hep-declarations/HepLean.AnomalyCancellation.SMNu.PlusU1.PlaneNonSols.jsonl

{"name":"SMRHN.PlusU1.ElevenPlane.isSolution_f1","declaration":"theorem SMRHN.PlusU1.ElevenPlane.isSolution_f1 (f : Fin 11 → ℚ) (hS : ACCSystem.IsSolution (SMRHN.PlusU1 3) (Finset.sum Finset.univ fun i => f i • SMRHN.PlusU1.ElevenPlane.B i)) : f 1 = 0"}
{"name":"SMRHN.PlusU1.ElevenPlane.B₁₀","declaration":"/-- A charge assignment forming one of the basis elements of the plane. -/\ndef SMRHN.PlusU1.ElevenPlane.B₁₀  : ACCSystemCharges.Charges (SMRHN.PlusU1 3).toACCSystemCharges"}
{"name":"SMRHN.PlusU1.ElevenPlane.isSolution_f10","declaration":"theorem SMRHN.PlusU1.ElevenPlane.isSolution_f10 (f : Fin 11 → ℚ) (hS : ACCSystem.IsSolution (SMRHN.PlusU1 3) (Finset.sum Finset.univ fun i => f i • SMRHN.PlusU1.ElevenPlane.B i)) : f 10 = 0"}
{"name":"SMRHN.PlusU1.ElevenPlane.B₂","declaration":"/-- A charge assignment forming one of the basis elements of the plane. -/\ndef SMRHN.PlusU1.ElevenPlane.B₂  : ACCSystemCharges.Charges (SMRHN.PlusU1 3).toACCSystemCharges"}
{"name":"SMRHN.PlusU1.ElevenPlane.Bi_Bj_quad","declaration":"theorem SMRHN.PlusU1.ElevenPlane.Bi_Bj_quad {i : Fin 11} {j : Fin 11} (hi : i ≠ j) : (SMνACCs.quadBiLin (SMRHN.PlusU1.ElevenPlane.B i)) (SMRHN.PlusU1.ElevenPlane.B j) = 0"}
{"name":"SMRHN.PlusU1.ElevenPlane.B₉","declaration":"/-- A charge assignment forming one of the basis elements of the plane. -/\ndef SMRHN.PlusU1.ElevenPlane.B₉  : ACCSystemCharges.Charges (SMRHN.PlusU1 3).toACCSystemCharges"}
{"name":"SMRHN.PlusU1.ElevenPlane.isSolution_f3","declaration":"theorem SMRHN.PlusU1.ElevenPlane.isSolution_f3 (f : Fin 11 → ℚ) (hS : ACCSystem.IsSolution (SMRHN.PlusU1 3) (Finset.sum Finset.univ fun i => f i • SMRHN.PlusU1.ElevenPlane.B i)) : f 3 = 0"}
{"name":"SMRHN.PlusU1.ElevenPlane.isSolution_f6","declaration":"theorem SMRHN.PlusU1.ElevenPlane.isSolution_f6 (f : Fin 11 → ℚ) (hS : ACCSystem.IsSolution (SMRHN.PlusU1 3) (Finset.sum Finset.univ fun i => f i • SMRHN.PlusU1.ElevenPlane.B i)) : f 6 = 0"}
{"name":"SMRHN.PlusU1.ElevenPlane.on_accQuad","declaration":"theorem SMRHN.PlusU1.ElevenPlane.on_accQuad (f : Fin 11 → ℚ) : SMνACCs.accQuad (Finset.sum Finset.univ fun i => f i • SMRHN.PlusU1.ElevenPlane.B i) =\n  Finset.sum Finset.univ fun i => SMRHN.PlusU1.ElevenPlane.quadCoeff i * f i ^ 2"}
{"name":"SMRHN.PlusU1.eleven_dim_plane_of_no_sols_exists","declaration":"theorem SMRHN.PlusU1.eleven_dim_plane_of_no_sols_exists  : ∃ B,\n  LinearIndependent ℚ B ∧\n    ∀ (f : Fin 11 → ℚ),\n      ACCSystem.IsSolution (SMRHN.PlusU1 3) (Finset.sum Finset.univ fun i => f i • B i) →\n        (Finset.sum Finset.univ fun i => f i • B i) = 0"}
{"name":"SMRHN.PlusU1.ElevenPlane.isSolution_f8","declaration":"theorem SMRHN.PlusU1.ElevenPlane.isSolution_f8 (f : Fin 11 → ℚ) (hS : ACCSystem.IsSolution (SMRHN.PlusU1 3) (Finset.sum Finset.univ fun i => f i • SMRHN.PlusU1.ElevenPlane.B i)) : f 8 = 0"}
{"name":"SMRHN.PlusU1.ElevenPlane.B₆","declaration":"/-- A charge assignment forming one of the basis elements of the plane. -/\ndef SMRHN.PlusU1.ElevenPlane.B₆  : ACCSystemCharges.Charges (SMRHN.PlusU1 3).toACCSystemCharges"}
{"name":"SMRHN.PlusU1.ElevenPlane.isSolution_f2","declaration":"theorem SMRHN.PlusU1.ElevenPlane.isSolution_f2 (f : Fin 11 → ℚ) (hS : ACCSystem.IsSolution (SMRHN.PlusU1 3) (Finset.sum Finset.univ fun i => f i • SMRHN.PlusU1.ElevenPlane.B i)) : f 2 = 0"}
{"name":"SMRHN.PlusU1.ElevenPlane.isSolution_f9","declaration":"theorem SMRHN.PlusU1.ElevenPlane.isSolution_f9 (f : Fin 11 → ℚ) (hS : ACCSystem.IsSolution (SMRHN.PlusU1 3) (Finset.sum Finset.univ fun i => f i • SMRHN.PlusU1.ElevenPlane.B i)) : f 9 = 0"}
{"name":"SMRHN.PlusU1.ElevenPlane.isSolution_f0","declaration":"theorem SMRHN.PlusU1.ElevenPlane.isSolution_f0 (f : Fin 11 → ℚ) (hS : ACCSystem.IsSolution (SMRHN.PlusU1 3) (Finset.sum Finset.univ fun i => f i • SMRHN.PlusU1.ElevenPlane.B i)) : f 0 = 0"}
{"name":"SMRHN.PlusU1.ElevenPlane.isSolution_f4","declaration":"theorem SMRHN.PlusU1.ElevenPlane.isSolution_f4 (f : Fin 11 → ℚ) (hS : ACCSystem.IsSolution (SMRHN.PlusU1 3) (Finset.sum Finset.univ fun i => f i • SMRHN.PlusU1.ElevenPlane.B i)) : f 4 = 0"}
{"name":"SMRHN.PlusU1.ElevenPlane.B","declaration":"/-- The charge assignment forming a basis of the plane. -/\ndef SMRHN.PlusU1.ElevenPlane.B  : Fin 11 → ACCSystemCharges.Charges (SMRHN.PlusU1 3).toACCSystemCharges"}
{"name":"SMRHN.PlusU1.ElevenPlane.isSolution_f_zero","declaration":"theorem SMRHN.PlusU1.ElevenPlane.isSolution_f_zero (f : Fin 11 → ℚ) (hS : ACCSystem.IsSolution (SMRHN.PlusU1 3) (Finset.sum Finset.univ fun i => f i • SMRHN.PlusU1.ElevenPlane.B i)) (k : Fin 11) : f k = 0"}
{"name":"SMRHN.PlusU1.ElevenPlane.isSolution_f7","declaration":"theorem SMRHN.PlusU1.ElevenPlane.isSolution_f7 (f : Fin 11 → ℚ) (hS : ACCSystem.IsSolution (SMRHN.PlusU1 3) (Finset.sum Finset.univ fun i => f i • SMRHN.PlusU1.ElevenPlane.B i)) : f 7 = 0"}
{"name":"SMRHN.PlusU1.ElevenPlane.isSolution_sum_part'","declaration":"theorem SMRHN.PlusU1.ElevenPlane.isSolution_sum_part' (f : Fin 11 → ℚ) (hS : ACCSystem.IsSolution (SMRHN.PlusU1 3) (Finset.sum Finset.univ fun i => f i • SMRHN.PlusU1.ElevenPlane.B i)) : (Finset.sum Finset.univ fun i => f i • SMRHN.PlusU1.ElevenPlane.B i) =\n  f 9 • SMRHN.PlusU1.ElevenPlane.B₉ + (-3 * f 9) • SMRHN.PlusU1.ElevenPlane.B₁₀"}
{"name":"SMRHN.PlusU1.ElevenPlane.B₅","declaration":"/-- A charge assignment forming one of the basis elements of the plane. -/\ndef SMRHN.PlusU1.ElevenPlane.B₅  : ACCSystemCharges.Charges (SMRHN.PlusU1 3).toACCSystemCharges"}
{"name":"SMRHN.PlusU1.ElevenPlane.isSolution_quadCoeff_f_sq_zero","declaration":"theorem SMRHN.PlusU1.ElevenPlane.isSolution_quadCoeff_f_sq_zero (f : Fin 11 → ℚ) (hS : ACCSystem.IsSolution (SMRHN.PlusU1 3) (Finset.sum Finset.univ fun i => f i • SMRHN.PlusU1.ElevenPlane.B i)) (k : Fin 11) : SMRHN.PlusU1.ElevenPlane.quadCoeff k * f k ^ 2 = 0"}
{"name":"SMRHN.PlusU1.ElevenPlane.B₈","declaration":"/-- A charge assignment forming one of the basis elements of the plane. -/\ndef SMRHN.PlusU1.ElevenPlane.B₈  : ACCSystemCharges.Charges (SMRHN.PlusU1 3).toACCSystemCharges"}
{"name":"SMRHN.PlusU1.ElevenPlane.isSolution_sum_part","declaration":"theorem SMRHN.PlusU1.ElevenPlane.isSolution_sum_part (f : Fin 11 → ℚ) (hS : ACCSystem.IsSolution (SMRHN.PlusU1 3) (Finset.sum Finset.univ fun i => f i • SMRHN.PlusU1.ElevenPlane.B i)) : (Finset.sum Finset.univ fun i => f i • SMRHN.PlusU1.ElevenPlane.B i) =\n  f 9 • SMRHN.PlusU1.ElevenPlane.B₉ + f 10 • SMRHN.PlusU1.ElevenPlane.B₁₀"}
{"name":"SMRHN.PlusU1.ElevenPlane.isSolution_grav","declaration":"theorem SMRHN.PlusU1.ElevenPlane.isSolution_grav (f : Fin 11 → ℚ) (hS : ACCSystem.IsSolution (SMRHN.PlusU1 3) (Finset.sum Finset.univ fun i => f i • SMRHN.PlusU1.ElevenPlane.B i)) : f 10 = -3 * f 9"}
{"name":"SMRHN.PlusU1.ElevenPlane.isSolution_only_if_zero","declaration":"theorem SMRHN.PlusU1.ElevenPlane.isSolution_only_if_zero (f : Fin 11 → ℚ) (hS : ACCSystem.IsSolution (SMRHN.PlusU1 3) (Finset.sum Finset.univ fun i => f i • SMRHN.PlusU1.ElevenPlane.B i)) : (Finset.sum Finset.univ fun i => f i • SMRHN.PlusU1.ElevenPlane.B i) = 0"}
{"name":"SMRHN.PlusU1.ElevenPlane.B₄","declaration":"/-- A charge assignment forming one of the basis elements of the plane. -/\ndef SMRHN.PlusU1.ElevenPlane.B₄  : ACCSystemCharges.Charges (SMRHN.PlusU1 3).toACCSystemCharges"}
{"name":"SMRHN.PlusU1.ElevenPlane.basis_linear_independent","declaration":"theorem SMRHN.PlusU1.ElevenPlane.basis_linear_independent  : LinearIndependent ℚ SMRHN.PlusU1.ElevenPlane.B"}
{"name":"SMRHN.PlusU1.ElevenPlane.Bi_sum_quad","declaration":"theorem SMRHN.PlusU1.ElevenPlane.Bi_sum_quad (i : Fin 11) (f : Fin 11 → ℚ) : (SMνACCs.quadBiLin (SMRHN.PlusU1.ElevenPlane.B i))\n    (Finset.sum Finset.univ fun k => f k • SMRHN.PlusU1.ElevenPlane.B k) =\n  f i * (SMνACCs.quadBiLin (SMRHN.PlusU1.ElevenPlane.B i)) (SMRHN.PlusU1.ElevenPlane.B i)"}
{"name":"SMRHN.PlusU1.ElevenPlane.quadCoeff","declaration":"/-- The coefficents of the quadratic equation in our basis. -/\ndef SMRHN.PlusU1.ElevenPlane.quadCoeff  : Fin 11 → ℚ"}
{"name":"SMRHN.PlusU1.ElevenPlane.B₃","declaration":"/-- A charge assignment forming one of the basis elements of the plane. -/\ndef SMRHN.PlusU1.ElevenPlane.B₃  : ACCSystemCharges.Charges (SMRHN.PlusU1 3).toACCSystemCharges"}
{"name":"SMRHN.PlusU1.ElevenPlane.quadCoeff_eq_bilinear","declaration":"theorem SMRHN.PlusU1.ElevenPlane.quadCoeff_eq_bilinear (i : Fin 11) : SMRHN.PlusU1.ElevenPlane.quadCoeff i = (SMνACCs.quadBiLin (SMRHN.PlusU1.ElevenPlane.B i)) (SMRHN.PlusU1.ElevenPlane.B i)"}
{"name":"SMRHN.PlusU1.ElevenPlane.B₀","declaration":"/-- A charge assignment forming one of the basis elements of the plane. -/\ndef SMRHN.PlusU1.ElevenPlane.B₀  : ACCSystemCharges.Charges (SMRHN.PlusU1 3).toACCSystemCharges"}
{"name":"SMRHN.PlusU1.ElevenPlane.isSolution_f5","declaration":"theorem SMRHN.PlusU1.ElevenPlane.isSolution_f5 (f : Fin 11 → ℚ) (hS : ACCSystem.IsSolution (SMRHN.PlusU1 3) (Finset.sum Finset.univ fun i => f i • SMRHN.PlusU1.ElevenPlane.B i)) : f 5 = 0"}
{"name":"SMRHN.PlusU1.ElevenPlane.B₇","declaration":"/-- A charge assignment forming one of the basis elements of the plane. -/\ndef SMRHN.PlusU1.ElevenPlane.B₇  : ACCSystemCharges.Charges (SMRHN.PlusU1 3).toACCSystemCharges"}
{"name":"SMRHN.PlusU1.ElevenPlane.B₁","declaration":"/-- A charge assignment forming one of the basis elements of the plane. -/\ndef SMRHN.PlusU1.ElevenPlane.B₁  : ACCSystemCharges.Charges (SMRHN.PlusU1 3).toACCSystemCharges"}