refactor: Change case of type and props
This commit is contained in:
jstoobysmith
parent
18b83f582e
commit
f7a638d32e
58 changed files with 695 additions and 696 deletions
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@ -34,17 +34,17 @@ variable {n : ℕ}
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/-- An equivalence between `(SMνCharges n).charges` and `(Fin 6 → Fin n → ℚ)`
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splitting the charges into species.-/
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@[simps!]
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def toSpeciesEquiv : (SMνCharges n).charges ≃ (Fin 6 → Fin n → ℚ) :=
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def toSpeciesEquiv : (SMνCharges n).Charges ≃ (Fin 6 → Fin n → ℚ) :=
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((Equiv.curry _ _ _).symm.trans ((@finProdFinEquiv 6 n).arrowCongr (Equiv.refl ℚ))).symm
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/-- Given an `i ∈ Fin 6`, the projection of charges onto a given species. -/
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@[simps!]
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def toSpecies (i : Fin 6) : (SMνCharges n).charges →ₗ[ℚ] (SMνSpecies n).charges where
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def toSpecies (i : Fin 6) : (SMνCharges n).Charges →ₗ[ℚ] (SMνSpecies n).Charges where
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toFun S := toSpeciesEquiv S i
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map_add' _ _ := by aesop
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map_smul' _ _ := by aesop
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lemma charges_eq_toSpecies_eq (S T : (SMνCharges n).charges) :
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lemma charges_eq_toSpecies_eq (S T : (SMνCharges n).Charges) :
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S = T ↔ ∀ i, toSpecies i S = toSpecies i T := by
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apply Iff.intro
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intro h
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@ -60,7 +60,7 @@ lemma toSMSpecies_toSpecies_inv (i : Fin 6) (f : (Fin 6 → Fin n → ℚ) ) :
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change (toSpeciesEquiv ∘ toSpeciesEquiv.symm ) _ i = f i
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simp
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lemma toSpecies_one (S : (SMνCharges 1).charges) (j : Fin 6) :
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lemma toSpecies_one (S : (SMνCharges 1).Charges) (j : Fin 6) :
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toSpecies j S ⟨0, by simp⟩ = S j := by
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match j with
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| 0 => rfl
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@ -94,7 +94,7 @@ variable {n : ℕ}
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/-- The gravitational anomaly equation. -/
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@[simp]
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def accGrav : (SMνCharges n).charges →ₗ[ℚ] ℚ where
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def accGrav : (SMνCharges n).Charges →ₗ[ℚ] ℚ where
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toFun S := ∑ i, (6 * Q S i + 3 * U S i + 3 * D S i + 2 * L S i + E S i + N S i)
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map_add' S T := by
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simp only
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@ -112,7 +112,7 @@ def accGrav : (SMνCharges n).charges →ₗ[ℚ] ℚ where
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ring
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lemma accGrav_decomp (S : (SMνCharges n).charges) :
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lemma accGrav_decomp (S : (SMνCharges n).Charges) :
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accGrav S = 6 * ∑ i, Q S i + 3 * ∑ i, U S i + 3 * ∑ i, D S i + 2 * ∑ i, L S i + ∑ i, E S i +
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∑ i, N S i := by
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simp only [accGrav, SMνSpecies_numberCharges, toSpecies_apply, Fin.isValue, LinearMap.coe_mk,
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@ -121,7 +121,7 @@ lemma accGrav_decomp (S : (SMνCharges n).charges) :
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repeat erw [← Finset.mul_sum]
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/-- Extensionality lemma for `accGrav`. -/
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lemma accGrav_ext {S T : (SMνCharges n).charges}
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lemma accGrav_ext {S T : (SMνCharges n).Charges}
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(hj : ∀ (j : Fin 6), ∑ i, (toSpecies j) S i = ∑ i, (toSpecies j) T i) :
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accGrav S = accGrav T := by
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rw [accGrav_decomp, accGrav_decomp]
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@ -130,7 +130,7 @@ lemma accGrav_ext {S T : (SMνCharges n).charges}
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/-- The `SU(2)` anomaly equation. -/
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@[simp]
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def accSU2 : (SMνCharges n).charges →ₗ[ℚ] ℚ where
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def accSU2 : (SMνCharges n).Charges →ₗ[ℚ] ℚ where
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toFun S := ∑ i, (3 * Q S i + L S i)
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map_add' S T := by
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simp only
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@ -147,7 +147,7 @@ def accSU2 : (SMνCharges n).charges →ₗ[ℚ] ℚ where
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-- rw [show Rat.cast a = a from rfl]
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ring
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lemma accSU2_decomp (S : (SMνCharges n).charges) :
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lemma accSU2_decomp (S : (SMνCharges n).Charges) :
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accSU2 S = 3 * ∑ i, Q S i + ∑ i, L S i := by
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simp only [accSU2, SMνSpecies_numberCharges, toSpecies_apply, Fin.isValue, LinearMap.coe_mk,
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AddHom.coe_mk]
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@ -155,7 +155,7 @@ lemma accSU2_decomp (S : (SMνCharges n).charges) :
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repeat erw [← Finset.mul_sum]
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/-- Extensionality lemma for `accSU2`. -/
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lemma accSU2_ext {S T : (SMνCharges n).charges}
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lemma accSU2_ext {S T : (SMνCharges n).Charges}
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(hj : ∀ (j : Fin 6), ∑ i, (toSpecies j) S i = ∑ i, (toSpecies j) T i) :
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accSU2 S = accSU2 T := by
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rw [accSU2_decomp, accSU2_decomp]
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@ -163,7 +163,7 @@ lemma accSU2_ext {S T : (SMνCharges n).charges}
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/-- The `SU(3)` anomaly equations. -/
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@[simp]
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def accSU3 : (SMνCharges n).charges →ₗ[ℚ] ℚ where
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def accSU3 : (SMνCharges n).Charges →ₗ[ℚ] ℚ where
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toFun S := ∑ i, (2 * Q S i + U S i + D S i)
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map_add' S T := by
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simp only
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@ -180,7 +180,7 @@ def accSU3 : (SMνCharges n).charges →ₗ[ℚ] ℚ where
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-- rw [show Rat.cast a = a from rfl]
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ring
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lemma accSU3_decomp (S : (SMνCharges n).charges) :
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lemma accSU3_decomp (S : (SMνCharges n).Charges) :
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accSU3 S = 2 * ∑ i, Q S i + ∑ i, U S i + ∑ i, D S i := by
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simp only [accSU3, SMνSpecies_numberCharges, toSpecies_apply, Fin.isValue, LinearMap.coe_mk,
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AddHom.coe_mk]
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@ -188,7 +188,7 @@ lemma accSU3_decomp (S : (SMνCharges n).charges) :
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repeat rw [← Finset.mul_sum]
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/-- Extensionality lemma for `accSU3`. -/
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lemma accSU3_ext {S T : (SMνCharges n).charges}
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lemma accSU3_ext {S T : (SMνCharges n).Charges}
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(hj : ∀ (j : Fin 6), ∑ i, (toSpecies j) S i = ∑ i, (toSpecies j) T i) :
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accSU3 S = accSU3 T := by
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rw [accSU3_decomp, accSU3_decomp]
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@ -196,7 +196,7 @@ lemma accSU3_ext {S T : (SMνCharges n).charges}
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/-- The `Y²` anomaly equation. -/
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@[simp]
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def accYY : (SMνCharges n).charges →ₗ[ℚ] ℚ where
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def accYY : (SMνCharges n).Charges →ₗ[ℚ] ℚ where
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toFun S := ∑ i, (Q S i + 8 * U S i + 2 * D S i + 3 * L S i
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+ 6 * E S i)
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map_add' S T := by
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@ -214,7 +214,7 @@ def accYY : (SMνCharges n).charges →ₗ[ℚ] ℚ where
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-- rw [show Rat.cast a = a from rfl]
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ring
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lemma accYY_decomp (S : (SMνCharges n).charges) :
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lemma accYY_decomp (S : (SMνCharges n).Charges) :
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accYY S = ∑ i, Q S i + 8 * ∑ i, U S i + 2 * ∑ i, D S i + 3 * ∑ i, L S i + 6 * ∑ i, E S i := by
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simp only [accYY, SMνSpecies_numberCharges, toSpecies_apply, Fin.isValue, LinearMap.coe_mk,
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AddHom.coe_mk]
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@ -222,7 +222,7 @@ lemma accYY_decomp (S : (SMνCharges n).charges) :
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repeat rw [← Finset.mul_sum]
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/-- Extensionality lemma for `accYY`. -/
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lemma accYY_ext {S T : (SMνCharges n).charges}
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lemma accYY_ext {S T : (SMνCharges n).Charges}
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(hj : ∀ (j : Fin 6), ∑ i, (toSpecies j) S i = ∑ i, (toSpecies j) T i) :
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accYY S = accYY T := by
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rw [accYY_decomp, accYY_decomp]
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@ -230,7 +230,7 @@ lemma accYY_ext {S T : (SMνCharges n).charges}
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/-- The quadratic bilinear map. -/
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@[simps!]
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def quadBiLin : BiLinearSymm (SMνCharges n).charges := BiLinearSymm.mk₂
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def quadBiLin : BiLinearSymm (SMνCharges n).Charges := BiLinearSymm.mk₂
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(fun S => ∑ i, (Q S.1 i * Q S.2 i +
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- 2 * (U S.1 i * U S.2 i) +
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D S.1 i * D S.2 i +
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@ -262,7 +262,7 @@ def quadBiLin : BiLinearSymm (SMνCharges n).charges := BiLinearSymm.mk₂
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intro i
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ring)
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lemma quadBiLin_decomp (S T : (SMνCharges n).charges) :
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lemma quadBiLin_decomp (S T : (SMνCharges n).Charges) :
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quadBiLin S T = ∑ i, Q S i * Q T i - 2 * ∑ i, U S i * U T i +
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∑ i, D S i * D T i - ∑ i, L S i * L T i + ∑ i, E S i * E T i := by
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erw [← quadBiLin.toFun_eq_coe]
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@ -275,17 +275,17 @@ lemma quadBiLin_decomp (S T : (SMνCharges n).charges) :
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/-- The quadratic anomaly cancellation condition. -/
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@[simp]
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def accQuad : HomogeneousQuadratic (SMνCharges n).charges :=
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def accQuad : HomogeneousQuadratic (SMνCharges n).Charges :=
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(@quadBiLin n).toHomogeneousQuad
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lemma accQuad_decomp (S : (SMνCharges n).charges) :
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lemma accQuad_decomp (S : (SMνCharges n).Charges) :
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accQuad S = ∑ i, (Q S i)^2 - 2 * ∑ i, (U S i)^2 + ∑ i, (D S i)^2 - ∑ i, (L S i)^2
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+ ∑ i, (E S i)^2 := by
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erw [quadBiLin_decomp]
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ring_nf
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/-- Extensionality lemma for `accQuad`. -/
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lemma accQuad_ext {S T : (SMνCharges n).charges}
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lemma accQuad_ext {S T : (SMνCharges n).Charges}
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(h : ∀ j, ∑ i, ((fun a => a^2) ∘ toSpecies j S) i =
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∑ i, ((fun a => a^2) ∘ toSpecies j T) i) :
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accQuad S = accQuad T := by
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@ -295,7 +295,7 @@ lemma accQuad_ext {S T : (SMνCharges n).charges}
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/-- The symmetric trilinear form used to define the cubic acc. -/
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@[simps!]
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def cubeTriLin : TriLinearSymm (SMνCharges n).charges := TriLinearSymm.mk₃
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def cubeTriLin : TriLinearSymm (SMνCharges n).Charges := TriLinearSymm.mk₃
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(fun S => ∑ i, (6 * ((Q S.1 i) * (Q S.2.1 i) * (Q S.2.2 i))
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+ 3 * ((U S.1 i) * (U S.2.1 i) * (U S.2.2 i))
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+ 3 * ((D S.1 i) * (D S.2.1 i) * (D S.2.2 i))
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@ -333,7 +333,7 @@ def cubeTriLin : TriLinearSymm (SMνCharges n).charges := TriLinearSymm.mk₃
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intro i
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ring)
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lemma cubeTriLin_decomp (S T R : (SMνCharges n).charges) :
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lemma cubeTriLin_decomp (S T R : (SMνCharges n).Charges) :
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cubeTriLin S T R = 6 * ∑ i, (Q S i * Q T i * Q R i) + 3 * ∑ i, (U S i * U T i * U R i) +
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3 * ∑ i, (D S i * D T i * D R i) + 2 * ∑ i, (L S i * L T i * L R i) +
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∑ i, (E S i * E T i * E R i) + ∑ i, (N S i * N T i * N R i) := by
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@ -347,16 +347,16 @@ lemma cubeTriLin_decomp (S T R : (SMνCharges n).charges) :
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/-- The cubic ACC. -/
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@[simp]
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def accCube : HomogeneousCubic (SMνCharges n).charges := cubeTriLin.toCubic
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def accCube : HomogeneousCubic (SMνCharges n).Charges := cubeTriLin.toCubic
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lemma accCube_decomp (S : (SMνCharges n).charges) :
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lemma accCube_decomp (S : (SMνCharges n).Charges) :
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accCube S = 6 * ∑ i, (Q S i)^3 + 3 * ∑ i, (U S i)^3 + 3 * ∑ i, (D S i)^3 + 2 * ∑ i, (L S i)^3 +
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∑ i, (E S i)^3 + ∑ i, (N S i)^3 := by
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erw [cubeTriLin_decomp]
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ring_nf
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/-- Extensionality lemma for `accCube`. -/
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lemma accCube_ext {S T : (SMνCharges n).charges}
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lemma accCube_ext {S T : (SMνCharges n).Charges}
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(h : ∀ j, ∑ i, ((fun a => a^3) ∘ toSpecies j S) i =
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∑ i, ((fun a => a^3) ∘ toSpecies j T) i) :
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accCube S = accCube T := by
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@ -19,8 +19,8 @@ open BigOperators
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/-- Given a map of for a generic species, the corresponding map for charges. -/
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@[simps!]
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def chargesMapOfSpeciesMap {n m : ℕ} (f : (SMνSpecies n).charges →ₗ[ℚ] (SMνSpecies m).charges) :
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(SMνCharges n).charges →ₗ[ℚ] (SMνCharges m).charges where
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def chargesMapOfSpeciesMap {n m : ℕ} (f : (SMνSpecies n).Charges →ₗ[ℚ] (SMνSpecies m).Charges) :
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(SMνCharges n).Charges →ₗ[ℚ] (SMνCharges m).Charges where
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toFun S := toSpeciesEquiv.symm (fun i => (LinearMap.comp f (toSpecies i)) S)
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map_add' S T := by
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rw [charges_eq_toSpecies_eq]
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@ -37,28 +37,28 @@ def chargesMapOfSpeciesMap {n m : ℕ} (f : (SMνSpecies n).charges →ₗ[ℚ]
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rfl
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lemma chargesMapOfSpeciesMap_toSpecies {n m : ℕ}
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(f : (SMνSpecies n).charges →ₗ[ℚ] (SMνSpecies m).charges)
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(S : (SMνCharges n).charges) (j : Fin 6) :
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(f : (SMνSpecies n).Charges →ₗ[ℚ] (SMνSpecies m).Charges)
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(S : (SMνCharges n).Charges) (j : Fin 6) :
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toSpecies j (chargesMapOfSpeciesMap f S) = (LinearMap.comp f (toSpecies j)) S := by
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erw [toSMSpecies_toSpecies_inv]
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/-- The projection of the `m`-family charges onto the first `n`-family charges for species. -/
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@[simps!]
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def speciesFamilyProj {m n : ℕ} (h : n ≤ m) :
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(SMνSpecies m).charges →ₗ[ℚ] (SMνSpecies n).charges where
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(SMνSpecies m).Charges →ₗ[ℚ] (SMνSpecies n).Charges where
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toFun S := S ∘ Fin.castLE h
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map_add' _ _ := rfl
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map_smul' _ _ := rfl
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/-- The projection of the `m`-family charges onto the first `n`-family charges. -/
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def familyProjection {m n : ℕ} (h : n ≤ m) : (SMνCharges m).charges →ₗ[ℚ] (SMνCharges n).charges :=
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def familyProjection {m n : ℕ} (h : n ≤ m) : (SMνCharges m).Charges →ₗ[ℚ] (SMνCharges n).Charges :=
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chargesMapOfSpeciesMap (speciesFamilyProj h)
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/-- For species, the embedding of the `m`-family charges onto the `n`-family charges, with all
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other charges zero. -/
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@[simps!]
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def speciesEmbed (m n : ℕ) :
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(SMνSpecies m).charges →ₗ[ℚ] (SMνSpecies n).charges where
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(SMνSpecies m).Charges →ₗ[ℚ] (SMνSpecies n).Charges where
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toFun S := fun i =>
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if hi : i.val < m then
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S ⟨i, hi⟩
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@ -82,29 +82,29 @@ def speciesEmbed (m n : ℕ) :
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/-- The embedding of the `m`-family charges onto the `n`-family charges, with all
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other charges zero. -/
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def familyEmbedding (m n : ℕ) : (SMνCharges m).charges →ₗ[ℚ] (SMνCharges n).charges :=
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def familyEmbedding (m n : ℕ) : (SMνCharges m).Charges →ₗ[ℚ] (SMνCharges n).Charges :=
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chargesMapOfSpeciesMap (speciesEmbed m n)
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/-- For species, the embedding of the `1`-family charges into the `n`-family charges in
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a universal manor. -/
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@[simps!]
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def speciesFamilyUniversial (n : ℕ) :
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(SMνSpecies 1).charges →ₗ[ℚ] (SMνSpecies n).charges where
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(SMνSpecies 1).Charges →ₗ[ℚ] (SMνSpecies n).Charges where
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toFun S _ := S ⟨0, by simp⟩
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map_add' S T := rfl
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map_smul' a S := rfl
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/-- The embedding of the `1`-family charges into the `n`-family charges in
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a universal manor. -/
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def familyUniversal (n : ℕ) : (SMνCharges 1).charges →ₗ[ℚ] (SMνCharges n).charges :=
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def familyUniversal (n : ℕ) : (SMνCharges 1).Charges →ₗ[ℚ] (SMνCharges n).Charges :=
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chargesMapOfSpeciesMap (speciesFamilyUniversial n)
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lemma toSpecies_familyUniversal {n : ℕ} (j : Fin 6) (S : (SMνCharges 1).charges)
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lemma toSpecies_familyUniversal {n : ℕ} (j : Fin 6) (S : (SMνCharges 1).Charges)
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(i : Fin n) : toSpecies j (familyUniversal n S) i = toSpecies j S ⟨0, by simp⟩ := by
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erw [chargesMapOfSpeciesMap_toSpecies]
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rfl
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lemma sum_familyUniversal {n : ℕ} (m : ℕ) (S : (SMνCharges 1).charges) (j : Fin 6) :
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lemma sum_familyUniversal {n : ℕ} (m : ℕ) (S : (SMνCharges 1).Charges) (j : Fin 6) :
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∑ i, ((fun a => a ^ m) ∘ toSpecies j (familyUniversal n S)) i =
|
||||
n * (toSpecies j S ⟨0, by simp⟩) ^ m := by
|
||||
simp only [SMνSpecies_numberCharges, Function.comp_apply, toSpecies_apply, Fin.zero_eta,
|
||||
|
|
@ -119,12 +119,12 @@ lemma sum_familyUniversal {n : ℕ} (m : ℕ) (S : (SMνCharges 1).charges) (j :
|
|||
intro i _
|
||||
erw [toSpecies_familyUniversal]
|
||||
|
||||
lemma sum_familyUniversal_one {n : ℕ} (S : (SMνCharges 1).charges) (j : Fin 6) :
|
||||
lemma sum_familyUniversal_one {n : ℕ} (S : (SMνCharges 1).Charges) (j : Fin 6) :
|
||||
∑ i, toSpecies j (familyUniversal n S) i = n * (toSpecies j S ⟨0, by simp⟩) := by
|
||||
simpa using @sum_familyUniversal n 1 S j
|
||||
|
||||
lemma sum_familyUniversal_two {n : ℕ} (S : (SMνCharges 1).charges)
|
||||
(T : (SMνCharges n).charges) (j : Fin 6) :
|
||||
lemma sum_familyUniversal_two {n : ℕ} (S : (SMνCharges 1).Charges)
|
||||
(T : (SMνCharges n).Charges) (j : Fin 6) :
|
||||
∑ i, (toSpecies j (familyUniversal n S) i * toSpecies j T i) =
|
||||
(toSpecies j S ⟨0, by simp⟩) * ∑ i, toSpecies j T i := by
|
||||
simp only [SMνSpecies_numberCharges, toSpecies_apply, Fin.zero_eta, Fin.isValue]
|
||||
|
|
@ -135,8 +135,8 @@ lemma sum_familyUniversal_two {n : ℕ} (S : (SMνCharges 1).charges)
|
|||
erw [toSpecies_familyUniversal]
|
||||
rfl
|
||||
|
||||
lemma sum_familyUniversal_three {n : ℕ} (S : (SMνCharges 1).charges)
|
||||
(T L : (SMνCharges n).charges) (j : Fin 6) :
|
||||
lemma sum_familyUniversal_three {n : ℕ} (S : (SMνCharges 1).Charges)
|
||||
(T L : (SMνCharges n).Charges) (j : Fin 6) :
|
||||
∑ i, (toSpecies j (familyUniversal n S) i * toSpecies j T i * toSpecies j L i) =
|
||||
(toSpecies j S ⟨0, by simp⟩) * ∑ i, toSpecies j T i * toSpecies j L i := by
|
||||
simp only [SMνSpecies_numberCharges, toSpecies_apply, Fin.zero_eta, Fin.isValue]
|
||||
|
|
@ -148,7 +148,7 @@ lemma sum_familyUniversal_three {n : ℕ} (S : (SMνCharges 1).charges)
|
|||
simp only [SMνSpecies_numberCharges, Fin.zero_eta, Fin.isValue, toSpecies_apply]
|
||||
ring
|
||||
|
||||
lemma familyUniversal_accGrav (S : (SMνCharges 1).charges) :
|
||||
lemma familyUniversal_accGrav (S : (SMνCharges 1).Charges) :
|
||||
accGrav (familyUniversal n S) = n * (accGrav S) := by
|
||||
rw [accGrav_decomp, accGrav_decomp]
|
||||
repeat rw [sum_familyUniversal_one]
|
||||
|
|
@ -156,7 +156,7 @@ lemma familyUniversal_accGrav (S : (SMνCharges 1).charges) :
|
|||
Finset.univ_unique, Fin.default_eq_zero, Finset.sum_singleton]
|
||||
ring
|
||||
|
||||
lemma familyUniversal_accSU2 (S : (SMνCharges 1).charges) :
|
||||
lemma familyUniversal_accSU2 (S : (SMνCharges 1).Charges) :
|
||||
accSU2 (familyUniversal n S) = n * (accSU2 S) := by
|
||||
rw [accSU2_decomp, accSU2_decomp]
|
||||
repeat rw [sum_familyUniversal_one]
|
||||
|
|
@ -164,7 +164,7 @@ lemma familyUniversal_accSU2 (S : (SMνCharges 1).charges) :
|
|||
Finset.univ_unique, Fin.default_eq_zero, Finset.sum_singleton]
|
||||
ring
|
||||
|
||||
lemma familyUniversal_accSU3 (S : (SMνCharges 1).charges) :
|
||||
lemma familyUniversal_accSU3 (S : (SMνCharges 1).Charges) :
|
||||
accSU3 (familyUniversal n S) = n * (accSU3 S) := by
|
||||
rw [accSU3_decomp, accSU3_decomp]
|
||||
repeat rw [sum_familyUniversal_one]
|
||||
|
|
@ -172,7 +172,7 @@ lemma familyUniversal_accSU3 (S : (SMνCharges 1).charges) :
|
|||
Finset.univ_unique, Fin.default_eq_zero, Finset.sum_singleton]
|
||||
ring
|
||||
|
||||
lemma familyUniversal_accYY (S : (SMνCharges 1).charges) :
|
||||
lemma familyUniversal_accYY (S : (SMνCharges 1).Charges) :
|
||||
accYY (familyUniversal n S) = n * (accYY S) := by
|
||||
rw [accYY_decomp, accYY_decomp]
|
||||
repeat rw [sum_familyUniversal_one]
|
||||
|
|
@ -180,7 +180,7 @@ lemma familyUniversal_accYY (S : (SMνCharges 1).charges) :
|
|||
Finset.univ_unique, Fin.default_eq_zero, Finset.sum_singleton]
|
||||
ring
|
||||
|
||||
lemma familyUniversal_quadBiLin (S : (SMνCharges 1).charges) (T : (SMνCharges n).charges) :
|
||||
lemma familyUniversal_quadBiLin (S : (SMνCharges 1).Charges) (T : (SMνCharges n).Charges) :
|
||||
quadBiLin (familyUniversal n S) T =
|
||||
S (0 : Fin 6) * ∑ i, Q T i - 2 * S (1 : Fin 6) * ∑ i, U T i + S (2 : Fin 6) *∑ i, D T i -
|
||||
S (3 : Fin 6) * ∑ i, L T i + S (4 : Fin 6) * ∑ i, E T i := by
|
||||
|
|
@ -191,7 +191,7 @@ lemma familyUniversal_quadBiLin (S : (SMνCharges 1).charges) (T : (SMνCharges
|
|||
sub_right_inj]
|
||||
ring
|
||||
|
||||
lemma familyUniversal_accQuad (S : (SMνCharges 1).charges) :
|
||||
lemma familyUniversal_accQuad (S : (SMνCharges 1).Charges) :
|
||||
accQuad (familyUniversal n S) = n * (accQuad S) := by
|
||||
rw [accQuad_decomp]
|
||||
repeat erw [sum_familyUniversal]
|
||||
|
|
@ -200,7 +200,7 @@ lemma familyUniversal_accQuad (S : (SMνCharges 1).charges) :
|
|||
Finset.univ_unique, Fin.default_eq_zero, Finset.sum_singleton]
|
||||
ring
|
||||
|
||||
lemma familyUniversal_cubeTriLin (S : (SMνCharges 1).charges) (T R : (SMνCharges n).charges) :
|
||||
lemma familyUniversal_cubeTriLin (S : (SMνCharges 1).Charges) (T R : (SMνCharges n).Charges) :
|
||||
cubeTriLin (familyUniversal n S) T R = 6 * S (0 : Fin 6) * ∑ i, (Q T i * Q R i) +
|
||||
3 * S (1 : Fin 6) * ∑ i, (U T i * U R i) + 3 * S (2 : Fin 6) * ∑ i, (D T i * D R i)
|
||||
+ 2 * S (3 : Fin 6) * ∑ i, (L T i * L R i) +
|
||||
|
|
@ -211,7 +211,7 @@ lemma familyUniversal_cubeTriLin (S : (SMνCharges 1).charges) (T R : (SMνCharg
|
|||
simp only [Fin.isValue, SMνSpecies_numberCharges, toSpecies_apply, add_left_inj]
|
||||
ring
|
||||
|
||||
lemma familyUniversal_cubeTriLin' (S T : (SMνCharges 1).charges) (R : (SMνCharges n).charges) :
|
||||
lemma familyUniversal_cubeTriLin' (S T : (SMνCharges 1).Charges) (R : (SMνCharges n).Charges) :
|
||||
cubeTriLin (familyUniversal n S) (familyUniversal n T) R =
|
||||
6 * S (0 : Fin 6) * T (0 : Fin 6) * ∑ i, Q R i +
|
||||
3 * S (1 : Fin 6) * T (1 : Fin 6) * ∑ i, U R i
|
||||
|
|
@ -224,7 +224,7 @@ lemma familyUniversal_cubeTriLin' (S T : (SMνCharges 1).charges) (R : (SMνChar
|
|||
simp only [Fin.isValue, SMνSpecies_numberCharges, toSpecies_apply]
|
||||
ring
|
||||
|
||||
lemma familyUniversal_accCube (S : (SMνCharges 1).charges) :
|
||||
lemma familyUniversal_accCube (S : (SMνCharges 1).Charges) :
|
||||
accCube (familyUniversal n S) = n * (accCube S) := by
|
||||
rw [accCube_decomp]
|
||||
repeat erw [sum_familyUniversal]
|
||||
|
|
|
|||
|
|
@ -53,7 +53,7 @@ lemma cubeSol (S : (SMNoGrav n).Sols) : accCube S.val = 0 := by
|
|||
|
||||
/-- An element of `charges` which satisfies the linear ACCs
|
||||
gives us a element of `LinSols`. -/
|
||||
def chargeToLinear (S : (SMNoGrav n).charges) (hSU2 : accSU2 S = 0) (hSU3 : accSU3 S = 0) :
|
||||
def chargeToLinear (S : (SMNoGrav n).Charges) (hSU2 : accSU2 S = 0) (hSU3 : accSU3 S = 0) :
|
||||
(SMNoGrav n).LinSols :=
|
||||
⟨S, by
|
||||
intro i
|
||||
|
|
@ -76,13 +76,13 @@ def quadToAF (S : (SMNoGrav n).QuadSols) (hc : accCube S.val = 0) :
|
|||
|
||||
/-- An element of `charges` which satisfies the linear and quadratic ACCs
|
||||
gives us a element of `QuadSols`. -/
|
||||
def chargeToQuad (S : (SMNoGrav n).charges) (hSU2 : accSU2 S = 0) (hSU3 : accSU3 S = 0) :
|
||||
def chargeToQuad (S : (SMNoGrav n).Charges) (hSU2 : accSU2 S = 0) (hSU3 : accSU3 S = 0) :
|
||||
(SMNoGrav n).QuadSols :=
|
||||
linearToQuad $ chargeToLinear S hSU2 hSU3
|
||||
|
||||
/-- An element of `charges` which satisfies the linear, quadratic and cubic ACCs
|
||||
gives us a element of `Sols`. -/
|
||||
def chargeToAF (S : (SMNoGrav n).charges) (hSU2 : accSU2 S = 0) (hSU3 : accSU3 S = 0)
|
||||
def chargeToAF (S : (SMNoGrav n).Charges) (hSU2 : accSU2 S = 0) (hSU3 : accSU3 S = 0)
|
||||
(hc : accCube S = 0) : (SMNoGrav n).Sols :=
|
||||
quadToAF (chargeToQuad S hSU2 hSU3) hc
|
||||
|
||||
|
|
@ -95,7 +95,7 @@ def linearToAF (S : (SMNoGrav n).LinSols)
|
|||
/-- The permutations acting on the ACC system corresponding to the SM with RHN,
|
||||
and no gravitational anomaly. -/
|
||||
def perm (n : ℕ) : ACCSystemGroupAction (SMNoGrav n) where
|
||||
group := permGroup n
|
||||
group := PermGroup n
|
||||
groupInst := inferInstance
|
||||
rep := repCharges
|
||||
linearInvariant := by
|
||||
|
|
|
|||
|
|
@ -60,7 +60,7 @@ lemma cubeSol (S : (SM n).Sols) : accCube S.val = 0 := by
|
|||
|
||||
/-- An element of `charges` which satisfies the linear ACCs
|
||||
gives us a element of `LinSols`. -/
|
||||
def chargeToLinear (S : (SM n).charges) (hGrav : accGrav S = 0)
|
||||
def chargeToLinear (S : (SM n).Charges) (hGrav : accGrav S = 0)
|
||||
(hSU2 : accSU2 S = 0) (hSU3 : accSU3 S = 0) : (SM n).LinSols :=
|
||||
⟨S, by
|
||||
intro i
|
||||
|
|
@ -84,14 +84,14 @@ def quadToAF (S : (SM n).QuadSols) (hc : accCube S.val = 0) :
|
|||
|
||||
/-- An element of `charges` which satisfies the linear and quadratic ACCs
|
||||
gives us a element of `QuadSols`. -/
|
||||
def chargeToQuad (S : (SM n).charges) (hGrav : accGrav S = 0)
|
||||
def chargeToQuad (S : (SM n).Charges) (hGrav : accGrav S = 0)
|
||||
(hSU2 : accSU2 S = 0) (hSU3 : accSU3 S = 0) :
|
||||
(SM n).QuadSols :=
|
||||
linearToQuad $ chargeToLinear S hGrav hSU2 hSU3
|
||||
|
||||
/-- An element of `charges` which satisfies the linear, quadratic and cubic ACCs
|
||||
gives us a element of `Sols`. -/
|
||||
def chargeToAF (S : (SM n).charges) (hGrav : accGrav S = 0) (hSU2 : accSU2 S = 0)
|
||||
def chargeToAF (S : (SM n).Charges) (hGrav : accGrav S = 0) (hSU2 : accSU2 S = 0)
|
||||
(hSU3 : accSU3 S = 0) (hc : accCube S = 0) : (SM n).Sols :=
|
||||
quadToAF (chargeToQuad S hGrav hSU2 hSU3) hc
|
||||
|
||||
|
|
@ -103,7 +103,7 @@ def linearToAF (S : (SM n).LinSols)
|
|||
|
||||
/-- The permutations acting on the ACC system corresponding to the SM with RHN. -/
|
||||
def perm (n : ℕ) : ACCSystemGroupAction (SM n) where
|
||||
group := permGroup n
|
||||
group := PermGroup n
|
||||
groupInst := inferInstance
|
||||
rep := repCharges
|
||||
linearInvariant := by
|
||||
|
|
|
|||
|
|
@ -24,102 +24,102 @@ open BigOperators
|
|||
namespace PlaneSeven
|
||||
|
||||
/-- A charge assignment forming one of the basis elements of the plane. -/
|
||||
def B₀ : (SM 3).charges := toSpeciesEquiv.invFun ( fun s => fun i =>
|
||||
def B₀ : (SM 3).Charges := toSpeciesEquiv.invFun ( fun s => fun i =>
|
||||
match s, i with
|
||||
| 0, 0 => 1
|
||||
| 0, 1 => - 1
|
||||
| _, _ => 0
|
||||
)
|
||||
|
||||
lemma B₀_cubic (S T : (SM 3).charges) : cubeTriLin B₀ S T =
|
||||
lemma B₀_cubic (S T : (SM 3).Charges) : cubeTriLin B₀ S T =
|
||||
6 * (S (0 : Fin 18) * T (0 : Fin 18) - S (1 : Fin 18) * T (1 : Fin 18)) := by
|
||||
simp [Fin.sum_univ_three, B₀, Fin.divNat, Fin.modNat, finProdFinEquiv]
|
||||
ring
|
||||
|
||||
/-- A charge assignment forming one of the basis elements of the plane. -/
|
||||
def B₁ : (SM 3).charges := toSpeciesEquiv.invFun ( fun s => fun i =>
|
||||
def B₁ : (SM 3).Charges := toSpeciesEquiv.invFun ( fun s => fun i =>
|
||||
match s, i with
|
||||
| 1, 0 => 1
|
||||
| 1, 1 => - 1
|
||||
| _, _ => 0
|
||||
)
|
||||
|
||||
lemma B₁_cubic (S T : (SM 3).charges) : cubeTriLin B₁ S T =
|
||||
lemma B₁_cubic (S T : (SM 3).Charges) : cubeTriLin B₁ S T =
|
||||
3 * (S (3 : Fin 18) * T (3 : Fin 18) - S (4 : Fin 18) * T (4 : Fin 18)) := by
|
||||
simp [Fin.sum_univ_three, B₁, Fin.divNat, Fin.modNat, finProdFinEquiv]
|
||||
ring
|
||||
|
||||
/-- A charge assignment forming one of the basis elements of the plane. -/
|
||||
def B₂ : (SM 3).charges := toSpeciesEquiv.invFun ( fun s => fun i =>
|
||||
def B₂ : (SM 3).Charges := toSpeciesEquiv.invFun ( fun s => fun i =>
|
||||
match s, i with
|
||||
| 2, 0 => 1
|
||||
| 2, 1 => - 1
|
||||
| _, _ => 0
|
||||
)
|
||||
|
||||
lemma B₂_cubic (S T : (SM 3).charges) : cubeTriLin B₂ S T =
|
||||
lemma B₂_cubic (S T : (SM 3).Charges) : cubeTriLin B₂ S T =
|
||||
3 * (S (6 : Fin 18) * T (6 : Fin 18) - S (7 : Fin 18) * T (7 : Fin 18)) := by
|
||||
simp [Fin.sum_univ_three, B₂, Fin.divNat, Fin.modNat, finProdFinEquiv]
|
||||
ring
|
||||
|
||||
/-- A charge assignment forming one of the basis elements of the plane. -/
|
||||
def B₃ : (SM 3).charges := toSpeciesEquiv.invFun ( fun s => fun i =>
|
||||
def B₃ : (SM 3).Charges := toSpeciesEquiv.invFun ( fun s => fun i =>
|
||||
match s, i with
|
||||
| 3, 0 => 1
|
||||
| 3, 1 => - 1
|
||||
| _, _ => 0
|
||||
)
|
||||
|
||||
lemma B₃_cubic (S T : (SM 3).charges) : cubeTriLin B₃ S T =
|
||||
lemma B₃_cubic (S T : (SM 3).Charges) : cubeTriLin B₃ S T =
|
||||
2 * (S (9 : Fin 18) * T (9 : Fin 18) - S (10 : Fin 18) * T (10 : Fin 18)) := by
|
||||
simp [Fin.sum_univ_three, B₃, Fin.divNat, Fin.modNat, finProdFinEquiv]
|
||||
ring_nf
|
||||
rfl
|
||||
|
||||
/-- A charge assignment forming one of the basis elements of the plane. -/
|
||||
def B₄ : (SM 3).charges := toSpeciesEquiv.invFun ( fun s => fun i =>
|
||||
def B₄ : (SM 3).Charges := toSpeciesEquiv.invFun ( fun s => fun i =>
|
||||
match s, i with
|
||||
| 4, 0 => 1
|
||||
| 4, 1 => - 1
|
||||
| _, _ => 0
|
||||
)
|
||||
|
||||
lemma B₄_cubic (S T : (SM 3).charges) : cubeTriLin B₄ S T =
|
||||
lemma B₄_cubic (S T : (SM 3).Charges) : cubeTriLin B₄ S T =
|
||||
(S (12 : Fin 18) * T (12 : Fin 18) - S (13 : Fin 18) * T (13 : Fin 18)) := by
|
||||
simp [Fin.sum_univ_three, B₄, Fin.divNat, Fin.modNat, finProdFinEquiv]
|
||||
ring_nf
|
||||
rfl
|
||||
|
||||
/-- A charge assignment forming one of the basis elements of the plane. -/
|
||||
def B₅ : (SM 3).charges := toSpeciesEquiv.invFun ( fun s => fun i =>
|
||||
def B₅ : (SM 3).Charges := toSpeciesEquiv.invFun ( fun s => fun i =>
|
||||
match s, i with
|
||||
| 5, 0 => 1
|
||||
| 5, 1 => - 1
|
||||
| _, _ => 0
|
||||
)
|
||||
|
||||
lemma B₅_cubic (S T : (SM 3).charges) : cubeTriLin B₅ S T =
|
||||
lemma B₅_cubic (S T : (SM 3).Charges) : cubeTriLin B₅ S T =
|
||||
(S (15 : Fin 18) * T (15 : Fin 18) - S (16 : Fin 18) * T (16 : Fin 18)) := by
|
||||
simp [Fin.sum_univ_three, B₅, Fin.divNat, Fin.modNat, finProdFinEquiv]
|
||||
ring_nf
|
||||
rfl
|
||||
|
||||
/-- A charge assignment forming one of the basis elements of the plane. -/
|
||||
def B₆ : (SM 3).charges := toSpeciesEquiv.invFun ( fun s => fun i =>
|
||||
def B₆ : (SM 3).Charges := toSpeciesEquiv.invFun ( fun s => fun i =>
|
||||
match s, i with
|
||||
| 1, 2 => 1
|
||||
| 2, 2 => -1
|
||||
| _, _ => 0
|
||||
)
|
||||
|
||||
lemma B₆_cubic (S T : (SM 3).charges) : cubeTriLin B₆ S T =
|
||||
lemma B₆_cubic (S T : (SM 3).Charges) : cubeTriLin B₆ S T =
|
||||
3* (S (5 : Fin 18) * T (5 : Fin 18) - S (8 : Fin 18) * T (8 : Fin 18)) := by
|
||||
simp [Fin.sum_univ_three, B₆, Fin.divNat, Fin.modNat, finProdFinEquiv]
|
||||
ring_nf
|
||||
|
||||
/-- The charge assignments forming a basis of the plane. -/
|
||||
@[simp]
|
||||
def B : Fin 7 → (SM 3).charges := fun i =>
|
||||
def B : Fin 7 → (SM 3).Charges := fun i =>
|
||||
match i with
|
||||
| 0 => B₀
|
||||
| 1 => B₁
|
||||
|
|
@ -129,7 +129,7 @@ def B : Fin 7 → (SM 3).charges := fun i =>
|
|||
| 5 => B₅
|
||||
| 6 => B₆
|
||||
|
||||
lemma B₀_Bi_cubic {i : Fin 7} (hi : 0 ≠ i) (S : (SM 3).charges) :
|
||||
lemma B₀_Bi_cubic {i : Fin 7} (hi : 0 ≠ i) (S : (SM 3).Charges) :
|
||||
cubeTriLin (B 0) (B i) S = 0 := by
|
||||
change cubeTriLin B₀ (B i) S = 0
|
||||
rw [B₀_cubic]
|
||||
|
|
@ -138,7 +138,7 @@ lemma B₀_Bi_cubic {i : Fin 7} (hi : 0 ≠ i) (S : (SM 3).charges) :
|
|||
simp [B₀, B₁, B₂, B₃, B₄, B₅, B₆, Fin.divNat, Fin.modNat, finProdFinEquiv]
|
||||
|
||||
|
||||
lemma B₁_Bi_cubic {i : Fin 7} (hi : 1 ≠ i) (S : (SM 3).charges) :
|
||||
lemma B₁_Bi_cubic {i : Fin 7} (hi : 1 ≠ i) (S : (SM 3).Charges) :
|
||||
cubeTriLin (B 1) (B i) S = 0 := by
|
||||
change cubeTriLin B₁ (B i) S = 0
|
||||
rw [B₁_cubic]
|
||||
|
|
@ -146,7 +146,7 @@ lemma B₁_Bi_cubic {i : Fin 7} (hi : 1 ≠ i) (S : (SM 3).charges) :
|
|||
simp at hi <;>
|
||||
simp [B₀, B₁, B₂, B₃, B₄, B₅, B₆, Fin.divNat, Fin.modNat, finProdFinEquiv]
|
||||
|
||||
lemma B₂_Bi_cubic {i : Fin 7} (hi : 2 ≠ i) (S : (SM 3).charges) :
|
||||
lemma B₂_Bi_cubic {i : Fin 7} (hi : 2 ≠ i) (S : (SM 3).Charges) :
|
||||
cubeTriLin (B 2) (B i) S = 0 := by
|
||||
change cubeTriLin B₂ (B i) S = 0
|
||||
rw [B₂_cubic]
|
||||
|
|
@ -154,7 +154,7 @@ lemma B₂_Bi_cubic {i : Fin 7} (hi : 2 ≠ i) (S : (SM 3).charges) :
|
|||
simp at hi <;>
|
||||
simp [B₀, B₁, B₂, B₃, B₄, B₅, B₆, Fin.divNat, Fin.modNat, finProdFinEquiv]
|
||||
|
||||
lemma B₃_Bi_cubic {i : Fin 7} (hi : 3 ≠ i) (S : (SM 3).charges) :
|
||||
lemma B₃_Bi_cubic {i : Fin 7} (hi : 3 ≠ i) (S : (SM 3).Charges) :
|
||||
cubeTriLin (B 3) (B i) S = 0 := by
|
||||
change cubeTriLin (B₃) (B i) S = 0
|
||||
rw [B₃_cubic]
|
||||
|
|
@ -162,7 +162,7 @@ lemma B₃_Bi_cubic {i : Fin 7} (hi : 3 ≠ i) (S : (SM 3).charges) :
|
|||
simp at hi <;>
|
||||
simp [B₀, B₁, B₂, B₃, B₄, B₅, B₆, Fin.divNat, Fin.modNat, finProdFinEquiv]
|
||||
|
||||
lemma B₄_Bi_cubic {i : Fin 7} (hi : 4 ≠ i) (S : (SM 3).charges) :
|
||||
lemma B₄_Bi_cubic {i : Fin 7} (hi : 4 ≠ i) (S : (SM 3).Charges) :
|
||||
cubeTriLin (B 4) (B i) S = 0 := by
|
||||
change cubeTriLin (B₄) (B i) S = 0
|
||||
rw [B₄_cubic]
|
||||
|
|
@ -170,7 +170,7 @@ lemma B₄_Bi_cubic {i : Fin 7} (hi : 4 ≠ i) (S : (SM 3).charges) :
|
|||
simp at hi <;>
|
||||
simp [B₀, B₁, B₂, B₃, B₄, B₅, B₆, Fin.divNat, Fin.modNat, finProdFinEquiv]
|
||||
|
||||
lemma B₅_Bi_cubic {i : Fin 7} (hi : 5 ≠ i) (S : (SM 3).charges) :
|
||||
lemma B₅_Bi_cubic {i : Fin 7} (hi : 5 ≠ i) (S : (SM 3).Charges) :
|
||||
cubeTriLin (B 5) (B i) S = 0 := by
|
||||
change cubeTriLin (B₅) (B i) S = 0
|
||||
rw [B₅_cubic]
|
||||
|
|
@ -178,7 +178,7 @@ lemma B₅_Bi_cubic {i : Fin 7} (hi : 5 ≠ i) (S : (SM 3).charges) :
|
|||
simp at hi <;>
|
||||
simp [B₀, B₁, B₂, B₃, B₄, B₅, B₆, Fin.divNat, Fin.modNat, finProdFinEquiv]
|
||||
|
||||
lemma B₆_Bi_cubic {i : Fin 7} (hi : 6 ≠ i) (S : (SM 3).charges) :
|
||||
lemma B₆_Bi_cubic {i : Fin 7} (hi : 6 ≠ i) (S : (SM 3).Charges) :
|
||||
cubeTriLin (B 6) (B i) S = 0 := by
|
||||
change cubeTriLin (B₆) (B i) S = 0
|
||||
rw [B₆_cubic]
|
||||
|
|
@ -186,7 +186,7 @@ lemma B₆_Bi_cubic {i : Fin 7} (hi : 6 ≠ i) (S : (SM 3).charges) :
|
|||
simp at hi <;>
|
||||
simp [B₀, B₁, B₂, B₃, B₄, B₅, B₆, Fin.divNat, Fin.modNat, finProdFinEquiv]
|
||||
|
||||
lemma Bi_Bj_ne_cubic {i j : Fin 7} (h : i ≠ j) (S : (SM 3).charges) :
|
||||
lemma Bi_Bj_ne_cubic {i j : Fin 7} (h : i ≠ j) (S : (SM 3).Charges) :
|
||||
cubeTriLin (B i) (B j) S = 0 := by
|
||||
fin_cases i
|
||||
exact B₀_Bi_cubic h S
|
||||
|
|
@ -279,7 +279,7 @@ theorem B_in_accCube (f : Fin 7 → ℚ) : accCube (∑ i, f i • B i) = 0 := b
|
|||
rw [Bi_Bj_Bk_cubic]
|
||||
simp
|
||||
|
||||
lemma B_sum_is_sol (f : Fin 7 → ℚ) : (SM 3).isSolution (∑ i, f i • B i) := by
|
||||
lemma B_sum_is_sol (f : Fin 7 → ℚ) : (SM 3).IsSolution (∑ i, f i • B i) := by
|
||||
let X := chargeToAF (∑ i, f i • B i) (by
|
||||
rw [map_sum]
|
||||
apply Fintype.sum_eq_zero
|
||||
|
|
@ -337,8 +337,8 @@ theorem basis_linear_independent : LinearIndependent ℚ B := by
|
|||
|
||||
end PlaneSeven
|
||||
|
||||
theorem seven_dim_plane_exists : ∃ (B : Fin 7 → (SM 3).charges),
|
||||
LinearIndependent ℚ B ∧ ∀ (f : Fin 7 → ℚ), (SM 3).isSolution (∑ i, f i • B i) := by
|
||||
theorem seven_dim_plane_exists : ∃ (B : Fin 7 → (SM 3).Charges),
|
||||
LinearIndependent ℚ B ∧ ∀ (f : Fin 7 → ℚ), (SM 3).IsSolution (∑ i, f i • B i) := by
|
||||
use PlaneSeven.B
|
||||
apply And.intro
|
||||
exact PlaneSeven.basis_linear_independent
|
||||
|
|
|
|||
|
|
@ -26,26 +26,26 @@ open BigOperators
|
|||
|
||||
/-- The group of `Sₙ` permutations for each species. -/
|
||||
@[simp]
|
||||
def permGroup (n : ℕ) := Fin 6 → Equiv.Perm (Fin n)
|
||||
def PermGroup (n : ℕ) := Fin 6 → Equiv.Perm (Fin n)
|
||||
|
||||
variable {n : ℕ}
|
||||
|
||||
@[simp]
|
||||
instance : Group (permGroup n) := Pi.group
|
||||
instance : Group (PermGroup n) := Pi.group
|
||||
|
||||
/-- The image of an element of `permGroup n` under the representation on charges. -/
|
||||
@[simps!]
|
||||
def chargeMap (f : permGroup n) : (SMνCharges n).charges →ₗ[ℚ] (SMνCharges n).charges where
|
||||
def chargeMap (f : PermGroup n) : (SMνCharges n).Charges →ₗ[ℚ] (SMνCharges n).Charges where
|
||||
toFun S := toSpeciesEquiv.symm (fun i => toSpecies i S ∘ f i )
|
||||
map_add' _ _ := rfl
|
||||
map_smul' _ _ := rfl
|
||||
|
||||
/-- The representation of `(permGroup n)` acting on the vector space of charges. -/
|
||||
@[simp]
|
||||
def repCharges {n : ℕ} : Representation ℚ (permGroup n) (SMνCharges n).charges where
|
||||
def repCharges {n : ℕ} : Representation ℚ (PermGroup n) (SMνCharges n).Charges where
|
||||
toFun f := chargeMap f⁻¹
|
||||
map_mul' f g := by
|
||||
simp only [permGroup, mul_inv_rev]
|
||||
simp only [PermGroup, mul_inv_rev]
|
||||
apply LinearMap.ext
|
||||
intro S
|
||||
rw [charges_eq_toSpecies_eq]
|
||||
|
|
@ -61,49 +61,49 @@ def repCharges {n : ℕ} : Representation ℚ (permGroup n) (SMνCharges n).char
|
|||
erw [toSMSpecies_toSpecies_inv]
|
||||
rfl
|
||||
|
||||
lemma repCharges_toSpecies (f : permGroup n) (S : (SMνCharges n).charges) (j : Fin 6) :
|
||||
lemma repCharges_toSpecies (f : PermGroup n) (S : (SMνCharges n).Charges) (j : Fin 6) :
|
||||
toSpecies j (repCharges f S) = toSpecies j S ∘ f⁻¹ j := by
|
||||
erw [toSMSpecies_toSpecies_inv]
|
||||
|
||||
|
||||
lemma toSpecies_sum_invariant (m : ℕ) (f : permGroup n) (S : (SMνCharges n).charges) (j : Fin 6) :
|
||||
lemma toSpecies_sum_invariant (m : ℕ) (f : PermGroup n) (S : (SMνCharges n).Charges) (j : Fin 6) :
|
||||
∑ i, ((fun a => a ^ m) ∘ toSpecies j (repCharges f S)) i =
|
||||
∑ i, ((fun a => a ^ m) ∘ toSpecies j S) i := by
|
||||
erw [repCharges_toSpecies]
|
||||
change ∑ i : Fin n, ((fun a => a ^ m) ∘ _) (⇑(f⁻¹ _) i) = ∑ i : Fin n, ((fun a => a ^ m) ∘ _) i
|
||||
refine Equiv.Perm.sum_comp _ _ _ ?_
|
||||
simp only [permGroup, Fin.isValue, Pi.inv_apply, ne_eq, coe_univ, Set.subset_univ]
|
||||
simp only [PermGroup, Fin.isValue, Pi.inv_apply, ne_eq, coe_univ, Set.subset_univ]
|
||||
|
||||
|
||||
lemma accGrav_invariant (f : permGroup n) (S : (SMνCharges n).charges) :
|
||||
lemma accGrav_invariant (f : PermGroup n) (S : (SMνCharges n).Charges) :
|
||||
accGrav (repCharges f S) = accGrav S :=
|
||||
accGrav_ext
|
||||
(by simpa using toSpecies_sum_invariant 1 f S)
|
||||
|
||||
|
||||
lemma accSU2_invariant (f : permGroup n) (S : (SMνCharges n).charges) :
|
||||
lemma accSU2_invariant (f : PermGroup n) (S : (SMνCharges n).Charges) :
|
||||
accSU2 (repCharges f S) = accSU2 S :=
|
||||
accSU2_ext
|
||||
(by simpa using toSpecies_sum_invariant 1 f S)
|
||||
|
||||
|
||||
lemma accSU3_invariant (f : permGroup n) (S : (SMνCharges n).charges) :
|
||||
lemma accSU3_invariant (f : PermGroup n) (S : (SMνCharges n).Charges) :
|
||||
accSU3 (repCharges f S) = accSU3 S :=
|
||||
accSU3_ext
|
||||
(by simpa using toSpecies_sum_invariant 1 f S)
|
||||
|
||||
lemma accYY_invariant (f : permGroup n) (S : (SMνCharges n).charges) :
|
||||
lemma accYY_invariant (f : PermGroup n) (S : (SMνCharges n).Charges) :
|
||||
accYY (repCharges f S) = accYY S :=
|
||||
accYY_ext
|
||||
(by simpa using toSpecies_sum_invariant 1 f S)
|
||||
|
||||
|
||||
lemma accQuad_invariant (f : permGroup n) (S : (SMνCharges n).charges) :
|
||||
lemma accQuad_invariant (f : PermGroup n) (S : (SMνCharges n).Charges) :
|
||||
accQuad (repCharges f S) = accQuad S :=
|
||||
accQuad_ext
|
||||
(toSpecies_sum_invariant 2 f S)
|
||||
|
||||
lemma accCube_invariant (f : permGroup n) (S : (SMνCharges n).charges) :
|
||||
lemma accCube_invariant (f : PermGroup n) (S : (SMνCharges n).Charges) :
|
||||
accCube (repCharges f S) = accCube S :=
|
||||
accCube_ext
|
||||
(by simpa using toSpecies_sum_invariant 3 f S)
|
||||
|
|
|
|||
|
|
@ -57,7 +57,7 @@ namespace BL
|
|||
|
||||
variable {n : ℕ}
|
||||
|
||||
lemma on_quadBiLin (S : (PlusU1 n).charges) :
|
||||
lemma on_quadBiLin (S : (PlusU1 n).Charges) :
|
||||
quadBiLin (BL n).val S = 1/2 * accYY S + 3/2 * accSU2 S - 2 * accSU3 S := by
|
||||
erw [familyUniversal_quadBiLin]
|
||||
rw [accYY_decomp, accSU2_decomp, accSU3_decomp]
|
||||
|
|
@ -90,7 +90,7 @@ lemma addQuad_zero (S : (PlusU1 n).QuadSols) (a : ℚ): addQuad S a 0 = a • S
|
|||
simp [addQuad, linearToQuad]
|
||||
rfl
|
||||
|
||||
lemma on_cubeTriLin (S : (PlusU1 n).charges) :
|
||||
lemma on_cubeTriLin (S : (PlusU1 n).Charges) :
|
||||
cubeTriLin (BL n).val (BL n).val S = 9 * accGrav S - 24 * accSU3 S := by
|
||||
erw [familyUniversal_cubeTriLin']
|
||||
rw [accGrav_decomp, accSU3_decomp]
|
||||
|
|
|
|||
|
|
@ -70,7 +70,7 @@ lemma cubeSol (S : (PlusU1 n).Sols) : accCube S.val = 0 := by
|
|||
|
||||
/-- An element of `charges` which satisfies the linear ACCs
|
||||
gives us a element of `LinSols`. -/
|
||||
def chargeToLinear (S : (PlusU1 n).charges) (hGrav : accGrav S = 0)
|
||||
def chargeToLinear (S : (PlusU1 n).Charges) (hGrav : accGrav S = 0)
|
||||
(hSU2 : accSU2 S = 0) (hSU3 : accSU3 S = 0) (hYY : accYY S = 0) :
|
||||
(PlusU1 n).LinSols :=
|
||||
⟨S, by
|
||||
|
|
@ -99,14 +99,14 @@ def quadToAF (S : (PlusU1 n).QuadSols) (hc : accCube S.val = 0) :
|
|||
|
||||
/-- An element of `charges` which satisfies the linear and quadratic ACCs
|
||||
gives us a element of `QuadSols`. -/
|
||||
def chargeToQuad (S : (PlusU1 n).charges) (hGrav : accGrav S = 0)
|
||||
def chargeToQuad (S : (PlusU1 n).Charges) (hGrav : accGrav S = 0)
|
||||
(hSU2 : accSU2 S = 0) (hSU3 : accSU3 S = 0) (hYY : accYY S = 0) (hQ : accQuad S = 0) :
|
||||
(PlusU1 n).QuadSols :=
|
||||
linearToQuad (chargeToLinear S hGrav hSU2 hSU3 hYY) hQ
|
||||
|
||||
/-- An element of `charges` which satisfies the linear, quadratic and cubic ACCs
|
||||
gives us a element of `Sols`. -/
|
||||
def chargeToAF (S : (PlusU1 n).charges) (hGrav : accGrav S = 0) (hSU2 : accSU2 S = 0)
|
||||
def chargeToAF (S : (PlusU1 n).Charges) (hGrav : accGrav S = 0) (hSU2 : accSU2 S = 0)
|
||||
(hSU3 : accSU3 S = 0) (hYY : accYY S = 0) (hQ : accQuad S = 0) (hc : accCube S = 0) :
|
||||
(PlusU1 n).Sols :=
|
||||
quadToAF (chargeToQuad S hGrav hSU2 hSU3 hYY hQ) hc
|
||||
|
|
@ -119,7 +119,7 @@ def linearToAF (S : (PlusU1 n).LinSols) (hQ : accQuad S.val = 0)
|
|||
|
||||
/-- The permutations acting on the ACC system corresponding to the SM with RHN. -/
|
||||
def perm (n : ℕ) : ACCSystemGroupAction (PlusU1 n) where
|
||||
group := permGroup n
|
||||
group := PermGroup n
|
||||
groupInst := inferInstance
|
||||
rep := repCharges
|
||||
linearInvariant := by
|
||||
|
|
|
|||
|
|
@ -23,11 +23,11 @@ open BigOperators
|
|||
|
||||
/-- A proposition which is true if for a given `n` a plane of charges of dimension `n` exists
|
||||
in which each point is a solution. -/
|
||||
def existsPlane (n : ℕ) : Prop := ∃ (B : Fin n → (PlusU1 3).charges),
|
||||
LinearIndependent ℚ B ∧ ∀ (f : Fin n → ℚ), (PlusU1 3).isSolution (∑ i, f i • B i)
|
||||
def ExistsPlane (n : ℕ) : Prop := ∃ (B : Fin n → (PlusU1 3).Charges),
|
||||
LinearIndependent ℚ B ∧ ∀ (f : Fin n → ℚ), (PlusU1 3).IsSolution (∑ i, f i • B i)
|
||||
|
||||
lemma exists_plane_exists_basis {n : ℕ} (hE : existsPlane n) :
|
||||
∃ (B : Fin 11 ⊕ Fin n → (PlusU1 3).charges), LinearIndependent ℚ B := by
|
||||
lemma exists_plane_exists_basis {n : ℕ} (hE : ExistsPlane n) :
|
||||
∃ (B : Fin 11 ⊕ Fin n → (PlusU1 3).Charges), LinearIndependent ℚ B := by
|
||||
obtain ⟨E, hE1, hE2⟩ := hE
|
||||
obtain ⟨B, hB1, hB2⟩ := eleven_dim_plane_of_no_sols_exists
|
||||
let Y := Sum.elim B E
|
||||
|
|
@ -59,11 +59,11 @@ lemma exists_plane_exists_basis {n : ℕ} (hE : existsPlane n) :
|
|||
| Sum.inr i => exact h4 i
|
||||
|
||||
|
||||
theorem plane_exists_dim_le_7 {n : ℕ} (hn : existsPlane n) : n ≤ 7 := by
|
||||
theorem plane_exists_dim_le_7 {n : ℕ} (hn : ExistsPlane n) : n ≤ 7 := by
|
||||
obtain ⟨B, hB⟩ := exists_plane_exists_basis hn
|
||||
have h1 := LinearIndependent.fintype_card_le_finrank hB
|
||||
simp at h1
|
||||
rw [show FiniteDimensional.finrank ℚ (PlusU1 3).charges = 18 from
|
||||
rw [show FiniteDimensional.finrank ℚ (PlusU1 3).Charges = 18 from
|
||||
FiniteDimensional.finrank_fin_fun ℚ] at h1
|
||||
exact Nat.le_of_add_le_add_left h1
|
||||
|
||||
|
|
|
|||
|
|
@ -55,7 +55,7 @@ namespace Y
|
|||
|
||||
variable {n : ℕ}
|
||||
|
||||
lemma on_quadBiLin (S : (PlusU1 n).charges) :
|
||||
lemma on_quadBiLin (S : (PlusU1 n).Charges) :
|
||||
quadBiLin (Y n).val S = accYY S := by
|
||||
erw [familyUniversal_quadBiLin]
|
||||
rw [accYY_decomp]
|
||||
|
|
@ -89,7 +89,7 @@ lemma addQuad_zero (S : (PlusU1 n).QuadSols) (a : ℚ): addQuad S a 0 = a • S
|
|||
simp [addQuad, linearToQuad]
|
||||
rfl
|
||||
|
||||
lemma on_cubeTriLin (S : (PlusU1 n).charges) :
|
||||
lemma on_cubeTriLin (S : (PlusU1 n).Charges) :
|
||||
cubeTriLin (Y n).val (Y n).val S = 6 * accYY S := by
|
||||
erw [familyUniversal_cubeTriLin']
|
||||
rw [accYY_decomp]
|
||||
|
|
@ -103,7 +103,7 @@ lemma on_cubeTriLin_AFL (S : (PlusU1 n).LinSols) :
|
|||
rw [YYsol S]
|
||||
simp
|
||||
|
||||
lemma on_cubeTriLin' (S : (PlusU1 n).charges) :
|
||||
lemma on_cubeTriLin' (S : (PlusU1 n).Charges) :
|
||||
cubeTriLin (Y n).val S S = 6 * accQuad S := by
|
||||
erw [familyUniversal_cubeTriLin]
|
||||
rw [accQuad_decomp]
|
||||
|
|
|
|||
|
|
@ -26,40 +26,40 @@ open BigOperators
|
|||
namespace ElevenPlane
|
||||
|
||||
/-- A charge assignment forming one of the basis elements of the plane. -/
|
||||
def B₀ : (PlusU1 3).charges := ![1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
|
||||
def B₀ : (PlusU1 3).Charges := ![1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
|
||||
|
||||
/-- A charge assignment forming one of the basis elements of the plane. -/
|
||||
def B₁ : (PlusU1 3).charges := ![0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
|
||||
def B₁ : (PlusU1 3).Charges := ![0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
|
||||
|
||||
/-- A charge assignment forming one of the basis elements of the plane. -/
|
||||
def B₂ : (PlusU1 3).charges := ![0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
|
||||
def B₂ : (PlusU1 3).Charges := ![0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
|
||||
|
||||
/-- A charge assignment forming one of the basis elements of the plane. -/
|
||||
def B₃ : (PlusU1 3).charges := ![0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
|
||||
def B₃ : (PlusU1 3).Charges := ![0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
|
||||
|
||||
/-- A charge assignment forming one of the basis elements of the plane. -/
|
||||
def B₄ : (PlusU1 3).charges := ![0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
|
||||
def B₄ : (PlusU1 3).Charges := ![0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
|
||||
|
||||
/-- A charge assignment forming one of the basis elements of the plane. -/
|
||||
def B₅ : (PlusU1 3).charges := ![0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0]
|
||||
def B₅ : (PlusU1 3).Charges := ![0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0]
|
||||
|
||||
/-- A charge assignment forming one of the basis elements of the plane. -/
|
||||
def B₆ : (PlusU1 3).charges := ![0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0]
|
||||
def B₆ : (PlusU1 3).Charges := ![0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0]
|
||||
|
||||
/-- A charge assignment forming one of the basis elements of the plane. -/
|
||||
def B₇ : (PlusU1 3).charges := ![0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0]
|
||||
def B₇ : (PlusU1 3).Charges := ![0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0]
|
||||
|
||||
/-- A charge assignment forming one of the basis elements of the plane. -/
|
||||
def B₈ : (PlusU1 3).charges := ![0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0]
|
||||
def B₈ : (PlusU1 3).Charges := ![0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0]
|
||||
|
||||
/-- A charge assignment forming one of the basis elements of the plane. -/
|
||||
def B₉ : (PlusU1 3).charges := ![0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 0]
|
||||
def B₉ : (PlusU1 3).Charges := ![0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 0]
|
||||
|
||||
/-- A charge assignment forming one of the basis elements of the plane. -/
|
||||
def B₁₀ : (PlusU1 3).charges := ![0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]
|
||||
def B₁₀ : (PlusU1 3).Charges := ![0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]
|
||||
|
||||
/-- The charge assignment forming a basis of the plane. -/
|
||||
def B : Fin 11 → (PlusU1 3).charges := fun i =>
|
||||
def B : Fin 11 → (PlusU1 3).Charges := fun i =>
|
||||
match i with
|
||||
| 0 => B₀
|
||||
| 1 => B₁
|
||||
|
|
@ -105,7 +105,7 @@ lemma on_accQuad (f : Fin 11 → ℚ) :
|
|||
ring
|
||||
|
||||
|
||||
lemma isSolution_quadCoeff_f_sq_zero (f : Fin 11 → ℚ) (hS : (PlusU1 3).isSolution (∑ i, f i • B i))
|
||||
lemma isSolution_quadCoeff_f_sq_zero (f : Fin 11 → ℚ) (hS : (PlusU1 3).IsSolution (∑ i, f i • B i))
|
||||
(k : Fin 11) : quadCoeff k * (f k)^2 = 0 := by
|
||||
obtain ⟨S, hS⟩ := hS
|
||||
have hQ := quadSol S.1
|
||||
|
|
@ -120,46 +120,46 @@ lemma isSolution_quadCoeff_f_sq_zero (f : Fin 11 → ℚ) (hS : (PlusU1 3).isSol
|
|||
fin_cases i <;> rfl
|
||||
exact sq_nonneg (f i)
|
||||
|
||||
lemma isSolution_f0 (f : Fin 11 → ℚ) (hS : (PlusU1 3).isSolution (∑ i, f i • B i)) : f 0 = 0 := by
|
||||
lemma isSolution_f0 (f : Fin 11 → ℚ) (hS : (PlusU1 3).IsSolution (∑ i, f i • B i)) : f 0 = 0 := by
|
||||
simpa using (isSolution_quadCoeff_f_sq_zero f hS 0)
|
||||
|
||||
lemma isSolution_f1 (f : Fin 11 → ℚ) (hS : (PlusU1 3).isSolution (∑ i, f i • B i)) : f 1 = 0 := by
|
||||
lemma isSolution_f1 (f : Fin 11 → ℚ) (hS : (PlusU1 3).IsSolution (∑ i, f i • B i)) : f 1 = 0 := by
|
||||
simpa using (isSolution_quadCoeff_f_sq_zero f hS 1)
|
||||
|
||||
lemma isSolution_f2 (f : Fin 11 → ℚ) (hS : (PlusU1 3).isSolution (∑ i, f i • B i)) : f 2 = 0 := by
|
||||
lemma isSolution_f2 (f : Fin 11 → ℚ) (hS : (PlusU1 3).IsSolution (∑ i, f i • B i)) : f 2 = 0 := by
|
||||
simpa using (isSolution_quadCoeff_f_sq_zero f hS 2)
|
||||
|
||||
lemma isSolution_f3 (f : Fin 11 → ℚ) (hS : (PlusU1 3).isSolution (∑ i, f i • B i)) : f 3 = 0 := by
|
||||
lemma isSolution_f3 (f : Fin 11 → ℚ) (hS : (PlusU1 3).IsSolution (∑ i, f i • B i)) : f 3 = 0 := by
|
||||
simpa using (isSolution_quadCoeff_f_sq_zero f hS 3)
|
||||
|
||||
lemma isSolution_f4 (f : Fin 11 → ℚ) (hS : (PlusU1 3).isSolution (∑ i, f i • B i)) : f 4 = 0 := by
|
||||
lemma isSolution_f4 (f : Fin 11 → ℚ) (hS : (PlusU1 3).IsSolution (∑ i, f i • B i)) : f 4 = 0 := by
|
||||
simpa using (isSolution_quadCoeff_f_sq_zero f hS 4)
|
||||
|
||||
lemma isSolution_f5 (f : Fin 11 → ℚ) (hS : (PlusU1 3).isSolution (∑ i, f i • B i)) : f 5 = 0 := by
|
||||
lemma isSolution_f5 (f : Fin 11 → ℚ) (hS : (PlusU1 3).IsSolution (∑ i, f i • B i)) : f 5 = 0 := by
|
||||
have h := isSolution_quadCoeff_f_sq_zero f hS 5
|
||||
rw [mul_eq_zero] at h
|
||||
change 1 = 0 ∨ _ = _ at h
|
||||
simpa using h
|
||||
|
||||
lemma isSolution_f6 (f : Fin 11 → ℚ) (hS : (PlusU1 3).isSolution (∑ i, f i • B i)) : f 6 = 0 := by
|
||||
lemma isSolution_f6 (f : Fin 11 → ℚ) (hS : (PlusU1 3).IsSolution (∑ i, f i • B i)) : f 6 = 0 := by
|
||||
have h := isSolution_quadCoeff_f_sq_zero f hS 6
|
||||
rw [mul_eq_zero] at h
|
||||
change 1 = 0 ∨ _ = _ at h
|
||||
simpa using h
|
||||
|
||||
lemma isSolution_f7 (f : Fin 11 → ℚ) (hS : (PlusU1 3).isSolution (∑ i, f i • B i)) : f 7 = 0 := by
|
||||
lemma isSolution_f7 (f : Fin 11 → ℚ) (hS : (PlusU1 3).IsSolution (∑ i, f i • B i)) : f 7 = 0 := by
|
||||
have h := isSolution_quadCoeff_f_sq_zero f hS 7
|
||||
rw [mul_eq_zero] at h
|
||||
change 1 = 0 ∨ _ = _ at h
|
||||
simpa using h
|
||||
|
||||
lemma isSolution_f8 (f : Fin 11 → ℚ) (hS : (PlusU1 3).isSolution (∑ i, f i • B i)) : f 8 = 0 := by
|
||||
lemma isSolution_f8 (f : Fin 11 → ℚ) (hS : (PlusU1 3).IsSolution (∑ i, f i • B i)) : f 8 = 0 := by
|
||||
have h := isSolution_quadCoeff_f_sq_zero f hS 8
|
||||
rw [mul_eq_zero] at h
|
||||
change 1 = 0 ∨ _ = _ at h
|
||||
simpa using h
|
||||
|
||||
lemma isSolution_sum_part (f : Fin 11 → ℚ) (hS : (PlusU1 3).isSolution (∑ i, f i • B i)) :
|
||||
lemma isSolution_sum_part (f : Fin 11 → ℚ) (hS : (PlusU1 3).IsSolution (∑ i, f i • B i)) :
|
||||
∑ i, f i • B i = f 9 • B₉ + f 10 • B₁₀ := by
|
||||
rw [Fin.sum_univ_castSucc, Fin.sum_univ_castSucc, Fin.sum_univ_castSucc, Fin.sum_univ_eight]
|
||||
change f 0 • B 0 + f 1 • B 1 + f 2 • B 2 + f 3 • B 3 + f 4 • B 4 + f 5 • B 5 + f 6 • B 6 +
|
||||
|
|
@ -170,7 +170,7 @@ lemma isSolution_sum_part (f : Fin 11 → ℚ) (hS : (PlusU1 3).isSolution (∑
|
|||
simp
|
||||
rfl
|
||||
|
||||
lemma isSolution_grav (f : Fin 11 → ℚ) (hS : (PlusU1 3).isSolution (∑ i, f i • B i)) :
|
||||
lemma isSolution_grav (f : Fin 11 → ℚ) (hS : (PlusU1 3).IsSolution (∑ i, f i • B i)) :
|
||||
f 10 = - 3 * f 9 := by
|
||||
have hx := isSolution_sum_part f hS
|
||||
obtain ⟨S, hS'⟩ := hS
|
||||
|
|
@ -183,12 +183,12 @@ lemma isSolution_grav (f : Fin 11 → ℚ) (hS : (PlusU1 3).isSolution (∑ i,
|
|||
simp at hg
|
||||
linear_combination hg
|
||||
|
||||
lemma isSolution_sum_part' (f : Fin 11 → ℚ) (hS : (PlusU1 3).isSolution (∑ i, f i • B i)) :
|
||||
lemma isSolution_sum_part' (f : Fin 11 → ℚ) (hS : (PlusU1 3).IsSolution (∑ i, f i • B i)) :
|
||||
∑ i, f i • B i = f 9 • B₉ + (- 3 * f 9) • B₁₀ := by
|
||||
rw [isSolution_sum_part f hS]
|
||||
rw [isSolution_grav f hS]
|
||||
|
||||
lemma isSolution_f9 (f : Fin 11 → ℚ) (hS : (PlusU1 3).isSolution (∑ i, f i • B i)) :
|
||||
lemma isSolution_f9 (f : Fin 11 → ℚ) (hS : (PlusU1 3).IsSolution (∑ i, f i • B i)) :
|
||||
f 9 = 0 := by
|
||||
have hx := isSolution_sum_part' f hS
|
||||
obtain ⟨S, hS'⟩ := hS
|
||||
|
|
@ -211,12 +211,12 @@ lemma isSolution_f9 (f : Fin 11 → ℚ) (hS : (PlusU1 3).isSolution (∑ i, f i
|
|||
rw [h1] at hc
|
||||
simpa using hc
|
||||
|
||||
lemma isSolution_f10 (f : Fin 11 → ℚ) (hS : (PlusU1 3).isSolution (∑ i, f i • B i)) :
|
||||
lemma isSolution_f10 (f : Fin 11 → ℚ) (hS : (PlusU1 3).IsSolution (∑ i, f i • B i)) :
|
||||
f 10 = 0 := by
|
||||
rw [isSolution_grav f hS, isSolution_f9 f hS]
|
||||
simp
|
||||
|
||||
lemma isSolution_f_zero (f : Fin 11 → ℚ) (hS : (PlusU1 3).isSolution (∑ i, f i • B i))
|
||||
lemma isSolution_f_zero (f : Fin 11 → ℚ) (hS : (PlusU1 3).IsSolution (∑ i, f i • B i))
|
||||
(k : Fin 11) : f k = 0 := by
|
||||
fin_cases k
|
||||
exact isSolution_f0 f hS
|
||||
|
|
@ -232,7 +232,7 @@ lemma isSolution_f_zero (f : Fin 11 → ℚ) (hS : (PlusU1 3).isSolution (∑ i,
|
|||
exact isSolution_f10 f hS
|
||||
|
||||
|
||||
lemma isSolution_only_if_zero (f : Fin 11 → ℚ) (hS : (PlusU1 3).isSolution (∑ i, f i • B i)) :
|
||||
lemma isSolution_only_if_zero (f : Fin 11 → ℚ) (hS : (PlusU1 3).IsSolution (∑ i, f i • B i)) :
|
||||
∑ i, f i • B i = 0 := by
|
||||
rw [isSolution_sum_part f hS]
|
||||
rw [isSolution_grav f hS]
|
||||
|
|
@ -244,7 +244,7 @@ theorem basis_linear_independent : LinearIndependent ℚ B := by
|
|||
apply Fintype.linearIndependent_iff.mpr
|
||||
intro f h
|
||||
let X : (PlusU1 3).Sols := chargeToAF 0 (by rfl) (by rfl) (by rfl) (by rfl) (by rfl) (by rfl)
|
||||
have hS : (PlusU1 3).isSolution (∑ i, f i • B i) := by
|
||||
have hS : (PlusU1 3).IsSolution (∑ i, f i • B i) := by
|
||||
use X
|
||||
rw [h]
|
||||
rfl
|
||||
|
|
@ -252,9 +252,9 @@ theorem basis_linear_independent : LinearIndependent ℚ B := by
|
|||
|
||||
end ElevenPlane
|
||||
|
||||
theorem eleven_dim_plane_of_no_sols_exists : ∃ (B : Fin 11 → (PlusU1 3).charges),
|
||||
theorem eleven_dim_plane_of_no_sols_exists : ∃ (B : Fin 11 → (PlusU1 3).Charges),
|
||||
LinearIndependent ℚ B ∧
|
||||
∀ (f : Fin 11 → ℚ), (PlusU1 3).isSolution (∑ i, f i • B i) → ∑ i, f i • B i = 0 := by
|
||||
∀ (f : Fin 11 → ℚ), (PlusU1 3).IsSolution (∑ i, f i • B i) → ∑ i, f i • B i = 0 := by
|
||||
use ElevenPlane.B
|
||||
apply And.intro
|
||||
exact ElevenPlane.basis_linear_independent
|
||||
|
|
|
|||
Loading…
Add table
Add a link
Reference in a new issue