refactor: multiple goal proves
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jstoobysmith
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12 changed files with 277 additions and 320 deletions
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@ -67,15 +67,13 @@ lemma lineEqPropSol_iff_lineEqCoeff_zero (T : MSSMACC.Sols) :
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lemma linEqPropSol_iff_proj_linEqProp (R : MSSMACC.Sols) :
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LineEqPropSol R ↔ LineEqProp (proj R.1.1) := by
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rw [lineEqPropSol_iff_lineEqCoeff_zero, lineEqCoeff, LineEqProp]
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apply Iff.intro
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intro h
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rw [show dot Y₃.val B₃.val = 108 by rfl] at h
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simp at h
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rw [α₁_proj, α₂_proj, h]
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simp only [neg_zero, Fin.isValue, Fin.reduceFinMk, zero_mul, and_self]
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intro h
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rw [h.2.2]
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simp
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refine Iff.intro (fun h => ?_) (fun h => ?_)
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· rw [show dot Y₃.val B₃.val = 108 by rfl] at h
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simp only [mul_eq_zero, OfNat.ofNat_ne_zero, false_or] at h
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rw [α₁_proj, α₂_proj, h]
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simp only [neg_zero, Fin.isValue, Fin.reduceFinMk, zero_mul, and_self]
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· rw [h.2.2]
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simp
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/-- A condition which is satisfied if the plane spanned by `R`, `Y₃` and `B₃` lies
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entirely in the quadratic surface. -/
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@ -97,37 +95,31 @@ def quadCoeff (T : MSSMACC.Sols) : ℚ :=
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(quadBiLin Y₃.val T.val ^ 2 + quadBiLin B₃.val T.val ^ 2)
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lemma inQuadSolProp_iff_quadCoeff_zero (T : MSSMACC.Sols) : InQuadSolProp T ↔ quadCoeff T = 0 := by
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apply Iff.intro
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intro h
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rw [quadCoeff, h.1, h.2]
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simp only [Fin.isValue, Fin.reduceFinMk, ne_eq, OfNat.ofNat_ne_zero, not_false_eq_true,
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zero_pow, add_zero, mul_zero]
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intro h
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rw [quadCoeff] at h
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rw [show dot Y₃.val B₃.val = 108 by rfl] at h
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simp only [Fin.isValue, Fin.reduceFinMk, mul_eq_zero, OfNat.ofNat_ne_zero, ne_eq,
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not_false_eq_true, pow_eq_zero_iff, or_self, false_or] at h
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apply (add_eq_zero_iff' (sq_nonneg _) (sq_nonneg _)).mp at h
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simp only [Fin.isValue, Fin.reduceFinMk, ne_eq, OfNat.ofNat_ne_zero,
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not_false_eq_true, pow_eq_zero_iff] at h
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exact h
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refine Iff.intro (fun h => ?_) (fun h => ?_)
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· rw [quadCoeff, h.1, h.2]
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simp only [Fin.isValue, Fin.reduceFinMk, ne_eq, OfNat.ofNat_ne_zero, not_false_eq_true,
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zero_pow, add_zero, mul_zero]
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· rw [quadCoeff, show dot Y₃.val B₃.val = 108 by rfl] at h
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simp only [Fin.isValue, Fin.reduceFinMk, mul_eq_zero, OfNat.ofNat_ne_zero, ne_eq,
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not_false_eq_true, pow_eq_zero_iff, or_self, false_or] at h
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apply (add_eq_zero_iff' (sq_nonneg _) (sq_nonneg _)).mp at h
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simp only [Fin.isValue, Fin.reduceFinMk, ne_eq, OfNat.ofNat_ne_zero,
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not_false_eq_true, pow_eq_zero_iff] at h
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exact h
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/-- The conditions `inQuadSolProp R` and `inQuadProp (proj R.1.1)` are equivalent. This is to be
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expected since both `R` and `proj R.1.1` define the same plane with `Y₃` and `B₃`. -/
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lemma inQuadSolProp_iff_proj_inQuadProp (R : MSSMACC.Sols) :
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InQuadSolProp R ↔ InQuadProp (proj R.1.1) := by
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rw [InQuadSolProp, InQuadProp]
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rw [quad_proj, quad_Y₃_proj, quad_B₃_proj]
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apply Iff.intro
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intro h
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rw [h.1, h.2]
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simp only [Fin.isValue, Fin.reduceFinMk, mul_zero, add_zero, and_self]
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intro h
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rw [show dot Y₃.val B₃.val = 108 by rfl] at h
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simp only [Fin.isValue, Fin.reduceFinMk, mul_eq_zero,
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OfNat.ofNat_ne_zero, or_self, false_or] at h
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rw [h.2.1, h.2.2]
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simp
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rw [InQuadSolProp, InQuadProp, quad_proj, quad_Y₃_proj, quad_B₃_proj]
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refine Iff.intro (fun h => ?_) (fun h => ?_)
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· rw [h.1, h.2]
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simp only [Fin.isValue, Fin.reduceFinMk, mul_zero, add_zero, and_self]
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· rw [show dot Y₃.val B₃.val = 108 by rfl] at h
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simp only [Fin.isValue, Fin.reduceFinMk, mul_eq_zero,
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OfNat.ofNat_ne_zero, or_self, false_or] at h
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rw [h.2.1, h.2.2]
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simp
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/-- A condition which is satisfied if the plane spanned by `R`, `Y₃` and `B₃` lies
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entirely in the cubic surface. -/
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@ -151,35 +143,30 @@ def cubicCoeff (T : MSSMACC.Sols) : ℚ :=
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lemma inCubeSolProp_iff_cubicCoeff_zero (T : MSSMACC.Sols) :
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InCubeSolProp T ↔ cubicCoeff T = 0 := by
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apply Iff.intro
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intro h
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rw [cubicCoeff, h.1, h.2]
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simp only [Fin.isValue, Fin.reduceFinMk, ne_eq, OfNat.ofNat_ne_zero, not_false_eq_true,
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zero_pow, add_zero, mul_zero]
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intro h
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rw [cubicCoeff] at h
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rw [show dot Y₃.val B₃.val = 108 by rfl] at h
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simp only [Fin.isValue, Fin.reduceFinMk, mul_eq_zero, OfNat.ofNat_ne_zero, ne_eq,
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not_false_eq_true, pow_eq_zero_iff, or_self, false_or] at h
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apply (add_eq_zero_iff' (sq_nonneg _) (sq_nonneg _)).mp at h
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simp only [Fin.isValue, Fin.reduceFinMk, ne_eq, OfNat.ofNat_ne_zero,
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not_false_eq_true, pow_eq_zero_iff] at h
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exact h.symm
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refine Iff.intro (fun h => ?_) (fun h => ?_)
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· rw [cubicCoeff, h.1, h.2]
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simp only [Fin.isValue, Fin.reduceFinMk, ne_eq, OfNat.ofNat_ne_zero, not_false_eq_true,
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zero_pow, add_zero, mul_zero]
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· rw [cubicCoeff, show dot Y₃.val B₃.val = 108 by rfl] at h
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simp only [Fin.isValue, Fin.reduceFinMk, mul_eq_zero, OfNat.ofNat_ne_zero, ne_eq,
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not_false_eq_true, pow_eq_zero_iff, or_self, false_or] at h
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apply (add_eq_zero_iff' (sq_nonneg _) (sq_nonneg _)).mp at h
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simp only [Fin.isValue, Fin.reduceFinMk, ne_eq, OfNat.ofNat_ne_zero,
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not_false_eq_true, pow_eq_zero_iff] at h
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exact h.symm
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lemma inCubeSolProp_iff_proj_inCubeProp (R : MSSMACC.Sols) :
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InCubeSolProp R ↔ InCubeProp (proj R.1.1) := by
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rw [InCubeSolProp, InCubeProp]
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rw [cube_proj, cube_proj_proj_Y₃, cube_proj_proj_B₃]
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apply Iff.intro
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intro h
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rw [h.1, h.2]
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simp only [Fin.isValue, Fin.reduceFinMk, mul_zero, add_zero, and_self]
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intro h
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rw [show dot Y₃.val B₃.val = 108 by rfl] at h
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simp only [Fin.isValue, Fin.reduceFinMk, mul_eq_zero, OfNat.ofNat_ne_zero, ne_eq,
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not_false_eq_true, pow_eq_zero_iff, or_self, false_or] at h
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rw [h.2.1, h.2.2]
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simp
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refine Iff.intro (fun h => ?_) (fun h => ?_)
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· rw [h.1, h.2]
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simp only [Fin.isValue, Fin.reduceFinMk, mul_zero, add_zero, and_self]
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· rw [show dot Y₃.val B₃.val = 108 by rfl] at h
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simp only [Fin.isValue, Fin.reduceFinMk, mul_eq_zero, OfNat.ofNat_ne_zero, ne_eq,
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not_false_eq_true, pow_eq_zero_iff, or_self, false_or] at h
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rw [h.2.1, h.2.2]
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simp
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/-- Those charge assignments perpendicular to `Y₃` and `B₃` which satisfy the condition
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`lineEqProp`. -/
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@ -384,9 +371,9 @@ lemma inQuadCubeToSol_proj (T : InQuadCubeSol) :
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ring_nf
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simp only [Fin.isValue, Fin.reduceFinMk, zero_smul, add_zero, zero_add]
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rw [← MulAction.mul_smul, mul_comm, mul_inv_cancel]
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simp only [one_smul]
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rw [show dot Y₃.val B₃.val = 108 by rfl]
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simp
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· simp only [one_smul]
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· rw [show dot Y₃.val B₃.val = 108 by rfl]
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simp
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/-- A solution from an element of `MSSMACC.AnomalyFreePerp × ℚ × ℚ × ℚ`. We will
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show that this map is a surjection. -/
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@ -443,15 +430,15 @@ lemma toSol_inQuadCube (T : InQuadCubeSol) : ∃ X, toSol X = T.val := by
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theorem toSol_surjective : Function.Surjective toSol := by
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intro T
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by_cases h₁ : ¬ LineEqPropSol T
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exact toSol_toSolNSProj ⟨T, h₁⟩
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simp at h₁
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by_cases h₂ : ¬ InQuadSolProp T
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exact toSol_inLineEq ⟨T, And.intro h₁ h₂⟩
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simp at h₂
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by_cases h₃ : ¬ InCubeSolProp T
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exact toSol_inQuad ⟨T, And.intro h₁ (And.intro h₂ h₃)⟩
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simp at h₃
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exact toSol_inQuadCube ⟨T, And.intro h₁ (And.intro h₂ h₃)⟩
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· exact toSol_toSolNSProj ⟨T, h₁⟩
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· simp at h₁
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by_cases h₂ : ¬ InQuadSolProp T
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· exact toSol_inLineEq ⟨T, And.intro h₁ h₂⟩
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· simp at h₂
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by_cases h₃ : ¬ InCubeSolProp T
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· exact toSol_inQuad ⟨T, And.intro h₁ (And.intro h₂ h₃)⟩
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· simp at h₃
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exact toSol_inQuadCube ⟨T, And.intro h₁ (And.intro h₂ h₃)⟩
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end AnomalyFreePerp
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