P_oddSnd, P!_oddShiftSnd]
ring
lemma P_linearACC (f : Fin n → ℚ) : (accGrav (2 * n + 1)) (P f) = 0 := by
rw [accGrav]
simp only [LinearMap.coe_mk, AddHom.coe_mk]
rw [sum_δ]
simp [P_δ₂, P_δ₁, P_δ₃]
rw [sum_odd]
simp [P_oddSnd, P_oddFst, P_oddMid]
lemma P!_linearACC (f : Fin n → ℚ) : (accGrav (2 * n + 1)) (P! f) = 0 := by
rw [accGrav]
simp only [LinearMap.coe_mk, AddHom.coe_mk]
rw [sum_δ!]
simp [P!_δ!₂, P!_δ!₁, P!_δ!₃]
rw [sum_oddShift]
simp [P!_oddShiftSnd, P!_oddShiftFst, P!_oddShiftZero]
lemma P_accCube (f : Fin n → ℚ) : accCube (2 * n +1) (P f) = 0 := by
rw [accCube_explicit, sum_δ, P_δ₃]
rw [accCube_explicit, sum_odd, P_oddMid]
simp only [ne_eq, OfNat.ofNat_ne_zero, not_false_eq_true, zero_pow, Function.comp_apply, zero_add]
apply Finset.sum_eq_zero
intro i _
simp only [P_δ₁, P_δ₂]
simp only [P_oddFst, P_oddSnd]
ring
lemma P!_accCube (f : Fin n → ℚ) : accCube (2 * n +1) (P! f) = 0 := by
rw [accCube_explicit, sum_δ!, P!_δ!₃]
rw [accCube_explicit, sum_oddShift, P!_oddShiftZero]
simp only [ne_eq, OfNat.ofNat_ne_zero, not_false_eq_true, zero_pow, Function.comp_apply, zero_add]
apply Finset.sum_eq_zero
intro i _
simp only [P!_δ!₁, P!_δ!₂]
simp only [P!_oddShiftFst, P!_oddShiftSnd]
ring
lemma P_P_P!_accCube (g : Fin n → ℚ) (j : Fin n) :
accCubeTriLinSymm (P g) (P g) (basis!AsCharges j)
= (P g (oddShiftFstEmbed j))^2 - (g j)^2 := by
= (P g (oddShiftFst j))^2 - (g j)^2 := by
simp only [accCubeTriLinSymm, PureU1Charges_numberCharges, TriLinearSymm.mk₃_toFun_apply_apply]
rw [sum_δ!, basis!_on_δ!₃]
rw [sum_oddShift, basis!_on_oddShiftZero]
simp only [mul_zero, Function.comp_apply, zero_add]
rw [Finset.sum_eq_single j, basis!_on_δ!₁_self, basis!_on_δ!₂_self]
· rw [← δ₂_δ!₂, P_δ₂]
rw [Finset.sum_eq_single j, basis!_on_oddShiftFst_self, basis!_on_oddShiftSnd_self]
· rw [← oddSnd_eq_oddShiftSnd, P_oddSnd]
ring
· intro k _ hkj
erw [basis!_on_δ!₁_other hkj.symm, basis!_on_δ!₂_other hkj.symm]
erw [basis!_on_oddShiftFst_other hkj.symm, basis!_on_oddShiftSnd_other hkj.symm]
simp only [mul_zero, add_zero]
· simp
lemma P_zero (f : Fin n → ℚ) (h : P f = 0) : ∀ i, f i = 0 := by
intro i
erw [← P_δ₁ f]
erw [← P_oddFst f]
rw [h]
rfl
lemma P!_zero (f : Fin n → ℚ) (h : P! f = 0) : ∀ i, f i = 0 := by
intro i
rw [← P!_δ!₁ f]
rw [← P!_oddShiftFst f]
rw [h]
rfl
lemma Pa_zero (f g : Fin n.succ → ℚ) (h : Pa f g = 0) :
∀ i, f i = 0 := by
have h₃ := Pa_δa₁ f g
have h₃ := Pa_oddShiftShiftZero f g
rw [h] at h₃
change 0 = _ at h₃
simp only at h₃
@ -516,11 +525,11 @@ lemma Pa_zero (f g : Fin n.succ → ℚ) (h : Pa f g = 0) :
rename_i iv hi
have hivi : iv < n.succ := lt_of_succ_lt hiv
have hi2 := hi hivi
have h1 := Pa_δa₄ f g ⟨iv, hivi⟩
have h1 := Pa_oddShiftShiftSnd f g ⟨iv, hivi⟩
rw [h, hi2] at h1
change 0 = _ at h1
simp only [neg_zero, succ_eq_add_one, zero_sub, zero_eq_neg] at h1
have h2 := Pa_δa₂ f g ⟨iv, succ_lt_succ_iff.mp hiv⟩
have h2 := Pa_oddShiftShiftFst f g ⟨iv, succ_lt_succ_iff.mp hiv⟩
simp only [succ_eq_add_one, h, Fin.succ_mk, Fin.castSucc_mk, h1, add_zero] at h2
exact h2.symm
exact hinduc i.val i.prop
@ -676,16 +685,16 @@ lemma span_basis (S : (PureU1 (2 * n.succ + 1)).LinSols) :
lemma span_basis_swap! {S : (PureU1 (2 * n.succ + 1)).LinSols} (j : Fin n.succ)
(hS : ((FamilyPermutations (2 * n.succ + 1)).linSolRep
(pairSwap (oddShiftFstEmbed j) (oddShiftSndEmbed j))) S = S') (g f : Fin n.succ → ℚ) (hS1 : S.val = P g + P! f) :
∃ (g' f' : Fin n.succ → ℚ),
(pairSwap (oddShiftFst j) (oddShiftSnd j))) S = S') (g f : Fin n.succ → ℚ)
(hS1 : S.val = P g + P! f) : ∃ (g' f' : Fin n.succ → ℚ),
S'.val = P g' + P! f' ∧ P! f' = P! f +
(S.val (oddShiftSndEmbed j) - S.val (oddShiftFstEmbed j)) • basis!AsCharges j ∧ g' = g := by
let X := P! f + (S.val (oddShiftSndEmbed j) - S.val (oddShiftFstEmbed j)) • basis!AsCharges j
(S.val (oddShiftSnd j) - S.val (oddShiftFst j)) • basis!AsCharges j ∧ g' = g := by
let X := P! f + (S.val (oddShiftSnd j) - S.val (oddShiftFst j)) • basis!AsCharges j
have hf : P! f ∈ Submodule.span ℚ (Set.range basis!AsCharges) := by
rw [(mem_span_range_iff_exists_fun ℚ)]
use f
rfl
have hP : (S.val (oddShiftSndEmbed j) - S.val (oddShiftFstEmbed j)) • basis!AsCharges j ∈
have hP : (S.val (oddShiftSnd j) - S.val (oddShiftFst j)) • basis!AsCharges j ∈
Submodule.span ℚ (Set.range basis!AsCharges) := by
apply Submodule.smul_mem
apply SetLike.mem_of_subset