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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
View file

@ -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)

32
HepLean/AnomalyCancellation/PureU1/Even/LineInCubic.lean
View file

@ -34,7 +34,7 @@ open BigOperators
variable {n : ℕ}
open VectorLikeEvenPlane
/-- A property on `LinSols`, satisfied if every point on the line between the two planes
/-- A property on `LinSols`, satisfied if every point on the line between the two planes
in the basis through that point is in the cubic. -/
def LineInCubic (S : (PureU1 (2 * n.succ)).LinSols) : Prop :=
∀ (g : Fin n.succ → ℚ) (f : Fin n → ℚ) (_ : S.val = Pa g f) (a b : ℚ) ,
@ -61,18 +61,18 @@ lemma lineInCubic_expand {S : (PureU1 (2 * n.succ)).LinSols} (h : LineInCubic S)
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),
∀ (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
/-- We say a `LinSol` satisfies `lineInCubicPerm` if all its permutations satisfy `lineInCubic`. -/
/-- We say a `LinSol` satisfies `lineInCubicPerm` if all its permutations satisfy `lineInCubic`. -/
def LineInCubicPerm (S : (PureU1 (2 * n.succ)).LinSols) : Prop :=
∀ (M : (FamilyPermutations (2 * n.succ)).group ),
LineInCubic ((FamilyPermutations (2 * n.succ)).linSolRep M S)
/-- If `lineInCubicPerm S` then `lineInCubic S`. -/
/-- If `lineInCubicPerm S` then `lineInCubic S`. -/
lemma lineInCubicPerm_self {S : (PureU1 (2 * n.succ)).LinSols}
(hS : LineInCubicPerm S) : LineInCubic S := hS 1
@ -94,7 +94,7 @@ lemma lineInCubicPerm_swap {S : (PureU1 (2 * n.succ)).LinSols}
(S.val (δ!₂ j) - S.val (δ!₁ j))
* accCubeTriLinSymm (P g) (P g) (basis!AsCharges j) = 0 := by
intro j g f h
let S' := (FamilyPermutations (2 * n.succ)).linSolRep (pairSwap (δ!₁ j) (δ!₂ j)) S
let S' := (FamilyPermutations (2 * n.succ)).linSolRep (pairSwap (δ!₁ j) (δ!₂ j)) S
have hSS' : ((FamilyPermutations (2 * n.succ)).linSolRep (pairSwap (δ!₁ j) (δ!₂ j))) S = S' := rfl
obtain ⟨g', f', hall⟩ := span_basis_swap! j hSS' g f h
have h1 := line_in_cubic_P_P_P! (lineInCubicPerm_self LIC) g f h
@ -106,21 +106,21 @@ lemma lineInCubicPerm_swap {S : (PureU1 (2 * n.succ)).LinSols}
lemma P_P_P!_accCube' {S : (PureU1 (2 * n.succ.succ )).LinSols}
(f : Fin n.succ.succ → ℚ) (g : Fin n.succ → ℚ) (hS : S.val = Pa f g) :
accCubeTriLinSymm (P f) (P f) (basis!AsCharges (Fin.last n)) =
accCubeTriLinSymm (P f) (P f) (basis!AsCharges (Fin.last n)) =
- (S.val (δ!₂ (Fin.last n)) + S.val (δ!₁ (Fin.last n))) * (2 * S.val δ!₄ +
S.val (δ!₂ (Fin.last n)) + S.val (δ!₁ (Fin.last n))) := by
rw [P_P_P!_accCube f (Fin.last n)]
S.val (δ!₂ (Fin.last n)) + S.val (δ!₁ (Fin.last n))) := by
rw [P_P_P!_accCube f (Fin.last n)]
have h1 := Pa_δ!₄ f g
have h2 := Pa_δ!₁ f g (Fin.last n)
have h3 := Pa_δ!₂ f g (Fin.last n)
simp at h1 h2 h3
have hl : f (Fin.succ (Fin.last (n ))) = - Pa f g δ!₄ := by
have hl : f (Fin.succ (Fin.last (n ))) = - Pa f g δ!₄ := by
simp_all only [Fin.succ_last, neg_neg]
erw [hl] at h2
have hg : g (Fin.last n) = Pa f g (δ!₁ (Fin.last n)) + Pa f g δ!₄ := by
have hg : g (Fin.last n) = Pa f g (δ!₁ (Fin.last n)) + Pa f g δ!₄ := by
linear_combination -(1 * h2)
have hll : f (Fin.castSucc (Fin.last (n ))) =
- (Pa f g (δ!₂ (Fin.last n)) + Pa f g (δ!₁ (Fin.last n)) + Pa f g δ!₄) := by
have hll : f (Fin.castSucc (Fin.last (n ))) =
- (Pa f g (δ!₂ (Fin.last n)) + Pa f g (δ!₁ (Fin.last n)) + Pa f g δ!₄) := by
linear_combination h3 - 1 * hg
rw [← hS] at hl hll
rw [hl, hll]
@ -145,7 +145,7 @@ lemma lineInCubicPerm_last_cond {S : (PureU1 (2 * n.succ.succ)).LinSols}
apply Or.inr
exact h1
lemma lineInCubicPerm_last_perm {S : (PureU1 (2 * n.succ.succ)).LinSols}
lemma lineInCubicPerm_last_perm {S : (PureU1 (2 * n.succ.succ)).LinSols}
(LIC : LineInCubicPerm S) : LineInPlaneCond S := by
refine @Prop_three (2 * n.succ.succ) LineInPlaneProp S (δ!₂ (Fin.last n)) (δ!₁ (Fin.last n))
δ!₄ ?_ ?_ ?_ ?_
@ -156,15 +156,15 @@ lemma lineInCubicPerm_last_perm {S : (PureU1 (2 * n.succ.succ)).LinSols}
intro M
exact lineInCubicPerm_last_cond (lineInCubicPerm_permute LIC M)
lemma lineInCubicPerm_constAbs {S : (PureU1 (2 * n.succ.succ)).Sols}
lemma lineInCubicPerm_constAbs {S : (PureU1 (2 * n.succ.succ)).Sols}
(LIC : LineInCubicPerm S.1.1) : ConstAbs S.val :=
linesInPlane_constAbs_AF S (lineInCubicPerm_last_perm LIC)
theorem lineInCubicPerm_vectorLike {S : (PureU1 (2 * n.succ.succ)).Sols}
theorem lineInCubicPerm_vectorLike {S : (PureU1 (2 * n.succ.succ)).Sols}
(LIC : LineInCubicPerm S.1.1) : VectorLikeEven S.val :=
ConstAbs.boundary_value_even S.1.1 (lineInCubicPerm_constAbs LIC)
theorem lineInCubicPerm_in_plane (S : (PureU1 (2 * n.succ.succ)).Sols)
theorem lineInCubicPerm_in_plane (S : (PureU1 (2 * n.succ.succ)).Sols)
(LIC : LineInCubicPerm S.1.1) : ∃ (M : (FamilyPermutations (2 * n.succ.succ)).group),
(FamilyPermutations (2 * n.succ.succ)).linSolRep M S.1.1
∈ Submodule.span ℚ (Set.range basis) :=

18
HepLean/AnomalyCancellation/PureU1/Even/Parameterization.lean
View file

@ -14,7 +14,7 @@ import Mathlib.Tactic.Polyrith
/-!
# Parameterization in even case
Given maps `g : Fin n.succ → ℚ`, `f : Fin n → ℚ` and `a : ℚ` we form a solution to the anomaly
Given maps `g : Fin n.succ → ℚ`, `f : Fin n → ℚ` and `a : ℚ` we form a solution to the anomaly
equations. We show that every solution can be got in this way, up to permutation, unless it, up to
permutation, lives in the plane spanned by the first part of the basis vector.
@ -60,7 +60,7 @@ lemma parameterizationCharge_cube (g : Fin n.succ → ℚ) (f : Fin n → ℚ) (
accCubeTriLinSymm.map_smul₃]
ring
/-- The construction of a `Sol` from a `Fin n.succ → ℚ`, a `Fin n → ℚ` and a `ℚ`. -/
/-- The construction of a `Sol` from a `Fin n.succ → ℚ`, a `Fin n → ℚ` and a `ℚ`. -/
def parameterization (g : Fin n.succ → ℚ) (f : Fin n → ℚ) (a : ℚ) :
(PureU1 (2 * n.succ)).Sols :=
⟨⟨parameterizationAsLinear g f a, by intro i; simp at i; exact Fin.elim0 i⟩,
@ -77,15 +77,15 @@ lemma anomalyFree_param {S : (PureU1 (2 * n.succ)).Sols}
erw [P!_accCube] at hC
linear_combination hC / 3
/-- A proposition on a solution which is true if `accCubeTriLinSymm (P g, P g, P! f) ≠ 0`.
/-- A proposition on a solution which is true if `accCubeTriLinSymm (P g, P g, P! f) ≠ 0`.
In this case our parameterization above will be able to recover this point. -/
def GenericCase (S : (PureU1 (2 * n.succ)).Sols) : Prop :=
∀ (g : Fin n.succ → ℚ) (f : Fin n → ℚ) (_ : S.val = P g + P! f) ,
accCubeTriLinSymm (P g) (P g) (P! f) ≠ 0
accCubeTriLinSymm (P g) (P g) (P! f) ≠ 0
lemma genericCase_exists (S : (PureU1 (2 * n.succ)).Sols)
(hs : ∃ (g : Fin n.succ → ℚ) (f : Fin n → ℚ), S.val = P g + P! f ∧
accCubeTriLinSymm (P g) (P g) (P! f) ≠ 0) : GenericCase S := by
accCubeTriLinSymm (P g) (P g) (P! f) ≠ 0) : GenericCase S := by
intro g f hS hC
obtain ⟨g', f', hS', hC'⟩ := hs
rw [hS] at hS'
@ -94,13 +94,13 @@ lemma genericCase_exists (S : (PureU1 (2 * n.succ)).Sols)
exact hC' hC
/-- A proposition on a solution which is true if `accCubeTriLinSymm (P g, P g, P! f) = 0`.-/
def SpecialCase (S : (PureU1 (2 * n.succ)).Sols) : Prop :=
def SpecialCase (S : (PureU1 (2 * n.succ)).Sols) : Prop :=
∀ (g : Fin n.succ → ℚ) (f : Fin n → ℚ) (_ : S.val = P g + P! f) ,
accCubeTriLinSymm (P g) (P g) (P! f) = 0
lemma specialCase_exists (S : (PureU1 (2 * n.succ)).Sols)
(hs : ∃ (g : Fin n.succ → ℚ) (f : Fin n → ℚ), S.val = P g + P! f ∧
accCubeTriLinSymm (P g) (P g) (P! f) = 0) : SpecialCase S := by
accCubeTriLinSymm (P g) (P g) (P! f) = 0) : SpecialCase S := by
intro g f hS
obtain ⟨g', f', hS', hC'⟩ := hs
rw [hS] at hS'
@ -111,7 +111,7 @@ lemma specialCase_exists (S : (PureU1 (2 * n.succ)).Sols)
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 ∨
have h1 : accCubeTriLinSymm (P g) (P g) (P! f) ≠ 0 ∨
accCubeTriLinSymm (P g) (P g) (P! f) = 0 := by
exact ne_or_eq _ _
cases h1 <;> rename_i h1
@ -119,7 +119,7 @@ lemma generic_or_special (S : (PureU1 (2 * n.succ)).Sols) :
exact Or.inr (specialCase_exists S ⟨g, f, h, h1⟩)
theorem generic_case {S : (PureU1 (2 * n.succ)).Sols} (h : GenericCase S) :
∃ g f a, S = parameterization g f a := by
∃ g f a, S = parameterization g f a := by
obtain ⟨g, f, hS⟩ := span_basis S.1.1
use g, f, (accCubeTriLinSymm (P! f) (P! f) (P g))⁻¹
rw [parameterization]