refactor: Lint
This commit is contained in:
jstoobysmith
parent
52e591fa7a
commit
9f27a3a9fd
46 changed files with 176 additions and 168 deletions
|
|
@ -550,7 +550,7 @@ def Pa' (f : (Fin n.succ) ⊕ (Fin n) → ℚ) : (PureU1 (2 * n.succ)).LinSols :
|
|||
∑ i, f i • basisa i
|
||||
|
||||
lemma Pa'_P'_P!' (f : (Fin n.succ) ⊕ (Fin n) → ℚ) :
|
||||
Pa' f = P' (f ∘ Sum.inl) + P!' (f ∘ Sum.inr):= by
|
||||
Pa' f = P' (f ∘ Sum.inl) + P!' (f ∘ Sum.inr) := by
|
||||
exact Fintype.sum_sum_type _
|
||||
|
||||
lemma P'_val (f : Fin n.succ → ℚ) : (P' f).val = P f := by
|
||||
|
|
@ -617,7 +617,7 @@ lemma Pa'_eq (f f' : (Fin n.succ) ⊕ (Fin n) → ℚ) : Pa' f = Pa' f' ↔ f =
|
|||
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
|
||||
(fun i => f i + -f' i) h1
|
||||
have h2i := h2 i
|
||||
linarith
|
||||
intro h
|
||||
|
|
@ -685,9 +685,9 @@ lemma span_basis (S : (PureU1 (2 * n.succ)).LinSols) :
|
|||
rfl
|
||||
|
||||
lemma P!_in_span (f : Fin n → ℚ) : P! f ∈ Submodule.span ℚ (Set.range basis!AsCharges) := by
|
||||
rw [(mem_span_range_iff_exists_fun ℚ)]
|
||||
use f
|
||||
rfl
|
||||
rw [(mem_span_range_iff_exists_fun ℚ)]
|
||||
use f
|
||||
rfl
|
||||
|
||||
lemma smul_basis!AsCharges_in_span (S : (PureU1 (2 * n.succ)).LinSols) (j : Fin n) :
|
||||
(S.val (δ!₂ j) - S.val (δ!₁ j)) • basis!AsCharges j ∈
|
||||
|
|
@ -700,11 +700,9 @@ lemma smul_basis!AsCharges_in_span (S : (PureU1 (2 * n.succ)).LinSols) (j : Fin
|
|||
lemma span_basis_swap! {S : (PureU1 (2 * n.succ)).LinSols} (j : Fin n)
|
||||
(hS : ((FamilyPermutations (2 * n.succ)).linSolRep
|
||||
(pairSwap (δ!₁ j) (δ!₂ j))) S = S') (g : Fin n.succ → ℚ) (f : Fin n → ℚ)
|
||||
(h : S.val = P g + P! f):
|
||||
∃
|
||||
(g' : Fin n.succ → ℚ) (f' : Fin n → ℚ),
|
||||
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
|
||||
(h : S.val = P g + P! f) : ∃ (g' : Fin n.succ → ℚ) (f' : Fin n → ℚ),
|
||||
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
|
||||
apply Submodule.add_mem
|
||||
|
|
@ -723,10 +721,8 @@ lemma span_basis_swap! {S : (PureU1 (2 * n.succ)).LinSols} (j : Fin n)
|
|||
exact hS
|
||||
|
||||
lemma vectorLikeEven_in_span (S : (PureU1 (2 * n.succ)).LinSols)
|
||||
(hS : VectorLikeEven S.val) :
|
||||
∃ (M : (FamilyPermutations (2 * n.succ)).group),
|
||||
(FamilyPermutations (2 * n.succ)).linSolRep M S
|
||||
∈ Submodule.span ℚ (Set.range basis) := by
|
||||
(hS : VectorLikeEven S.val) : ∃ (M : (FamilyPermutations (2 * n.succ)).group),
|
||||
(FamilyPermutations (2 * n.succ)).linSolRep M S ∈ Submodule.span ℚ (Set.range basis) := by
|
||||
use (Tuple.sort S.val).symm
|
||||
change sortAFL S ∈ Submodule.span ℚ (Set.range basis)
|
||||
rw [mem_span_range_iff_exists_fun ℚ]
|
||||
|
|
|
|||
|
|
@ -49,23 +49,23 @@ lemma lineInCubic_expand {S : (PureU1 (2 * n.succ)).LinSols} (h : LineInCubic S)
|
|||
change accCubeTriLinSymm.toCubic (a • P g + b • P! f) = 0 at h1
|
||||
simp only [TriLinearSymm.toCubic_add] at h1
|
||||
simp only [HomogeneousCubic.map_smul,
|
||||
accCubeTriLinSymm.map_smul₁, accCubeTriLinSymm.map_smul₂, accCubeTriLinSymm.map_smul₃] at h1
|
||||
accCubeTriLinSymm.map_smul₁, accCubeTriLinSymm.map_smul₂, accCubeTriLinSymm.map_smul₃] at h1
|
||||
erw [P_accCube, P!_accCube] at h1
|
||||
rw [← h1]
|
||||
ring
|
||||
|
||||
/--
|
||||
This lemma states that for a given `S` of type `(PureU1 (2 * n.succ)).AnomalyFreeLinear` and
|
||||
a proof `h` that the line through `S` lies on a cubic curve,
|
||||
for any functions `g : Fin n.succ → ℚ` and `f : Fin n → ℚ`, if `S.val = P g + P! f`,
|
||||
then `accCubeTriLinSymm.toFun (P g, P g, P! f) = 0`.
|
||||
This lemma states that for a given `S` of type `(PureU1 (2 * n.succ)).AnomalyFreeLinear` and
|
||||
a proof `h` that the line through `S` lies on a cubic curve,
|
||||
for any functions `g : Fin n.succ → ℚ` and `f : Fin n → ℚ`, if `S.val = P g + P! f`,
|
||||
then `accCubeTriLinSymm.toFun (P g, P g, P! f) = 0`.
|
||||
-/
|
||||
lemma line_in_cubic_P_P_P! {S : (PureU1 (2 * n.succ)).LinSols} (h : LineInCubic S) :
|
||||
∀ (g : Fin n.succ → ℚ) (f : Fin n → ℚ) (_ : S.val = P g + P! f),
|
||||
accCubeTriLinSymm (P g) (P g) (P! f) = 0 := by
|
||||
intro g f hS
|
||||
linear_combination 2 / 3 * (lineInCubic_expand h g f hS 1 1) -
|
||||
(lineInCubic_expand h g f hS 1 2) / 6
|
||||
(lineInCubic_expand h g f hS 1 2) / 6
|
||||
|
||||
/-- A `LinSol` satisfies `LineInCubicPerm` if all its permutations satisfy `lineInCubic`. -/
|
||||
def LineInCubicPerm (S : (PureU1 (2 * n.succ)).LinSols) : Prop :=
|
||||
|
|
|
|||
|
|
@ -49,15 +49,15 @@ lemma parameterizationAsLinear_val (g : Fin n.succ → ℚ) (f : Fin n → ℚ)
|
|||
change a • (_ • (P' g).val + _ • (P!' f).val) = _
|
||||
rw [P'_val, P!'_val]
|
||||
|
||||
lemma parameterizationCharge_cube (g : Fin n.succ → ℚ) (f : Fin n → ℚ) (a : ℚ):
|
||||
lemma parameterizationCharge_cube (g : Fin n.succ → ℚ) (f : Fin n → ℚ) (a : ℚ) :
|
||||
accCube (2* n.succ) (parameterizationAsLinear g f a).val = 0 := by
|
||||
change accCubeTriLinSymm.toCubic _ = 0
|
||||
rw [parameterizationAsLinear_val, HomogeneousCubic.map_smul,
|
||||
TriLinearSymm.toCubic_add, HomogeneousCubic.map_smul, HomogeneousCubic.map_smul]
|
||||
erw [P_accCube, P!_accCube]
|
||||
rw [accCubeTriLinSymm.map_smul₁, accCubeTriLinSymm.map_smul₂,
|
||||
accCubeTriLinSymm.map_smul₃, accCubeTriLinSymm.map_smul₁, accCubeTriLinSymm.map_smul₂,
|
||||
accCubeTriLinSymm.map_smul₃]
|
||||
accCubeTriLinSymm.map_smul₃, accCubeTriLinSymm.map_smul₁, accCubeTriLinSymm.map_smul₂,
|
||||
accCubeTriLinSymm.map_smul₃]
|
||||
ring
|
||||
|
||||
/-- The construction of a `Sol` from a `Fin n.succ → ℚ`, a `Fin n → ℚ` and a `ℚ`. -/
|
||||
|
|
@ -112,7 +112,7 @@ lemma generic_or_special (S : (PureU1 (2 * n.succ)).Sols) :
|
|||
GenericCase S ∨ SpecialCase S := by
|
||||
obtain ⟨g, f, h⟩ := span_basis S.1.1
|
||||
have h1 : accCubeTriLinSymm (P g) (P g) (P! f) ≠ 0 ∨
|
||||
accCubeTriLinSymm (P g) (P g) (P! f) = 0 := by
|
||||
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⟩)
|
||||
|
|
@ -142,8 +142,8 @@ lemma special_case_lineInCubic {S : (PureU1 (2 * n.succ)).Sols}
|
|||
erw [P_accCube, P!_accCube]
|
||||
have h := h g f hS
|
||||
rw [accCubeTriLinSymm.map_smul₁, accCubeTriLinSymm.map_smul₂,
|
||||
accCubeTriLinSymm.map_smul₃, accCubeTriLinSymm.map_smul₁, accCubeTriLinSymm.map_smul₂,
|
||||
accCubeTriLinSymm.map_smul₃]
|
||||
accCubeTriLinSymm.map_smul₃, accCubeTriLinSymm.map_smul₁, accCubeTriLinSymm.map_smul₂,
|
||||
accCubeTriLinSymm.map_smul₃]
|
||||
rw [h]
|
||||
rw [anomalyFree_param _ _ hS] at h
|
||||
simp at h
|
||||
|
|
|
|||
Loading…
Add table
Add a link
Reference in a new issue